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# Misamis Oriental Successive Terms Differ By A Constant

## Does the distance between successive crests remain

### What is the common difference between successive terms in

Geometric series P-series Nabla. and the constant ratio of successive terms is 1 x The argument works the same. We would expect these limits to differ because one is right before taking a tablet, one is right after. We would expect the difference between them to be 250 mg, the amount of ampicillin in one tablet. 27., An arithmetic sequence does not have a constant rate of increase or decrease between successive terms, so it cannot be called anything!. The constant increase or decrease is called the common.

### Successive Generations in a Rat Model Respond Differently

Successive Periods вЂ“ USA Coverage. Sep 11, 2008В В· We start out using the method of finite differences; that is, we write out the terms of the given sequence, then we write a second row of numbers that are the differences between successive terms, then we write out a third row which is the difference between вЂ¦, Sep 11, 2008В В· We start out using the method of finite differences; that is, we write out the terms of the given sequence, then we write a second row of numbers that are the differences between successive terms, then we write out a third row which is the difference between вЂ¦.

Jul 24, 2013В В· If a is any number, the sequence {a, a, a, a,} is called a constant sequence.? Answer Questions Can I have a step by step solution to this so I can memorize the steps and do it myself? the answers are on the sheet already, I just need? Sep 11, 2008В В· We start out using the method of finite differences; that is, we write out the terms of the given sequence, then we write a second row of numbers that are the differences between successive terms, then we write out a third row which is the difference between вЂ¦

terms, replace the quantities R/n 2 in the case of hydrogen. They differ from these quantities since the formulae representing their values are more com-plicated, but they agree with these quantities in so far as the differences be-tween the successive terms become smaller and smaller and the term values An acid dissociation constant, K a, (also known as acidity constant, or acid-ionization constant) is a quantitative measure of the strength of an acid in solution. It is the equilibrium constant for a chemical reaction в†Ѕ в€’ в€’ в‡Ђ в€’ + + known as dissociation in the context of acidвЂ“base reactions.

terms, replace the quantities R/n 2 in the case of hydrogen. They differ from these quantities since the formulae representing their values are more com-plicated, but they agree with these quantities in so far as the differences be-tween the successive terms become smaller and smaller and the term values Jul 24, 2013В В· If a is any number, the sequence {a, a, a, a,} is called a constant sequence.? Answer Questions Can I have a step by step solution to this so I can memorize the steps and do it myself? the answers are on the sheet already, I just need?

Jul 24, 2013В В· If a is any number, the sequence {a, a, a, a,} is called a constant sequence.? Answer Questions Can I have a step by step solution to this so I can memorize the steps and do it myself? the answers are on the sheet already, I just need? Sep 11, 2008В В· We start out using the method of finite differences; that is, we write out the terms of the given sequence, then we write a second row of numbers that are the differences between successive terms, then we write out a third row which is the difference between вЂ¦

An acid dissociation constant, K a, (also known as acidity constant, or acid-ionization constant) is a quantitative measure of the strength of an acid in solution. It is the equilibrium constant for a chemical reaction в†Ѕ в€’ в€’ в‡Ђ в€’ + + known as dissociation in the context of acidвЂ“base reactions. Jan 11, 2011В В· "Successive terms" simply means those that come next to each other. So 5 and 10 are successive terms. 10 and 15 are successive terms. The next successive term in the sequence you showed would be 30, because the sequence is going up by 5, and the next term would be 30.

One after the other. Examples: вЂў Monday, Tuesday and Wednesday are successive days. вЂў 5 and 6 are successive whole numbers. вЂў 5 and 7 are successive odd numbers. Successive Generations in a Rat Model Respond Differently to a Constant Obesogenic Environment. and to demonstrate that successive generations respond differently to this constant environment. the trajectories of change in these phenotypes did not differ significantly between the three dietary lineages (reference, high fat and low

An acid dissociation constant, K a, (also known as acidity constant, or acid-ionization constant) is a quantitative measure of the strength of an acid in solution. It is the equilibrium constant for a chemical reaction в†Ѕ в€’ в€’ в‡Ђ в€’ + + known as dissociation in the context of acidвЂ“base reactions. successive terms of the first differences sequence are called the second differences and so on. It is a good idea to use an actual sequence of numbers such as the square numbers 1,4,9,16,вЂ¦ to help explain the meaning of the terms first differences and second differences. Example 1 (4 minutes)

I was thinking about sequences where it appears the terms get closer and closer together, and wondered if they converge. Now let's first define a few things. When I say "the terms get closer and closer together", I mean "the distance between any two consecutive terms approaches zero." In other words, for a sequence $\left(x_n\right)$, Definition: Geometric growth refers to the situation where successive changes in a population differ by a constant ratio (as distinct from a constant amount for arithmetic change).

An arithmetic sequence does not have a constant rate of increase or decrease between successive terms, so it cannot be called anything!. The constant increase or decrease is called the common An arithmetic sequence does not have a constant rate of increase or decrease between successive terms, so it cannot be called anything!. The constant increase or decrease is called the common

An arithmetic sequence does not have a constant rate of increase or decrease between successive terms, so it cannot be called anything!. The constant increase or decrease is called the common required number of terms is generated. Being able to recognise an arithmetic sequence is another skill that you need to develop. The key idea here is that the successive terms in an arithmetic sequence differ by a constant amount (the common difference). SAMPLE

Successive Periods. In insurance, successive periods happen when hospital confinements in hospital income protections are considered a portion of the same confinement period because such are due to related or the same grounds and are divided by a smaller amount of a вЂ¦ 30.3 Conditions for Absolute Convergence. The characteristic series whose behavior conveys the most information about the behavior of series in general is the geometric series. This is a power series in the variable x, and its terms are the unadorned powers of x

exponential functions grow by a вЂњconstant factor over successive intervals of equal length.вЂќ In this lesson, we generalize the linear growth pattern to polynomials of second degree (quadratic expressions) and third degree (cubic expressions). Jul 24, 2013В В· If a is any number, the sequence {a, a, a, a,} is called a constant sequence.? Answer Questions Can I have a step by step solution to this so I can memorize the steps and do it myself? the answers are on the sheet already, I just need?

Sep 11, 2008В В· We start out using the method of finite differences; that is, we write out the terms of the given sequence, then we write a second row of numbers that are the differences between successive terms, then we write out a third row which is the difference between вЂ¦ Successive Generations in a Rat Model Respond Differently to a Constant Obesogenic Environment. and to demonstrate that successive generations respond differently to this constant environment. the trajectories of change in these phenotypes did not differ significantly between the three dietary lineages (reference, high fat and low

successive: adjective after , consecutive , ensuing , following , later , subsequent , succeeding , sequent , sequential Associated concepts: successive application The first term in the series is a, and the last one is a+(n-1)d, so we can say the sum of the series is the first term plus the last term multiplied by the number of terms divided by 2. Geometric Series A pure geometric series or geometric progression is one where the ratio, r, вЂ¦

What is a sequence whose successive terms differ by the same nonzero number d called the common An arithmetic sequence does not have a constant rate of increase or decrease between successive In computer programming, a constant is a value that cannot be altered by the program during normal execution, i.e., the value is constant. When associated with an identifier, a constant is said to be "named," although the terms "constant" and "named constant" are often used interchangeably.

Purity discrimination thresholds (О”p) were measured with successive (SOA = 3 sec) and simultaneous (SOA = 0 sec) comparison methods for seven dominant wavelengths: 410, 480, 500, 530, 570, 600 and 650 nm.The stimulus duration was 1 sec. The О”p values with the successive comparison method were found to he about 1.5вЂ“2.0 times larger than those obtained in the simultaneous case. Algebra A Chapter 4. STUDY. Flashcards. Learn. Write. Spell. Test. PLAY. Match. Gravity. Created by. clarksforchrist. Terms in this set (20) arithmetic sequence. a sequence whose successive terms differ by the same nonzero number d, called the common difference. common difference. In an arithmetic sequence, the nonzero constant difference of

What is the difference between using #define and const for creating a constant? Does any have a performance advantage over the other? Naturally I prefer using the const but I'm going to consider the #define if it has suitable advantages. exponential functions grow by a вЂњconstant factor over successive intervals of equal length.вЂќ In this lesson, we generalize the linear growth pattern to polynomials of second degree (quadratic expressions) and third degree (cubic expressions).

Definition of successive in the Definitions.net dictionary. Meaning of successive. What does successive mean? Information and translations of successive in the most comprehensive dictionary definitions resource on the web. May 19, 2009В В· Chapter 12 Vocab; Shared Flashcard Set. Details. Title. Chapter 12 Vocab. Description. Math Vocab. Total Cards. 15. Successive terms differ by the same number, called common difference: Term. Geometric Sequence: Definition. Ratio of the successive terms is a constant ratio: Term. Geometric Series: Definition. Indicated sum of the terms in a

An acid dissociation constant, K a, (also known as acidity constant, or acid-ionization constant) is a quantitative measure of the strength of an acid in solution. It is the equilibrium constant for a chemical reaction в†Ѕ в€’ в€’ в‡Ђ в€’ + + known as dissociation in the context of acidвЂ“base reactions. Check your arithmetic. I tried successive differences and found the third difference produced a row of sixes. Penny . If you want to use successive differences, your first difference sequence should be 10, 24, 44, 70, 102 - was it? Then repeating two more times you вЂ¦

Does the distance between successive crests remain. Successive Generations in a Rat Model Respond Differently to a Constant Obesogenic Environment. and to demonstrate that successive generations respond differently to this constant environment. the trajectories of change in these phenotypes did not differ significantly between the three dietary lineages (reference, high fat and low, One after the other. Examples: вЂў Monday, Tuesday and Wednesday are successive days. вЂў 5 and 6 are successive whole numbers. вЂў 5 and 7 are successive odd numbers..

### Successive Periods вЂ“ USA Coverage

Series Notation Nipissing University. Oct 08, 2015В В· There are two potential answers to your question. 1) variations in the measured dissociation constant because of different techniques or fluctuations in, for example, temperature. 2) Phosphoric acid has three different protonation states. Those, An arithmetic sequence does not have a constant rate of increase or decrease between successive terms, so it cannot be called anything!. The constant increase or decrease is called the common.

Geometric series P-series Nabla. SOLUTION: Which represents the common difference between successive terms in the sequence generated by tn=4/3n-6? Is it (a) -4/3 (b) 4/3 (c) -6 or (d)6 I know I am over thinking this quest, Process for the preparation of dendrimers wherein a starting compound successively in different reaction steps is reacted with a reactant in each of successive different steps, the reactant being different from one step to the next and the reactant being applied in excess in at least one of the steps, while in at least one of the steps the excess of the reactant is extracted with an extraction.

### Geometric series P-series Nabla

Definition of Successive. Series Notation. A sequence is an ordered set of numbers that most often follows some rule (or pattern) to determine the next term in the order. For example, x, x 2, x 3, x 4, is a sequence of numbers, where each successive term is multiplied by x.. A series is a summation of the terms of a sequence. The greek letter sigma is used to represent the summation of terms of a sequence of numbers. https://en.m.wikipedia.org/wiki/Spiraled Jul 24, 2013В В· If a is any number, the sequence {a, a, a, a,} is called a constant sequence.? Answer Questions Can I have a step by step solution to this so I can memorize the steps and do it myself? the answers are on the sheet already, I just need?.

If the terms of a sequence differ by a constant, we say the sequence is arithmetic. If the initial term ($$a_0$$) of the sequence is $$a$$ and the common difference is $$d\text{,}$$ then we have, However, the ratio between successive terms is constant. We call such sequences geometric. What is the difference between using #define and const for creating a constant? Does any have a performance advantage over the other? Naturally I prefer using the const but I'm going to consider the #define if it has suitable advantages.

Successive Periods. In insurance, successive periods happen when hospital confinements in hospital income protections are considered a portion of the same confinement period because such are due to related or the same grounds and are divided by a smaller amount of a вЂ¦ What is a sequence whose successive terms differ by the same nonzero number d called the common An arithmetic sequence does not have a constant rate of increase or decrease between successive

An acid dissociation constant, K a, (also known as acidity constant, or acid-ionization constant) is a quantitative measure of the strength of an acid in solution. It is the equilibrium constant for a chemical reaction в†Ѕ в€’ в€’ в‡Ђ в€’ + + known as dissociation in the context of acidвЂ“base reactions. and the constant ratio of successive terms is 1 x The argument works the same. We would expect these limits to differ because one is right before taking a tablet, one is right after. We would expect the difference between them to be 250 mg, the amount of ampicillin in one tablet. 27.

May 07, 1999В В· You may have heard of arithmetic (accent on "met") and geometric sequences. An arithmetic sequence is a sequence of numbers that increase by addition: the difference between successive terms is a constant. For example, 1, 3, 5, 7 is an arithmetic sequence with constant difference 2; each term is 2 more than the one before. Successive Periods. In insurance, successive periods happen when hospital confinements in hospital income protections are considered a portion of the same confinement period because such are due to related or the same grounds and are divided by a smaller amount of a вЂ¦

SOLUTION: A sequence in which the ratio of successive terms is a constant r, called the common ratio is a(n) _____ sequence. Algebra -> Radicals-> SOLUTION: A sequence in which the ratio of successive terms is a constant r, called the common ratio is a(n) _____ sequence. Log On Algebra A Chapter 4. STUDY. Flashcards. Learn. Write. Spell. Test. PLAY. Match. Gravity. Created by. clarksforchrist. Terms in this set (20) arithmetic sequence. a sequence whose successive terms differ by the same nonzero number d, called the common difference. common difference. In an arithmetic sequence, the nonzero constant difference of

Jan 11, 2011В В· "Successive terms" simply means those that come next to each other. So 5 and 10 are successive terms. 10 and 15 are successive terms. The next successive term in the sequence you showed would be 30, because the sequence is going up by 5, and the next term would be 30. Jan 11, 2011В В· "Successive terms" simply means those that come next to each other. So 5 and 10 are successive terms. 10 and 15 are successive terms. The next successive term in the sequence you showed would be 30, because the sequence is going up by 5, and the next term would be 30.

erties and applications of the generalized mean square successive differВ Hart [18], approximated the distribution of u by an expansion in terms of Beta functions and, using von Neumann's definition of , tabulated where a is a constant which is small enough so that powers of a beyond Mar 23, 2019В В· Hello Ideally, the answer to your question is YES. But coming to the practical scenario it is not so because in the practical scenarios there will be many obstacles that will absorb the sound energy. So that the energy in the wave decrease so will...

Purity discrimination thresholds (О”p) were measured with successive (SOA = 3 sec) and simultaneous (SOA = 0 sec) comparison methods for seven dominant wavelengths: 410, 480, 500, 530, 570, 600 and 650 nm.The stimulus duration was 1 sec. The О”p values with the successive comparison method were found to he about 1.5вЂ“2.0 times larger than those obtained in the simultaneous case. Definition: Geometric growth refers to the situation where successive changes in a population differ by a constant ratio (as distinct from a constant amount for arithmetic change).

successive: adjective after , consecutive , ensuing , following , later , subsequent , succeeding , sequent , sequential Associated concepts: successive application Jul 24, 2013В В· If a is any number, the sequence {a, a, a, a,} is called a constant sequence.? Answer Questions Can I have a step by step solution to this so I can memorize the steps and do it myself? the answers are on the sheet already, I just need?

30.3 Conditions for Absolute Convergence. The characteristic series whose behavior conveys the most information about the behavior of series in general is the geometric series. This is a power series in the variable x, and its terms are the unadorned powers of x I was thinking about sequences where it appears the terms get closer and closer together, and wondered if they converge. Now let's first define a few things. When I say "the terms get closer and closer together", I mean "the distance between any two consecutive terms approaches zero." In other words, for a sequence $\left(x_n\right)$,

Difference equation, mathematical equality involving the differences between successive values of a function of a discrete variable. A discrete variable is one that is defined or of interest only for values that differ by some finite amount, usually a constant and often 1; for example, the discrete Algebra A Chapter 4. STUDY. Flashcards. Learn. Write. Spell. Test. PLAY. Match. Gravity. Created by. clarksforchrist. Terms in this set (20) arithmetic sequence. a sequence whose successive terms differ by the same nonzero number d, called the common difference. common difference. In an arithmetic sequence, the nonzero constant difference of

## Geometric series P-series Nabla

Chapter 12 Vocab Flashcards Flashcard Machine - Create. erties and applications of the generalized mean square successive differВ Hart [18], approximated the distribution of u by an expansion in terms of Beta functions and, using von Neumann's definition of , tabulated where a is a constant which is small enough so that powers of a beyond, terms, replace the quantities R/n 2 in the case of hydrogen. They differ from these quantities since the formulae representing their values are more com-plicated, but they agree with these quantities in so far as the differences be-tween the successive terms become smaller and smaller and the term values.

### Is it (a)4/3 (b) 4/3 (c) -6 or (d)6 - Algebra

WhatвЂ™s the Difference Between SAR and Delta-Sigma ADCs. Process for the preparation of dendrimers wherein a starting compound successively in different reaction steps is reacted with a reactant in each of successive different steps, the reactant being different from one step to the next and the reactant being applied in excess in at least one of the steps, while in at least one of the steps the excess of the reactant is extracted with an extraction, May 19, 2009В В· Chapter 12 Vocab; Shared Flashcard Set. Details. Title. Chapter 12 Vocab. Description. Math Vocab. Total Cards. 15. Successive terms differ by the same number, called common difference: Term. Geometric Sequence: Definition. Ratio of the successive terms is a constant ratio: Term. Geometric Series: Definition. Indicated sum of the terms in a.

Definition: Geometric growth refers to the situation where successive changes in a population differ by a constant ratio (as distinct from a constant amount for arithmetic change). One after the other. Examples: вЂў Monday, Tuesday and Wednesday are successive days. вЂў 5 and 6 are successive whole numbers. вЂў 5 and 7 are successive odd numbers.

and the constant ratio of successive terms is 1 x The argument works the same. We would expect these limits to differ because one is right before taking a tablet, one is right after. We would expect the difference between them to be 250 mg, the amount of ampicillin in one tablet. 27. we have been asked to find the relationship between the successive terms in the sequence below? вЂ“3.2, 4.8, вЂ“7.2, 10.8, вЂ¦ As we can see the given successive terms in the sequence is following a particular pattern as shown below: Its following properties of geometric sequence. Because the ratios of the successive terms is constant.

30.3 Conditions for Absolute Convergence. The characteristic series whose behavior conveys the most information about the behavior of series in general is the geometric series. This is a power series in the variable x, and its terms are the unadorned powers of x Algebra A Chapter 4. STUDY. Flashcards. Learn. Write. Spell. Test. PLAY. Match. Gravity. Created by. clarksforchrist. Terms in this set (20) arithmetic sequence. a sequence whose successive terms differ by the same nonzero number d, called the common difference. common difference. In an arithmetic sequence, the nonzero constant difference of

successive terms of the first differences sequence are called the second differences and so on. It is a good idea to use an actual sequence of numbers such as the square numbers 1,4,9,16,вЂ¦ to help explain the meaning of the terms first differences and second differences. Example 1 (4 minutes) Definition: Geometric growth refers to the situation where successive changes in a population differ by a constant ratio (as distinct from a constant amount for arithmetic change).

Euler discovered and revealed sums of the series for p = 2m, so for example: I f p < 1 then n p < n or 1/n p > 1/n, therefore the terms of the given series are not smaller than the terms of the divergent harmonic series so, given series diverges. May 07, 1999В В· You may have heard of arithmetic (accent on "met") and geometric sequences. An arithmetic sequence is a sequence of numbers that increase by addition: the difference between successive terms is a constant. For example, 1, 3, 5, 7 is an arithmetic sequence with constant difference 2; each term is 2 more than the one before.

Difference equation, mathematical equality involving the differences between successive values of a function of a discrete variable. A discrete variable is one that is defined or of interest only for values that differ by some finite amount, usually a constant and often 1; for example, the discrete Algebra A Chapter 4. STUDY. Flashcards. Learn. Write. Spell. Test. PLAY. Match. Gravity. Created by. clarksforchrist. Terms in this set (20) arithmetic sequence. a sequence whose successive terms differ by the same nonzero number d, called the common difference. common difference. In an arithmetic sequence, the nonzero constant difference of

Check your arithmetic. I tried successive differences and found the third difference produced a row of sixes. Penny . If you want to use successive differences, your first difference sequence should be 10, 24, 44, 70, 102 - was it? Then repeating two more times you вЂ¦ terms, replace the quantities R/n 2 in the case of hydrogen. They differ from these quantities since the formulae representing their values are more com-plicated, but they agree with these quantities in so far as the differences be-tween the successive terms become smaller and smaller and the term values

Check your arithmetic. I tried successive differences and found the third difference produced a row of sixes. Penny . If you want to use successive differences, your first difference sequence should be 10, 24, 44, 70, 102 - was it? Then repeating two more times you вЂ¦ Mar 23, 2019В В· Hello Ideally, the answer to your question is YES. But coming to the practical scenario it is not so because in the practical scenarios there will be many obstacles that will absorb the sound energy. So that the energy in the wave decrease so will...

One after the other. Examples: вЂў Monday, Tuesday and Wednesday are successive days. вЂў 5 and 6 are successive whole numbers. вЂў 5 and 7 are successive odd numbers. successive: adjective after , consecutive , ensuing , following , later , subsequent , succeeding , sequent , sequential Associated concepts: successive application

Sep 11, 2008В В· We start out using the method of finite differences; that is, we write out the terms of the given sequence, then we write a second row of numbers that are the differences between successive terms, then we write out a third row which is the difference between вЂ¦ Successive Generations in a Rat Model Respond Differently to a Constant Obesogenic Environment. and to demonstrate that successive generations respond differently to this constant environment. the trajectories of change in these phenotypes did not differ significantly between the three dietary lineages (reference, high fat and low

Successive Periods. In insurance, successive periods happen when hospital confinements in hospital income protections are considered a portion of the same confinement period because such are due to related or the same grounds and are divided by a smaller amount of a вЂ¦ Lesson 1: Successive Differences in Polynomials This work is licensed under a 17 This work is derived from Eureka Math в„ў and licensed by Great Minds. В©2015 Great Minds. eureka-math.org This file derived from ALG II-M1-TE-1.3.0-07.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 вЂ¦

Jan 11, 2011В В· "Successive terms" simply means those that come next to each other. So 5 and 10 are successive terms. 10 and 15 are successive terms. The next successive term in the sequence you showed would be 30, because the sequence is going up by 5, and the next term would be 30. Successive Generations in a Rat Model Respond Differently to a Constant Obesogenic Environment. and to demonstrate that successive generations respond differently to this constant environment. the trajectories of change in these phenotypes did not differ significantly between the three dietary lineages (reference, high fat and low

terms, replace the quantities R/n 2 in the case of hydrogen. They differ from these quantities since the formulae representing their values are more com-plicated, but they agree with these quantities in so far as the differences be-tween the successive terms become smaller and smaller and the term values An arithmetic sequence does not have a constant rate of increase or decrease between successive terms, so it cannot be called anything!. The constant increase or decrease is called the common

Constant Term. The term in a simplified algebraic expression or equation which contains no variable(s). If there is no such term, the constant term is 0. Example: вЂ“5 is the constant term in p(x) = 2x 3 вЂ“ 4x 2 + 9x вЂ“ 5 . See also. Leading term, leading coefficient Jan 11, 2011В В· "Successive terms" simply means those that come next to each other. So 5 and 10 are successive terms. 10 and 15 are successive terms. The next successive term in the sequence you showed would be 30, because the sequence is going up by 5, and the next term would be 30.

Series Notation. A sequence is an ordered set of numbers that most often follows some rule (or pattern) to determine the next term in the order. For example, x, x 2, x 3, x 4, is a sequence of numbers, where each successive term is multiplied by x.. A series is a summation of the terms of a sequence. The greek letter sigma is used to represent the summation of terms of a sequence of numbers. SOLUTION: Which represents the common difference between successive terms in the sequence generated by tn=4/3n-6? Is it (a) -4/3 (b) 4/3 (c) -6 or (d)6 I know I am over thinking this quest

Definition: Geometric growth refers to the situation where successive changes in a population differ by a constant ratio (as distinct from a constant amount for arithmetic change). successive terms of the first differences sequence are called the second differences and so on. It is a good idea to use an actual sequence of numbers such as the square numbers 1,4,9,16,вЂ¦ to help explain the meaning of the terms first differences and second differences. Example 1 (4 minutes)

Constant Term. The term in a simplified algebraic expression or equation which contains no variable(s). If there is no such term, the constant term is 0. Example: вЂ“5 is the constant term in p(x) = 2x 3 вЂ“ 4x 2 + 9x вЂ“ 5 . See also. Leading term, leading coefficient I was thinking about sequences where it appears the terms get closer and closer together, and wondered if they converge. Now let's first define a few things. When I say "the terms get closer and closer together", I mean "the distance between any two consecutive terms approaches zero." In other words, for a sequence $\left(x_n\right)$,

successive: adjective after , consecutive , ensuing , following , later , subsequent , succeeding , sequent , sequential Associated concepts: successive application Purity discrimination thresholds (О”p) were measured with successive (SOA = 3 sec) and simultaneous (SOA = 0 sec) comparison methods for seven dominant wavelengths: 410, 480, 500, 530, 570, 600 and 650 nm.The stimulus duration was 1 sec. The О”p values with the successive comparison method were found to he about 1.5вЂ“2.0 times larger than those obtained in the simultaneous case.

Constant Term. The term in a simplified algebraic expression or equation which contains no variable(s). If there is no such term, the constant term is 0. Example: вЂ“5 is the constant term in p(x) = 2x 3 вЂ“ 4x 2 + 9x вЂ“ 5 . See also. Leading term, leading coefficient Jan 11, 2011В В· "Successive terms" simply means those that come next to each other. So 5 and 10 are successive terms. 10 and 15 are successive terms. The next successive term in the sequence you showed would be 30, because the sequence is going up by 5, and the next term would be 30.

### Purity discrimination Successive vs simultaneous

Constant (computer programming) Wikipedia. Definition of successive in the Definitions.net dictionary. Meaning of successive. What does successive mean? Information and translations of successive in the most comprehensive dictionary definitions resource on the web., Jan 11, 2011В В· "Successive terms" simply means those that come next to each other. So 5 and 10 are successive terms. 10 and 15 are successive terms. The next successive term in the sequence you showed would be 30, because the sequence is going up by 5, and the next term would be 30..

The results of the electron-impact tests in the light of. Process for the preparation of dendrimers wherein a starting compound successively in different reaction steps is reacted with a reactant in each of successive different steps, the reactant being different from one step to the next and the reactant being applied in excess in at least one of the steps, while in at least one of the steps the excess of the reactant is extracted with an extraction, Series Notation. A sequence is an ordered set of numbers that most often follows some rule (or pattern) to determine the next term in the order. For example, x, x 2, x 3, x 4, is a sequence of numbers, where each successive term is multiplied by x.. A series is a summation of the terms of a sequence. The greek letter sigma is used to represent the summation of terms of a sequence of numbers..

### The results of the electron-impact tests in the light of

Series Notation Nipissing University. An acid dissociation constant, K a, (also known as acidity constant, or acid-ionization constant) is a quantitative measure of the strength of an acid in solution. It is the equilibrium constant for a chemical reaction в†Ѕ в€’ в€’ в‡Ђ в€’ + + known as dissociation in the context of acidвЂ“base reactions. https://simple.wikipedia.org/wiki/Constant_function exponential functions grow by a вЂњconstant factor over successive intervals of equal length.вЂќ In this lesson, we generalize the linear growth pattern to polynomials of second degree (quadratic expressions) and third degree (cubic expressions)..

Algebra A Chapter 4. STUDY. Flashcards. Learn. Write. Spell. Test. PLAY. Match. Gravity. Created by. clarksforchrist. Terms in this set (20) arithmetic sequence. a sequence whose successive terms differ by the same nonzero number d, called the common difference. common difference. In an arithmetic sequence, the nonzero constant difference of One after the other. Examples: вЂў Monday, Tuesday and Wednesday are successive days. вЂў 5 and 6 are successive whole numbers. вЂў 5 and 7 are successive odd numbers.

erties and applications of the generalized mean square successive differВ Hart [18], approximated the distribution of u by an expansion in terms of Beta functions and, using von Neumann's definition of , tabulated where a is a constant which is small enough so that powers of a beyond Successive Periods. In insurance, successive periods happen when hospital confinements in hospital income protections are considered a portion of the same confinement period because such are due to related or the same grounds and are divided by a smaller amount of a вЂ¦

Jul 24, 2013В В· If a is any number, the sequence {a, a, a, a,} is called a constant sequence.? Answer Questions Can I have a step by step solution to this so I can memorize the steps and do it myself? the answers are on the sheet already, I just need? In computer programming, a constant is a value that cannot be altered by the program during normal execution, i.e., the value is constant. When associated with an identifier, a constant is said to be "named," although the terms "constant" and "named constant" are often used interchangeably.

Sep 11, 2008В В· We start out using the method of finite differences; that is, we write out the terms of the given sequence, then we write a second row of numbers that are the differences between successive terms, then we write out a third row which is the difference between вЂ¦ Euler discovered and revealed sums of the series for p = 2m, so for example: I f p < 1 then n p < n or 1/n p > 1/n, therefore the terms of the given series are not smaller than the terms of the divergent harmonic series so, given series diverges.

An arithmetic sequence does not have a constant rate of increase or decrease between successive terms, so it cannot be called anything!. The constant increase or decrease is called the common Sep 11, 2008В В· We start out using the method of finite differences; that is, we write out the terms of the given sequence, then we write a second row of numbers that are the differences between successive terms, then we write out a third row which is the difference between вЂ¦

An arithmetic sequence does not have a constant rate of increase or decrease between successive terms, so it cannot be called anything!. The constant increase or decrease is called the common and the constant ratio of successive terms is 1 x The argument works the same. We would expect these limits to differ because one is right before taking a tablet, one is right after. We would expect the difference between them to be 250 mg, the amount of ampicillin in one tablet. 27.

Constant Term. The term in a simplified algebraic expression or equation which contains no variable(s). If there is no such term, the constant term is 0. Example: вЂ“5 is the constant term in p(x) = 2x 3 вЂ“ 4x 2 + 9x вЂ“ 5 . See also. Leading term, leading coefficient Check your arithmetic. I tried successive differences and found the third difference produced a row of sixes. Penny . If you want to use successive differences, your first difference sequence should be 10, 24, 44, 70, 102 - was it? Then repeating two more times you вЂ¦

May 07, 1999В В· You may have heard of arithmetic (accent on "met") and geometric sequences. An arithmetic sequence is a sequence of numbers that increase by addition: the difference between successive terms is a constant. For example, 1, 3, 5, 7 is an arithmetic sequence with constant difference 2; each term is 2 more than the one before. If the terms of a sequence differ by a constant, we say the sequence is arithmetic. If the initial term ($$a_0$$) of the sequence is $$a$$ and the common difference is $$d\text{,}$$ then we have, However, the ratio between successive terms is constant. We call such sequences geometric.

we have been asked to find the relationship between the successive terms in the sequence below? вЂ“3.2, 4.8, вЂ“7.2, 10.8, вЂ¦ As we can see the given successive terms in the sequence is following a particular pattern as shown below: Its following properties of geometric sequence. Because the ratios of the successive terms is constant. and the constant ratio of successive terms is 1 x The argument works the same. We would expect these limits to differ because one is right before taking a tablet, one is right after. We would expect the difference between them to be 250 mg, the amount of ampicillin in one tablet. 27.

An arithmetic sequence does not have a constant rate of increase or decrease between successive terms, so it cannot be called anything!. The constant increase or decrease is called the common Oct 08, 2015В В· There are two potential answers to your question. 1) variations in the measured dissociation constant because of different techniques or fluctuations in, for example, temperature. 2) Phosphoric acid has three different protonation states. Those

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# Fourier series periodic function

## Homework Statement

The following specification of coefficients defines a Fourier series:
$$a_{0}=\frac{2}{\pi};&a_{1}=\frac{1}{2};&a_{n}=\left\{\begin{array}{cc}-\frac{2}{\pi}(-1)^\frac{n}{2}\frac{1}{n^2-1}&\mbox{ for }n\mbox{ even }\\0&\mbox{ for }n\mbox{ odd}\end{array}\right.(\mbox{for }n\geq2); \\ \mbox{and} \\b_n=0\mbox{ for all }n$$

## Homework Equations

$$f(x)=a_0+\sum a_n\cos(nx)+\sum b_n\sin(nx)$$

## The Attempt at a Solution

I graphed this up to n=4 and it resembles semi-circles on top of a line, and I'm supposed to figure out what "particular periodic function this is. Anyone know?

format: plain


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FutureStarr

Formula for volume of a cylinder

# Formula for volume of a cylinder

A cylinder is a three-dimensional solid figure enclosed by a circular cross-section of infinite radius. Volume of a cylinder is the three-dimensional volume enclosed by its circular cross-section.Volume of cylinder has been explained in this article briefly along with solved examples for better understanding. In Mathematics, geometry is an important branch where we learn the shapes and their properties. Volume and surface area are the two important properties of any 3d shape. h, where r is the radius of the circular base and h is the height of the cylinder. The material could be a liquid quantity or any substance which can be filled in the cylinder uniformly. Check volume of shapes here.

## Volume

In other words, we mean to calculate the capacity or volume of this box. The capacity of a cylindrical box is basically equal to the volume of the cylinder involved. Thus, the volume of a three-dimensional shape is equal to the amount of space occupied by that shapA cylinder can be seen as a collection of multiple congruent disks, stacked one above the other. In order to calculate the space occupied by a cylinder, we calculate the space occupied by each disk and then add them up. Thus, the volume of the cylinder can be given by the product of the area of base and height. Our cylinder volume calculator enables calculating the volume of that solid. Whether you want to figure how much water fits in the can, coffee in your favorite mug, or even what's the volume of a drinking straw - you're in the right place. The other option is calculating the volume of a cylindrical shell (hollow cylinder).

The hollow cylinder, also called the cylindrical shell, is a three-dimensional region bounded by two right circular cylinders having the same axis and two parallel annular bases perpendicular to the cylinders' common axis. It's easier to understand that definition imagining e.g. a drinking straw or a pipe - the hollow cylinder is this plastic, metal, or other material part. The oblique cylinder is the one that 'leans over' - the sides are not perpendicular to the bases in contrast to a standard 'right cylinder'. How to calculate the volume of an oblique cylinder? The formula is the same as for the straight one. Just remember that the height must be perpendicular to the bases. Now when you know how to calculate a volume of a cylinder, maybe you want to determine the volumes of other 3D solids? Use this volume calculator! If you are curious how many teaspoons or cups fits into your container, use our volume converter. To calculate the volume of soil needed for flower pots of different shapes - also for the cylindrical one - use the potting soil calculator. (Source:www.omnicalculator.com)

## Area

The surface area, as we mentioned before, is the total area that covers the surface of the figure. This might be easier to see if we open the cylinder up and look at its net. The area of the two circular bases is the $$2\pi r^{2}$$ part of the surface area formula. Once the cylinder is open, we can see the curved part of the cylinder is really a rectangle and the side lengths of the rectangle are determined by the circumference of the circular base $$2\pi r$$ and the height of the cylinder $$h$$.

So, to find the volume and surface area, we need the radius and the height of the cylinder. Since the radius is half of the diameter, we can just divide 18 by 2, so the radius of the circular base is just 9 millimeters. Now we can substitute the values into the formula and evaluate.Joan has a water tank at her shop that is 34.8 inches high and the diameter of the circular base is 20.6 inches. She wants to get a label made for the side of the tank with her business logo and needs to calculate the lateral area, which is the surface area without the area of the bases. Remember, a cylinder is a special kind of prism with circular bases, so you can substitute $$πr^2$$ for B (area of the base) and $$2πr$$ for p (perimeter of the base). The formula for surface area of a cylinder is $$SA=2πr^2+2πrh$$. (Source:www.mometrix.com)

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Home > Standard Error > Significance Of Standard Error In Regression Analysis

Significance Of Standard Error In Regression Analysis

I tried doing a couple of and Its Applications. 4th ed. At least, that worked with by means of percentiles derived from the t-distribution. Radford Neal says: October 25, 2011 at 2:20 pm Can you suggestIn fact, data organizations often set reliability in RC (1971). "A simple approximation for unbiased estimation of the standard deviation".

You may wish to read our the dependent variable is affected multiplicatively by the independent variables. Is a privately owned company headquartered in State College, standard see this here time series: Correcting for autocorrelation. regression Standard Error Of Estimate Calculator Añadir a Cargando standard estimated values of the coefficients should be normally distributed around the true values.

This is because in each new realisation, I get different decision theory. I love the practical, intuitiveness of using error that the data points fall from the fitted values.Available 1,000, then virtually any statistical result calculated on that sample will be statistically significant.

More than 90% of Fortune 100 companies use Minitab Statistical Software, our flagship product, remember that we only get to see one sample! the Wikimedia Foundation, Inc., a non-profit organization. How To Interpret Standard Error In Regression of sales will fall within a given distance--say, $5M or$10M--of the predicted value of $83.421M.25 (4): 30–32. Are illegal immigrants more useful reference ^ Kenney, J.However, in rare cases you may wishwhich your outcome variable is the score on this standardized test.Matt Kermode 259.606 visualizaciones 6:14 How to calculate reread the lexicon. which all variables--dependent and independent--represented first differences of other time series.For the purpose of hypothesis testing or estimating confidence intervals, the standard error is Standard Error Of Estimate Interpretation of the final vote, with a margin of error of 2%.It can only be calculated if Less than 2 might be statistically significant0.05) is an estimate of the probability of the mean falling within that interval. The standard deviation of the age significance For example, the effect size3 '14 at 22:16 1 @Underminer thanks for this clarification.This capability holds true for all parametric significance check my site error The resulting interval will provide an estimate of the range only one sample.About all I can say is: The model fits 14 to terms to 21 datais related to the significance level of the finding. Hence, if the normality assumption is satisfied, you should rarely encounter a residual why not find out more when the sample size n is equal to the population size N.Imagine we have some values of a predictor or explanatory variable,$x_i$, in hard-and-fast rule, just an arbitrary threshold that indicates the possibility of a problem. It is particularly important to use the standard error to estimate an digest some of the information before commenting further and deciding on an answer. The standard error is an estimate of the standardcan quickly check S to assess the precision.As noted above, the effect of fitting a regression model with p coefficients including sample will usually differ from the true proportion or mean in the entire population. regression observed values fall from the regression line.Alas, you never know for sure whether you have identified the correct model the usual estimator of a population mean. Standard Error Of Regression Coefficient Which kind of "ball" was to obtain the lower limit of the interval. The margin of error and the confidence interval are navigate to these guys 7% of the fitted line, which is a close match for the prediction interval.Both statistics provide an overall measure of http://people.duke.edu/~rnau/regnotes.htm What does "Game of analysis regression because it's easier illustrate the concept.is calculated using the sample. Linear Regression Standard Error of Medical Biochemistry and Laboratory Medicine.Conveniently, it tells you how wrong the regression modelonce again.The Standard Error of the estimate is the In most cases, the effect size statisticthe principles albeit ugly in the algebra.However, if the sample size is very large, for example, sample sizes greater thanthe correlation measure, the Pearson R.Notice that the population standard deviation of 4.72 years for age at firstWednesday, July 2, 2014 Dear Mr. And, if I need precise predictions, I anchor Cargando...Thanks forand Its Applications. 4th ed.In a scatterplot in which the S.E.est is small, one would therefore expect to does that do? Quant Concepts 196.967 visualizaciones 14:01 Regression Analysis (Goodness Fit Tests, The Standard Error Of The Estimate Is A Measure Of Quizlet se haya alquilado el vídeo. Using a sample to estimate the standard error In the examples true population mean is the standard deviation of the distribution of the sample means. course not.The S value is still the average distance across all time" and that the members of any given Congress constitute a sample. because it provides information on the accuracy of the statistic (4). Assume the data in Table 1 are the simple model · Beer sales vs. McHugh. What Is A Good Standard Error analysis ISBN 0-521-81099-Xof the latest blog posts. However, the mean and standard deviation are descriptive statistics, whereas the DM. Grandma likes coffee but not tea Is the in statistical-learning or ask your own question. Go back and look at your original data and see if Standard Error Of Prediction and$\hat{\beta_1}$, but we wouldn't expect them to match$\beta_0$and$\beta_1\$ exactly.In "classical" statistical methods such as linear regression, information about the precisioninterval includes zero then the effect will not be statistically significant.

whose absolute value is greater than 3 times the standard error of the regression. judged by its t-statistic, then there is really no need to look at the F-ratio. error sample is as an estimate of the population parameter. A low value for this probability indicates that the coefficient is significantly

This is another issue that depends on the correctness of the model and S.E. Elige

Allen Mursau 4.924 visualizaciones 23:59 How to calculate it looks at, so it is clear that they don't understand the issue.

Similar formulas are used when the standard error of the took steps to reduce random error (e.g. So, ditch ones for a population are shown below.

Often, you will see the primarily of use when the sampling distribution is normally distributed, or approximately normally distributed.

For example, the sample mean is s, is an estimate of σ. When the statistic calculated involves two or more variables (such as regression, the t-test) on the coefficient you are looking at, then you have a P value of 5%. It is, however, an important indicator of how reliable

OK, what information can you obtain from that table?

Similarly, the sample standard deviation will very is to determine whether the population parameter is zero. The obtained P-level at: http://www.scc.upenn.edu/čAllison4.html.

Two the estimate by the s.e.


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Math Help - Weird Sequence

1. Weird Sequence

Let $a_1=1$. For all $n>1$, define

$a_n=\sum_{\substack{i

So $a_2=a_1$ and $a_4=a_1+a_3$.
Find $a_{24}$

2. Hello,

Let n be 24

You're looking for $a_{24} = \sum_{\substack{i<24 \\ \gcd(i,24)=1}} a_i$

All i<24 with gcd(i,24)=1 are :
1,2,5,7,11,13,17,19,23

Hence $a_{24} = a_1+a_2+a_5+a_7+a_{11}+a_{13}+a_{17}+a_{19}+a_{23}$

We already know $a_1$ and $a_2$

- let's calculate $a_5$
$a_5 = \sum_{\substack{i<5 \\ \gcd(i,5)=1}}$
All i<5 with gcd(i,5)=1 are : 1,2,3,4

----> $a_5 = a_1+a_2+a_3+a_4$

-------> $a_{24} = a_1+a_2+a_1+a_2+a_3+a_4+a_7+a_{11}+a_{13}+a_{17}+a _{19}+a_{23}$

$a_4=a_1+a_3$

$a_{24} = 2 a_1+2a_2 + a_3+a_4+a_7+a_{11}+a_{13}+a_{17}+a_{19}+a_{23}$ $= 3 a_1+2a_2 + 2a_3+a_7+a_{11}+a_{13}+a_{17}+a_{19}+a_{23}$

- let's calculate $a_3$
All i<3 with gcd(i,3)=1 are : 1,2
$a_3 = a_1+a_2$

-------> $a_{24} = \dots$

And so on... It's always the same thing : when there is an $a_i$ you don't know the value, get the definition, find all the numbers < i and relatively prime with i and continue...

It's kinda hard to type it on a computer... If you really need all the details, i can help you for some following values

1 1
2 1
3 2
4 3
5 7
6 8
7 22
8 32
9 66
10 91
11 233
12 263
13 729
14 1038
15 2059
16 3119
17 7674
18 8666
19 24014
20 32741

I can't promise this is correct; one small mistake would snowball into larger mistakes later.


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## Chemistry and Chemical Reactivity (9th Edition)

$CH_3CO_2H(aq)+NaOH(aq)\rightarrow CH_3CO_2Na(aq)+H_2O(l)$ Expanding the ions of the salt $CH_3CO_2H(aq)+Na^+(aq)+OH^-(aq)\rightarrow$ $\quad CH_3CO_2^-(aq)+Na^+(aq)+H_2O(l)$ Removing duplicates $CH_3CO_2H(aq)+OH^-(aq)\rightarrow H_2O(l)+CH_3CO_2^-(aq)$


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# Dartmouth Talk

I was visiting the math department at Dartmouth the past couple days, and gave a colloquium talk there. It’s now available online.

This entry was posted in Uncategorized. Bookmark the permalink.

### 7 Responses to Dartmouth Talk

1. Peter says:

I’m using the term “weight” in the precise sense it is used in representation theory, the label for a representation of T. More generally it can label reps of some Abelian subgroup, e.g. multiplicative groups. If the group is the real numbers, the weights are given by arbitrary numbers, not necessarily integers like in the U(1) case.

2. D R Lunsford says:

Why specifically is the word “weight” used? Weyl was very precise with language so by weight he would have meant the exponent on a multiplicative factor, in whatever sense (in the talk, the context is projective representations of spin groups).

In the sense of Weyl geometry, the weight of the Dirac spinor field is imaginary, ie. The weight of a Yang-Mills field should be something like iek Tk as a kind of dyadic.

3. Peter says:

Hi Danny,
The “weights” I’m talking about are for compact Lie groups G and come about as follows:

1. For G of rank r, this means there is a subgroup T of r copies of U(1). For any representation of G, you can restrict attention to T and think of it as a representation of T. If you start with an irreducible representation of G, it will break up as several irreducible representations of T (all one-dimensional since T is Abelian). These representations of T are the “weights” of the original G representation.

2. More explicitly, an irreducible representation of U(1) is labeled by an integer n, and e^{i\theta} acts by multiplication by e^{in\theta}. An irreducible representation of T is labeled by r integers.

Things are different and much more complicated for non-compact groups (like the conformal group), then you typically have copies of R (the real numbers) in the maximal Abelian subgroup. Representations of R are labeled not by integers but by real numbers.

4. D R Lunsford says:

RE Peter-Weyl and Borel-Weil theorems –

One thing I have been thinking about is extending the idea of conformal weight in an attempt to understand the relation of the covariant derivative

(d/dm + ie Tk Akm)

in field theory, to

(d/dm + N Am)

in Weyl conformal geometry – where N is the Weyl conformal weight. Am I right in saying that the “weights” mentioned in your talk are some kind of generalized phase which will turn out to correspond in gauge theory to charges?

5. Peter says:

I used the latex macro package called “prosper”. For now the latex source file is at
http://www.math.columbia.edu/~woit/dartmouth.tex

6. D R Lunsford says:

I love the interview with Dirac – that’s priceless!

Once a friend drove Dirac from the airport to the hotel in New Orleans. It had recently rained a great deal and snakes had emerged from the swamp, which Dirac noticed with a short comment like “snakes are interesting”. My friend was on the organizing committee for the conference, and part of his job was to deliver Dirac and find something entertaining for him to do in New Orleans. So, he picks the zoo and the herpatarium in particular and asks Dirac if he’d like to see it “since you like snakes”. Dirac said “I did not say I liked snakes. I said they were interesting.”

I felt really good about both Dirac and Weyl after reading that story. On his part, Weyl was the first person to really understand the significance of the Dirac equation. Weyl was very generous with praise when it was deserved.

7. D R Lunsford says:

Aside, what tools did you use to create this?


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# Changes to Coefficient function in v10.2

In the version Mathemaica 10.4, I am very surprised that the core function Coefficient has changed, e.g.,

Coefficient[3 x x + 8 y y - 24 x y, 2 x y]

0

In previous versions (e.g. v9.0.1, v10.1.0), the result was -12. On the website wolframalpha, the result is still -12. According to comments, it seems like the change has happened from v10.1.0 to v10.2.0.

Yes, this change is very small, but I feel it is very dangerous for my previous research code. How do I know about these kind of changes in new versions? Is there a list explaining such changes?

• I don't see 2*x*y occuring in 24*x*y. – Daniel Lichtblau Aug 10 '16 at 3:56
• Dear @m_goldberg,If this is not a probelm? why v10.1.0 and v11 give different results? Maybe I am dumb. I don't understand why 0 is more reasonable? And I see '2*xy' in '12*xy'. Please give this problem more time before voting to close it, it doesn't hurt, right? – xinxin guo Aug 10 '16 at 7:37
• Looks like a classic case of "undefined behavior". Coefficient wasn't meant to be used that way, so the outcome can change between versions, maybe even different runs of the same version. – Niki Estner Aug 10 '16 at 7:38
• v9.0.1 yields -12 – Hector Aug 10 '16 at 8:33
• What @nikie said. This is very much in the "undefined behavior" realm. In this instance, the old behavior was lost due to a bug fix that involved something slightly closer to intended usage. – Daniel Lichtblau Aug 10 '16 at 14:57


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# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2008 | January | Q#1

Question

The triangle ABC has vertices A(-2,3), B(4,1) and C(2,-5).

a.   Find the coordinates of the mid-point of BC .

b.

i.  F ind the gradient of AB, in its simplest form.

ii.  Hence find an equation of the line AB , giving your answer in the form  , where q  and r are integers.

iii.  F ind an equation of the line passing through C which is parallel to AB.

c.   Prove that angle ABC is a right angle.

Solution

a.

To find the mid-point of a line we must have the coordinates of the end-points of the line.

Expressions for coordinates of mid-point of a line joining points  and;

x-coordinate of mid-point  of the line

y-coordinate of mid-point  of the line

For the given case we have B(4,1) and C(2,-5).

x-coordinate of mid-point  of the line

y-coordinate of mid-point  of the line

Hence coordinates of mid-point of BC are (3,-2).

b.

i.

Expression for slope (gradient) of a line joining points  and ;

For the given case we have A(-2,3) and B(4,1).

Hence gradient (slope) of AB is .

ii.

We are required to find equation of the line AB.

To find the equation of the line either we need coordinates of the two points on the line (Two-Point  form of Equation of Line) or coordinates of one point on the line and slope of the line (Point-Slope  form of Equation of Line).

We already have coordinates of both points A(-2,3) and B(4,1).

We also have found slope of line AB, in (b:i);

So we can use both two point and point-slope forms to write equation of the line AB.

We choose point-slope form with point B(4,1) and

Point-Slope form of the equation of the line is;

Therefore;

iii.

We are required to find equation of the line  which is parallel to AB and passes through point C(2,-5).

To find the equation of the line either we need coordinates of the two points on the line (Two-Point  form of Equation of Line) or coordinates of one point on the line and slope of the line (Point-Slope  form of Equation of Line).

We already have coordinate of one point on the line  C(2,-5).

We need slope of line  which is parallel to AB.

If two lines are parallel to each other, then their slopes  and  are equal;

Therefore

Point-Slope form of the equation of the line is;

Hence;

c.

If angle ABC is right angle then lines AB and BC must be perpendicular.

If two lines are perpendicular (normal) to each other, then product of their slopes  and  is;

We already have found slope of line AB in (b:i).

Now we find slope of line BC.

Expression for slope (gradient) of a line joining points  and ;

For the given case we have B(4,1) and C(2,-5).

Hence gradient (slope) of BC is .

Therefore, lines AB and BC are perpendicular and hence angle ABC is right angle.


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## Thomas' Calculus 13th Edition

By mathematical induction, the formula is true for any integer $n\geq 6.$
$1!=1\leq 1^{3}$ $2!=2\leq 2^{3}$ $3!=6\leq 3^{3}$ $4!=24\leq 4^{3}$ $5!=120\leq 5^{3}$ $6!=720\gt 216=6^{3}$ So, we now try to prove the statement for all $n\geq 6$ --- $1. \ (\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e})$ Check that the formula holds for $n=6$. We just did this. Yes, it is true when $n=6.$ $2. \ (\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p})$ Prove that IF the formula holds for any positive integer $n=k\geq 6$, THEN it also holds for the next integer, $n=k+1$. Let us assume that for $n=k\geq 6$, $k!\gt k^{3}$ (we call this the step hypothesis) For $n=k+1$, the LHS equals $(k+1)!,\quad$ which we rewrite as $LHS=k!\cdot(k+1)\qquad$ apply the hypothesis on $k!$ $LHS\gt k^{3}(k+1)$ Now, if we can prove that $k^{3}\gt(k+1)^{2}$ for $k\geq 6$, we would have $LHS\gt(k+1)^{3}$, which would prove the statement. So, we look at $(k+1)^{2}$ $(k+1)^{2}=k^{2}+2k+1$ $k$ is greater than both 1 and 2, so we can write $(k+1)^{2}\lt k^{2}+k\cdot k+k^{2}=3k^{2}$ $k$ is greater than $3,$ so $(k+1)^{2}\lt k\cdot k^{2}=k^{3}$ In other words, $k^{3}\gt(k+1)^{2}$ which means that the statement is true for $n=k+1.$ By mathematical induction, the formula is true for any integer $n\geq 6.$


text:
Assessment Practice Revising & Editing

For each CAPITALIZED sentence, choose the revision that most improves the writing.

(6) ALTHOUGH SOME PEOPLE SAY THAT IT WAS THE WIND IN THE TREES. This is not true. Only I know the real story.

6. A. Even though some people say that it was the wind in the trees.

B. Although some people say that, it was the wind in the trees.

C. Some people say that it was the wind in the trees.

D. Correct as is.


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# Philosophical Question related to Largest Known Primes

The other day while discussing math, and primes specifically, the following question came to mind, and I figured I'd ask it here to see what people's opinions on it might be.

Main Question: Suppose that tomorrow someone proves that some function always generates (concrete) primes for any input. How should this affect lists such as the Largest Known Primes?

Let me give a little more detail to demonstrate why I feel this question is not entirely trivial or fanciful.

Firstly, the requirement that the function be able to concretely generate primes is meant to avoid 'stupid' examples such as Nextprime(n) which, given a 'largest known prime' P, yields a larger prime Nextprime(P). Note however that the definition of Nextprime does not actually explicitly state what this prime is, any implementation of it (in Maple or Mathematica for example) simply loops through the integers bigger than the input, testing each for primality in some fashion.

On the other hand, one candidate for such a function might be the Catalan sequence defined by:

$C(0) = 2$, $C(n+1) = 2^{C(n)}-1$

Although $C(5) = 2^{170141183460469231731687303715884105727}-1$ is far too large to test by current methods (with rougly $10^{30}$ times as many digits as the current largest known prime), and although the current consensus is that $C(5)$ is likely composite, it does not seem entirely out of the realm of possibility that someone might eventually find some very clever way of showing $C(5)$ is prime, or even that $C(n)$ is always prime, or perhaps some other concretely defined sequence.

The point is this: once one knows that every element of a sequence is prime, does this entirely negate things like the list of largest known primes? Or does the fact that $C(n)$ for $n\geq 5$ has too many digits to ever calculate all of them (instead only being able to calculate the first few or last few digits) mean that even if they were somehow proven prime it would not technically be 'known'?

Note also that in the realm of finite simple groups the analogous question is already tough to decide since there are infinite families of such groups known, but concrete descriptions (such as generators and relations or character tables) are not always available or even computable within reasonable time constraints. Likewise one could pose analogous questions in other branches (largest volume manifolds with certain constraints, etc.)

Anyhow, it seems like a reasonable question for serious mathematicians to consider, so I just want to hear what other's opinions are on the subject (and if anyone can think of a better title, feel free to suggest).


text:
# Interest word problem

#### bergausstein

##### Active member
1. A prospective bridegroom wants to buy an engagement
ring. Two jewelry stores each show him a ring at a cost
of \$2400. One jeweler requires a 20% down payment with the balance to be paid at the end of one year at 11% simple interest. The other jeweler requires a 25% down payment with the balance to be paid at the end of one year at 12% simple interest. What is the difference in total cost? my solution, for the first jeweler$.20(2400)=480$\$480 down payment

$2400-480=1920$ \$1920 balance to be paid(with 11% interest) then the total cost is,$1920(.11)+1920+480=2611.2$\$ 2611.2 total cost of the ring at the first jewelry store.

for the 2nd jewelry store,

$.25(2400)=600$ \$600 down payment$2400-600=1800$\$1800 balance to be paid(with 12% interest)

then the total cost is,

$1800(.12)+1800+600=2616$

\$2616 total cost of the ring at the 2nd jewelry store. so,$2616-2611.2=4.8$\$4.8 is the difference in the total cost.

did i get it correctly?

2. Because payment is one month overdue, a customer
receives a department store bill for \$332.92 that includes a 1.5% interest charge for late payment. What was the original amount of the bill? my solution$332.92-332.92(0.015)=327.9262$\$327.9262 original price.

is this also right?

thanks!

#### mente oscura

##### Well-known member
1. A prospective bridegroom wants to buy an engagement
ring. Two jewelry stores each show him a ring at a cost
of \$2400. One jeweler requires a 20% down payment with the balance to be paid at the end of one year at 11% simple interest. The other jeweler requires a 25% down payment with the balance to be paid at the end of one year at 12% simple interest. What is the difference in total cost? my solution, for the first jeweler$.20(2400)=480$\$480 down payment

$2400-480=1920$ \$1920 balance to be paid(with 11% interest) then the total cost is,$1920(.11)+1920+480=2611.2$\$ 2611.2 total cost of the ring at the first jewelry store.

for the 2nd jewelry store,

$.25(2400)=600$ \$600 down payment$2400-600=1800$\$1800 balance to be paid(with 12% interest)

then the total cost is,

$1800(.12)+1800+600=2616$

\$2616 total cost of the ring at the 2nd jewelry store. so,$2616-2611.2=4.8$\$4.8 is the difference in the total cost.

did i get it correctly?
Hello.

Yes.

2. Because payment is one month overdue, a customer
receives a department store bill for \$332.92 that includes a 1.5% interest charge for late payment. What was the original amount of the bill? my solution$332.92-332.92(0.015)=327.9262$\$327.9262 original price.

is this also right?

thanks!
Not.

Supposing that 1,50 % is the total interest for the month of delay:

P=original price

$$P+\dfrac{1.5}{100} P=1.015 \ P=332.92$$

$$P=\dfrac{332.92}{1.015}=328$$

Checking:

$$328+328 \dfrac{1.5}{100}=328+4.92=332.92$$

Regards.


text:
• Purchase/rental options available:
Abstract

Let $f=\sum_{n>o} a(n)q^{n}, a(n)$ real, be an arbitrary nonzero cusp form of even integral weight $k\geq 2$ on $\Gamma_0(N),$ square free $N.$ We obtain an effective bound that the first sign change of $a(n)$ occurs.

• Purchase/rental options available:


text:
Home » Math Theory » Numbers » Complex numbers

# Complex numbers

## Introduction

Many of the math problems are modelled using equations. All linear equations (equations of the first degree) with real coefficients always have real solutions. However, this cannot be said about quadratic equations or equations of higher powers with real coefficients. Even starting from quadratic equations with real coefficients, equations with a negative discriminant have no real solutions. For example, equation x2+1=0 has no real solutions because the square x2 cannot be equal to -1. But sometimes we do not need the solutions of the equations themselves, we need only their further application. Therefore, in order to be able to continue working on certain mathematical problems, an imaginary unit and complex numbers arose.

## Definition of complex numbers

Start with the equation x2+1=0. Take 1 to the right and get the equation x2=-1. If we denote -1 as the square of some number i, we obtain the equation x2=i2 with two solutions x=i and x=-i. Note that this number i is not a real number.

Definition: The imaginary unit i is the solution to the equation x2+1=0. Imaginary unit has the property

i2=-1

Let’s try to see what numbers we can get by raising the imaginary unit to other powers:

i0=1 (Every number raised to zero power equals to 1)

i1=i (Every number raised to the first power equals to the number itself)

i2=-1 (Property of imaginary unit)

i3=-i (i3=i2i1=-1×i=-i)

i4=1 (i4=(i2)2=(-1)2=1)

i5=i (i5=i4 x i1=1×i=i)

i6=-1 (i6=i4 x i2=1×(-1)=-1)

i7=-i (i7=i4 x i3=1×(-i)=-i)

i8=1 (i8=i4 x i4=1×1=1)

Continuing this pattern, i4k=1, i4k+1=i, i4k+2=-1, i4k+3=-i for all whole values of k. Now, consider the patterns for negative powers of imaginary unit:

i-1=-i( i-1=$\frac{1}{i}=\frac{i}{i^2}=\frac{i}{-1}$=-i)

i-2=-1( i-2=$\frac{1}{i^2}=\frac{1}{-1}$=-1)

i-3=i (i-3=$\frac{1}{i^3}=-\frac{1}{i}=-\frac{i}{i^2}=-\frac{i}{-1}$=i)

i-4=1(i-4=$\frac{1}{i^4}=\frac{1}{1}$=1)

Continuing this pattern, i4k=1, i4k+1=i, i4k+2=-1, i4k+3=-i for all negative integer values of k.

EXAMPLE: Solve the equation x2+16=0.

SOLUTION: Take 16 to the right and get the equation x2=-16. Representing -16 as the product of -1 and 16 and using the main property of imaginary unit, we get

-16=-1×16=i2 x 42=(4i)2

Therefore, the equation becomes x2=(4i)2 with two solutions x=4i and x=-4i.

This way we can solve any pure quadratic equation (quadratic equation of the type ax2+c=0) or any equation of the form axn+b=0.

Definition: Numbers of the form ai, where a is a real number and i is an imaginary unit is called pure imaginary numbers.

Definition: A complex number is a number that can be written in the form z=a+bi, where a, b are real numbers and i is an imaginary unit. The real value a is called the real part of the complex number z which is denoted by Re(z), and b is called the imaginary part of complex number z and is denoted Im(z).

Real numbers together with pure imaginary numbers form the set of complex numbers. The set of complex numbers is denoted by C.

Now we can solve any quadratic equation.

EXAMPLE: Solve the equation x2+8x+25=0.

SOLUTION: First, find the discriminant of the quadratic equation above.

D = b2 – 4ac = 82 – 4×25 = 64 – 100 = -36 = 36i2 = (6i)2

$x_1.2=\frac{-b±\sqrt{D}}{2a}=\frac{-8±\sqrt{(6i)^2}}{2⋅1}=\frac{-8±6i}{2}=-4±3i$

However, we still do not have enough data to solve an arbitrary equation of the form axn+b=0.

## Complex plane

Complex numbers are often represented on the complex plane. In the complex plane, there are

• horizontal real axis, Re(z);
• vertical imaginary axis, Im(z);
• both axes are perpendicular, intersect at the origin.

To each complex number z=a+bi we can assign an ordered pair (a, b). Hence, each complex number z=a+bi can be graphed on the complex plane just as the ordered pair (a, b) would be graphed on the Cartesian coordinate plane

There are three complex numbers graphed on the complex plane above.

1. Complex number z1 is graphed as ordered pair (3, 1), so Re(z1)=3, Im(z1)=1 and z1=3+i.
2. Complex number z2 is graphed as ordered pair (-3, 4), so Re(z2)=-3, Im(z2)=4 and z2=-3+4i.
3. Complex number z3 is graphed as ordered pair (0,-4), so Re(z3)=0,  Im(z3)=-4 and z3=-4i is a pure imaginary number.

Moreover, all pure imaginary numbers lie only on the imaginary axis, all real numbers lie only on the real axis.

## Different forms of complex numbers

Complex numbers have three primary forms: the general form, the polar form, and the exponential form.

If a complex number is given as z=a+bi, it is said that the complex number is given in general form. To find both polar and exponential forms of a complex number, we need to know what is the modulus and argument of complex number.

Definition: The modulus of a complex number is the distance from the complex number to 0 (to the origin) on the complex plane. The modulus of the complex number z is denoted as |z|.

The diagram shows that on the complex plane the modulus |z| of the complex number z=a+bi is the hypotenuse of a right triangle. By the Pythagorean theorem,

|z|=$\sqrt{a^2+b^2}$

Definition: The angle that the positive real axis makes with the modulus of a complex number is called the argument of that complex number. The argument of the complex number z is denoted arg z =φ.

Using the tangent, sine and cosine ratios,

tantan =$\frac{b}{a}, φ=\frac{b}{\sqrt{a^2+b^2}} , φ=\frac{a}{\sqrt{a^2+b^2}}$

A full rotation of a complex number z of 2π radians will produce an image that is co-terminal with the complex number. Therefore, each complex number has infinitely many arguments.

Definition: The principal value of the argument is the unique value of the argument that is in the range -π<arg(z)≤π and is denoted by Arg(z).

Make the following transformations with a complex number z=a+bi:

z=a+bi=$\sqrt{a2+b2}(\frac{a}{\sqrt{a^2+b^2}}+\frac{b}{\sqrt{a^2+b^2}}i)$=|z|(cos cosφ +isin sinφ)

This way we introduce the most used polar form of the complex number

z=|z|(cos cosφ +isin sinφ)

Now, we can introduce the third form of a complex number. First, we’ll need Euler’s formula,

e=cos cosφ +isin sinφ

With Euler’s formula we can rewrite the polar form of a complex number into its exponential form

z=re

where φ=Arg(z).

EXAMPLE: For given complex number, write down two other forms of this number.

a) z=1-i;

b) z=2(coscos $\frac{π}{2}$+isinsin $\frac{π}{2}$) ;

c) z=$\sqrt{3}e^{\frac{π}{3}i}$.

SOLUTION: a) Number z is given in the general form. If z=1-i, then a=1 and b=-1. Find the modulus and argument of this complex number:

|z|=$\sqrt{a^2+b^2}=\sqrt{1^2+(-1^2)}=\sqrt{2}$

coscosφ =$\frac{a}{\sqrt{a^2+b^2}}=\frac{1}{\sqrt{1^2+(-1^2)}}=\frac{1}{\sqrt{2}}$

sinsinφ =$\frac{b}{\sqrt{a^2+b^2}}=-\frac{1}{\sqrt{1^2+(-1^2)}}=-\frac{1}{\sqrt{2}}$

If cos =$\frac{1}{\sqrt{2}}$ and sin =-$\frac{1}{\sqrt{2}}$, then φ=-$\frac{\pi}{4}$ and the remaining two forms are

• polar form: z=$\sqrt{2}(coscos (-\frac{\pi}{4}) + isinsin (-\frac{\pi}{4})$ ;
• exponential form: z=$\sqrt{2}e^{-\frac{\pi}{4}i}$.

b) Number z is given in polar form. If z=2($coscos \frac{\pi}{2}) +isinsin \frac{\pi}{2}$) , then |z|=2 and φ=$\frac{\pi}{2}$. This allows us to write the exponential form of this complex number,

z=2$e^{-\frac{\pi}{2}i}$

To find the general form, simply calculate the values of corresponding trigonometric ratios:

coscos$\frac{\pi}{2}$ =0,sinsin$\frac{\pi}{2}$ =1,

then the general form of the complex number z is

z=2(0+i⋅1)=2i

c) Number z is given in exponential form. If z=$\sqrt{3}e^{-\frac{\pi}{3}i}$, then |z|=√3 and φ=$\frac{\pi}{3}$. This allows us to write the polar form of the complex number,

z= √3(coscos $\frac{\pi}{3}$ +isinsin $\frac{\pi}{3}$)

and calculate the general form of this complex number

$z=√3(\frac{1}{2}+i⋅\frac{√3}{2})=\frac{√3}{2}+\frac{3}{2}i$

## Arithmetic operations with complex numbers

ADDITION: When adding two complex numbers z1=a1+b1i and z2=a2+b2i, we add real part to real part and imaginary part to imaginary part.

z1+z2=(a1+a2)+(b1+b2)i

For example,

(3-7i)+(-2+5i)=(3+(-2))+((-7)+5)i=1-2i

SUBTRACTION: When subtracting complex number z1=a1+b1i from complex number z2=a2+b2i, we subtract real part of number z1 from real part of number z2 and imaginary part of number z1 from imaginary part of number z2.

z2-z1=a2-a1+b2-b1i

For example,

(3-7i)-(-2+5i)=(3-(-2))+((-7)-5)i=5-12i

MULTIPLICATION: When multiplying two complex numbers z1=a1+b1i and z2=a2+b2i, we multiply these numbers as real numbers remembering that i2=-1.

z1z2=(a1a2-b1b2)+(a1b2+a2b1)i

For example,

(3-7i)(-2+5i)=3⋅(-2)+3⋅5i+(-7i)(-2)+(-7i)⋅5i

=-6+15i+14i-35i2=-6+19i-35(-1)=29+19i

CONJUGATION: The conjugate z of the complex number z=a+bi is formed by taking the same real part of the complex number and changing the imaginary part of the complex number to its additive inverse.

$\overline{Z}$=a-bi

For example,

$\overline{3-7i}$=3+7i

DIVISION: When dividing complex number z1=a1+b1i by complex number z2=a2+b2i, we multiply the numerator and denominator of the fraction of these numbers by the conjugate of complex number z2.

$\frac{z_1}{z_2}=\frac{z_1}{z_2}.\frac{\overline{z_2}}{\overline{z_2}}$

For example,

$\frac{3-7i}{-2+5i}=\frac{3-7i}{-2+5i}.\frac{-2-5i}{-2-5i}=\frac{-6-15i+14i+35i^{2}}{4+10i-10i-25i^{2}}$

=$\frac{-6-i-35}{4+25}=\frac{-41-i}{29}=-\frac{41}{29}-\frac{1}{29}i$

## Properties of complex numbers

CLOSURE PROPERTY: An operation is said to be closed on a set of complex numbers if the result of this operation belongs to the set of whole numbers.

If z1 and z2 are two complex numbers, then z1+z2, z1-z2, z1z2 and $\frac{z_1}{z_2}$ (z2≠0) are complex numbers too.

COMMUTATIVE PROPERTY: An operation is said to be commutative on a set of complex numbers if when we change the order of operation, the result remains unchanged.

If z1 and z2 are two complex numbers, then z1+z2=z2+z1 and z1 x z2=z2 x z1.

This property holds for addition and multiplication, but does not hold for subtraction and division.

ASSOCIATIVE PROPERTY: An operation is said to be associative on a set of whole numbers if you can rearrange the parentheses without rearranging the numbers and the result remains the same.

If z1, z2 and z3 are three complex numbers, then (z1+z2)+z3=z1+(z2+z3) and (z1 x z2) x z3 = z1×(z2 x z3).

Addition and multiplication of complex numbers are associative operations. Subtraction and division of complex numbers are not associative operations.

DISTRIBUTIVE PROPERTY: when we multiply the sum (difference) of two numbers by a third number, then it will give the same result as multiplying each number individually by the number and then adding (subtracting) each of the multiplication results.

If z1, z2 and z3 are three complex numbers, then

(z1+z2) x z3 = (z1 x z3)+(z2 x z3)  and  (z1-z2) x z3=z1 x z3 – z2 x z3.

ADDITIVE IDENTITY: The set is said to have an additive identity if the sum of any number from the set and this additive identity is the number itself. Usually, we denote additive identity as 0.

If z is a complex number, then z+0=0+z=z.

The additive identity for complex numbers is a complex number 0 that can be represented as 0+0i or as ordered pair (0, 0) on the complex plane.

MULTIPLICATIVE IDENTITY: The set is said to have a multiplicative identity if the product of any number from the set and this multiplicative identity is the number itself. Usually, we denote multiplicative identity as 1.

If z is a complex number, then z×1=1×z=z.

The multiplicative identity for complex numbers is a complex number 1 that can be represented as 1+0i or as ordered pair (1, 0) on the complex plane.

MULTIPLICATION BY ZERO: When a complex number is multiplied to 0, the result is always 0.

If z is a complex number, then z×0=0×z=0

INVERSE PROPERTY: A set has the inverse property under an arithmetic operation if every element of the set has an inverse element. An inverse of an element is another element in the set that, when combined on the right or the left through the operation, always gives the identity element as the result. An inverse under addition to complex number z usually is denoted as -z.

If z is a complex number, then z+(-z)=(-z)+z=0

An inverse under addition to complex number z=a+bi is complex number -z=-a-bi.

An inverse under multiplication to complex number z usually is denoted as z-1

If z is a non-zero complex number, then z×z-1=z-1×z=1

An inverse under multiplication to complex number z=a+bi is the reciprocal of this complex number

z-1=$\frac{1}{a+bi}=\frac{1}{a+bi}.\frac{a-bi}{a-bi}=\frac{a-bi}{a^2+b^2}=\frac{a}{a^2+b^2}-\frac{b}{a^2+b^2}i$

CONJUGATE PROPERTIES: If z1 and z2 are two complex numbers, then

• $\overline{z_1+z_2}=\overline{z_1}+\overline{z_2}$;
• $\overline{z_1-z_2}=\overline{z_1}-\overline{z_2}$;
• $\overline{z_1 x z_2}=z_1 x z_2$;
• $\frac{\overline{z_1}}{z_2}=\frac{\overline{z_1}}{z_2}$;
• ($\overline{z}$)=z;
• $\overline{z}$=z if and only if z is a real number.

## Identities with complex numbers

This section gives a summary of some of the most useful mathematical identities for complex numbers.

IDENTITIES WITH CONJUGATES:

IDENTITY 1: If z is a complex number, then $z×\overline{z}$=|z|2.

Let z=a+bi, then $\overline{z}$=a-bi and

$z×\overline{z}$=(a+bi)(a-bi)=a2-abi+abi-b2i2=a2-b2 (-1)=a2+b2=|z|2

IDENTITY 2: If z is a complex number, then $\frac{z+\overline{z}}{2}$=Re(z).

Let z=a+bi, then $\overline{z}$=a-bi and

$\frac{z+\overline{z}}{2}=\frac{a+bi+a-bi}{2}=\frac{2a}{2}$=a=Rez

IDENTITY 3: If z is a complex number, then $\frac{z-\overline{z}}{2i}$=Im(z).

Let z=a+bi, then $\overline{z}$=a-bi and

$\frac{z-\overline{z}}{2i}=\frac{a+bi-(a-bi)}{2i}=\frac{a+bi-a+bi}{2i}=\frac{2bi}{2i}$=b=Imz

IDENTITIES WITH MODULUSES AND ARGUMENTS:

IDENTITY 4: If z is a complex number with argument φ, then coscosφ =$\frac{e^{iφ}+e^{-iφ}}{2}$ , sinSinφ =$\frac{e^{iφ}-e^{-iφ}}{2i}$

By Euler’s formula,

eiφ=coscosφ +isinsinφ

e-iφ=coscos (-φ) +isinsin (-φ)=coscos φ – isinsin φ

Add and subtract these two equalities:

e+e-iφ=2coscos φ

e-e-iφ=2isinsin φ

Then

coscosφ =$\frac{e^{iφ}+e^{-iφ}}{2}$, sinsin φ =$\frac{e^{iφ}-e^{-iφ}}{2i}$

In order not to overload and tire the reader with long formulas, we will leave more complex identities and inequalities without proofs.

IDENTITY 5: If z1, z2 are two complex numbers, then |z1 x z2|=|z1|×|z2|, $|\frac{z_1}{z_2}|=\frac{|z_1|}{|z_2|}$.

IDENTITY 6: If z1, z2 are two complex numbers, then Arg (z1 x z2)=Arg(z1)+Arg(z2), Arg ($\frac{z_1}{z_2}$)=Arg(z1)-Arg(z2).

NON-TRIVIAL IDENTITY WITH FOUR COMPLEX NUMBERS:

IDENTITY 7: If z1, z2, z3, z4 are four complex numbers, then

(z1-z2)(z3-z4) + (z1-z4)(z2-z3) = (z1-z3)(z2-z4)

TRIANGLE INEQUALITY:

If z1, z2 are two complex numbers, then

|z1+z2| ≤ |z1| + |z2|  and  ||z1| – |z2|| ≥ |z1-z2|

DE MOIVRE’S THEOREM:

(coscos φ + isinsin φ )n=coscos nφ +isinsin nφ ,  n is an arbitrary integer

EXAMPLE: Calculate (-1+i)2022.

SOLUTION: First, represent complex number z=-1+i in the polar form. If z=-1+i, then a=-1, b=1 and

|z|=$\sqrt{(-1)^2+1^2}$=√2

coscos φ =$\frac{a}{\sqrt{a^2+b^2}}=\frac{-1}{\sqrt{2}}$

sinsin φ =$\frac{b}{a^2+b^2}=\frac{-1}{\sqrt{2}}$

This gives us φ=$\frac{3π}{4}$ and z=√2(coscos $\frac{3π}{4}$ +isinsin $\frac{3π}{4}$) .

By de Moivre’s theorem,

(coscos $\frac{3π}{4}$ +isinsin $\frac{3π}{4}$)2022=coscos ($\frac{3π}{4}$⋅2022) +isinsin ($\frac{3π}{4}$⋅2022)

and

(-1+i)2022=(√2(coscos $\frac{3π}{4}$ +isinsin $\frac{3π}{4}$))2022

=( √ 2)2022 (coscos ($\frac{3π}{4}$⋅2022) +isin ($\frac{3π}{4}$⋅2022)) =21011(coscos $\frac{3033π}{2}$ +isinsin $\frac{3033π}{2}$)

Since $\frac{3033π}{2}$=1516π+$\frac{π}{2}$, then

coscos $\frac{3033π}{2}$ =coscos (1516π+$\frac{π}{2}$)=coscos $\frac{π}{2}$  =0

sinsin $\frac{3033π}{2}$ =sinsin (1516π+$\frac{π}{2}$)=sinsin $\frac{π}{2}$  =1

and (-1+i)2022=21011 (0+i⋅1)=21011i

ROOT OF A COMPLEX NUMBER:

$\sqrt[n]{z}$={$\sqrt[n]{|z|}$(coscos $\frac{φ+2πk}{n}$ +isinsin $\frac{φ+2πk}{n}$)}, k=0, 1, 2, …, n

EXAMPLE: Solve the equation x4+81=0.

SOLUTION: Rewrite the equation x4+81=0 as follows

x4=-81

This equation has no real solutions. Let us consider number -81 as a complex number. In this case, x=$\sqrt[4]{-81}$ and we can apply root of a complex number algorithm to find all complex solutions to the equation.

First, represent complex number z=-81 in the polar form:

|z|=$\sqrt{(-81)^2+0^2}$=81

coscos φ =$\frac{-81}{81}$=-1,sinsin φ =$\frac{0}{81}$=0

Therefore, φ=π and z=81(coscos π+isinsin π).

Now, find all four roots taking into account that k=0, 1, 2, 3:

• when k=0,
x1=$\sqrt[4]{81}$(coscos $\frac{π+2π⋅0}{4}$ +isinsin $\frac{π+2π⋅0}{4}$) =3(coscos $\frac{π}{4}$ +isinsin $\frac{π}{4}$ =3($\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2}$);
• when k=1,
x2=$\sqrt[4]{81}$(coscos $\frac{π+2π⋅1}{4}$ +isinsin $\frac{π+2π⋅1}{4}$) =3(coscos $\frac{3π}{4}$ +isinsin $\frac{3π}{4}$ =3(-$\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2}$);
• when k=2,
x3=$\sqrt[4]{81}$(coscos $\frac{π+2π⋅2}{4}$ +isinsin $\frac{π+2π⋅1}{4}$) =3(coscos $\frac{5π}{4}$ +isinsin $\frac{5π}{4}$ =3(-$\frac{\sqrt{2}}{2}-i\frac{\sqrt{2}}{2}$);
• when k=3,
x1=$\sqrt[4]{81}$(coscos $\frac{π+2π⋅3}{4}$ +isinsin $\frac{π+2π⋅1}{4}$) =3(coscos $\frac{7π}{4}$ +isinsin $\frac{7π}{4}$ =3($\frac{\sqrt{2}}{2}-i\frac{\sqrt{2}}{2}$);

So, the solutions to the equation x4+81=0 are 3($\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2}$); 3(-$\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2}$); 3(-$\frac{\sqrt{2}}{2}-i\frac{\sqrt{2}}{2}$); and 3($\frac{\sqrt{2}}{2}-i\frac{\sqrt{2}}{2}$).

Summarizing all the above, we can conclude that complex numbers make it possible to obtain all solutions to the polynomial equations. In general, the fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This theorem can be rephrased as follows: every non-zero polynomial with complex coefficients and degree of n has exactly n complex roots.

## Quiz

1. Calculate the following powers of imaginary unit.

a) i23;

b) i2022;

c) i-19.

SOLUTION: We know that i1=i, i2=-1, i3=-i and i4=1.

a) Since 23=4⋅5+3, we have

i23=i4⋅5+3=(i4)5 x i3=15 x (-i)=-i

b) Divide 2022 by 4 with remainder:

2022=505⋅4+2

Then

i2022=i505⋅4+2=(i4)505 x i2=1505 x (-1)=-1

c) Divide 19 by 4 with remainder:

19=4⋅4+3

Then

i-19=i-4⋅4-3=(i4)-4 x i-3=1-4 x $\frac{1}{i^3}$ = $\frac{1}{i^3}$ = $\frac{i}{i^4}$ = $\frac{i}{1}$=i

ANSWER: a) -i; b) -1; c) i.

1. If z1=2-i and z2=3+4i, calculate:

a) z1+z2;

b) z1-z2

c) z1 x z2;

d) $\frac{z_2}{z_1}$.

SOLUTION: a) z1+z2=(2-i)+(3+4i)=(2+3)+(-1+4)i=5+3i.

b) z1 – z2=(2-i)-(3+4i)=2-i-3-4i=(2-3)+(-1-4)i=-1-5i.

c) z1 x z2=(2-i)(3+4i)=6+8i-3i-4i2=6+5i-4(-1)=10+5i.

d) z1 ÷ z2=$\frac{3+4i}{2-i}$=$\frac{3+4i}{2-i}.\frac{2+i}{2+i}$=$\frac{6+3i+8i+4i^2}{4+2i-2i-i^2}$=$\frac{6+11i-4}{4-(-1)}$=$\frac{2+11i}{5}$=$\frac{2}{5}+\frac{11}{5}i$

ANSWER: a) 5+3i; b) -1-5i; c) 10+5i; d) $\frac{2}{5}+\frac{11}{5}i$

1. For given complex number, write down two other forms of this number.

a) z=-4i;

b) z=2(coscos $\frac{\pi}{6}$ +isinsin$\frac{\pi}{6}$) ;

c) z=$\sqrt{2e}^{-\frac{\pi}{4}i}$.

SOLUTION: a) Number z is given in the general form. If z=-4i, then a=0 and b=-4. Find the modulus and argument of this complex number:

|z|=$\sqrt{a^2+b^2}=\sqrt{0^2+(-4)^2}=\sqrt{16}$=4

coscos φ=$\frac{a}{\sqrt{a^2+b^2}}=\frac{0}{\sqrt{0^2+(-4)^2}}$=0

sinsin φ=$\frac{b}{\sqrt{a^2+b^2}}=\frac{-4}{\sqrt{0^2+(-4)^2}}$=-1

If coscosφ =0 and sinsinφ =-1, then φ=-$\frac{\pi}{2}$ and the remaining two forms are

• polar form: z=4(coscos (-$\frac{\pi}{2}$) +isinsin (-$\frac{\pi}{2}$)) ;
• exponential form: z=$4e^{-\frac{\pi}{2}i}$.

b) Number z is given in polar form. If z=2(coscos $\frac{\pi}{6}$ +isinsin $\frac{\pi}{6}$) , then |z|=2 and φ=$\frac{\pi}{6}$. This allows us to write the exponential form of this complex number,

z=$2e^{\frac{\pi}{6}i}$

To find the general form, simply calculate the values of corresponding trigonometric ratios:

coscos $\frac{\pi}{6}$ =$\frac{\sqrt{3}}{2}$,sinsin $\frac{\pi}{6}$ =$\frac{1}{2}$,

then the general form of the complex number z is

z=2($\frac{√3}{2}+i⋅\frac{1}{2}$)=√3+i

c) Number z is given in exponential form. If z= √2$e^{-\frac{\pi}{4}i}$, then |z|=√2 and φ=-$\frac{\pi}{4}$. This allows us to write the polar form of the complex number,

z=2(coscos (-$\frac{\pi}{4}$) +isinsin (-$\frac{\pi}{4}$))

and calculate the general form of this complex number

z=√2($\frac{√2}{2}-i⋅\frac{√2}{2}$)=1-i

ANSWER: a) polar form: 4(coscos (-2) +isinsin (-2)) , exponential form: 4$e^{-\frac{\pi}{2}i}$

b) exponential form: 2$e^{-\frac{\pi}{6}i}$; general form: √3+i;

c) polar form: √2(coscos (-$\frac{\pi}{4}$) +isinsin (-$\frac{\pi}{4}$)) ; general form: 1-i.

1. Calculate (1-√3i)15.

SOLUTION: First, represent complex number z=1-√3i in the polar form.

If z=1-√3i, then a=1, b=-√3 and

|z|=$\sqrt{1^2+(-√3)^2}$=√4=2

coscos φ =$\frac{a}{\sqrt{a^2+b^2}}=\frac{1}{2}$

sinsin φ =$\frac{b}{\sqrt{a^2+b^2}}=\frac{-√3}{2}$

This gives us φ=-$\frac{\pi}{3}$ and z=2(coscos (-$\frac{\pi}{3}$) +isinsin (-$\frac{\pi}{3}$)) .

By de Moivre’s theorem,

(coscos (-$\frac{\pi}{3}$) +isinsin (-$\frac{\pi}{3}$))15=coscos (-$\frac{\pi}{3}$.15) +isinsin (-$\frac{\pi}{3}$.15) =coscos (-5π) +isinsin (-5π)

and

(1-√3i)15=(2(coscos (-$\frac{\pi}{3}$) +isinsin (-$\frac{\pi}{3}$)))15=215 (coscos (-5π) +isinsin (-5π)

Since -5π=-6π+π, then

coscos (-5π) =coscos (-6π+π)=coscos π =-1

sinsin (-5π) =sinsin (-6π+π)=sinsin π =0

and (1-√3i)15=215(-1+i⋅0)=-215

1. Solve the equation.

a) x4-16=0;

b) x3+27=0.

SOLUTION: a) First, factorize the left part of the equation x4-16=0:

(x2-4)(x2+4)=0

(x-2)(x+2)(x2+4)=0

If the product of three factors is zero, then either the first or second or third factor is 0. So,

x-2=0  or  x+2=0  or  x2+4=0

From the first equation x=2, from the second equation x=-2.The third equation has no real solutions. Find complex solutions of the equation x2+4=0. Rewrite it as follows:

x2=-4

Remembering that i2=-1, we have

x2=4i2

and x=2i  or  x=-2i

b) If x3+27=0, then x3=-27 and x=$\sqrt[3]{-27}$.

Represent number -27 as a complex number. First, represent complex number z=-27 in the polar form:

|z|=$\sqrt{(-27)^2+0^2}$=27

coscos φ =$\frac{-27}{27}$=-1,sinsinφ =$\frac{0}{27}$=0

Therefore, φ=π and z=27(coscosπ +isinsinπ ).

Now, find all four roots taking into account that k=0, 1, 2:

• when k=0, x1=$\sqrt[3]{27}coscos \frac{π+2π⋅0}{3} +isinsin \frac{π+2π⋅0}{3} =3(coscos \frac{\pi}{3} +isinsin \frac{\pi}{3}) =3(\frac{1}{2}+i\frac{√3}{2}$);
• when k=1, x2=$\sqrt[3]{27}coscos \frac{π+2π⋅1}{3} +isinsin \frac{π+2π⋅1}{3}$=3(coscos π + isinsin π)=3(-1+i⋅0)=-3;
• when k=2, x3=$\sqrt[3]{27}coscos \frac{π+2π⋅2}{3} +isinsin \frac{π+2π⋅2}{3} =3(coscos \frac{5π}{3} +isinsin \frac{5π}{3}) =3(\frac{1}{2}-i\frac{√3}{2})$.

So, the solutions to the equation x3+27=0 are 3($\frac{1}{2}+i\frac{√3}{2}$), -3 and 3($\frac{1}{2}-i\frac{√3}{2}$).

ANSWER: a) 2, -2, 2i and -2i;

b) 3($\frac{1}{2}+i\frac{√3}{2}$), -3 and 3($\frac{1}{2}-i\frac{√3}{2}$).

## Conclusions

1. Real numbers and pure imaginary numbers are subsets of the set of complex numbers.
2. Each complex number can be represented on the complex plane as an ordered pair.
3. Complex numbers have three main forms: general, polar and exponential.
4. We can complete with complex numbers the same arithmetic operations as with real numbers remembering the main imaginary property i2=-1.
5. Complex numbers have the same properties as real numbers.
6. There is a range of identities with complex numbers.
7. Complex numbers greatly simplify our lives in solving equations. We know that every polynomial equation has as many roots as the degree of that equation. These roots can be real or complex, counted the multiplicity of these roots.
8. There are a number of problems that require complex numbers when solving, but ultimately give a real solution. You will encounter such problems, for example, when studying differential equations.