text:
Cookbook:Syrup

Syrup, in cooking, refers to a fluid with a high sugar content. It can range in consistency from thin to viscous, depending on the exact sugar content. Syrups can have a range of flavors.

List
The following is a list of common syrups:

 * Agave syrup
 * Barley malt syrup
 * Breakfast/pancake syrup
 * Brown rice syrup
 * Cider syrup
 * Chocolate syrup
 * Corn syrup
 * Date syrup
 * Glucose syrup
 * Golden syrup
 * Grenadine
 * Simple syrup
 * Maple syrup
 * Molasses
 * Treacle

title: Cookbook:Syrup


text:
Algebra/Chapter 1

Chapter 1: Elementary Arithmetic

Introduction
Many of the most important aspects of modern day mathematics have their roots in the world of arithmetic, including algebra. This chapter is meant to re-visit, or for some introduce, those roots that are often overlooked, and review of all of the prerequisites that are necessary for understanding algebra.

Sections

 * 1) /The Number Line/  - (Completed 12 October 2023)
 * 2) Operations in Arithmetic  - 10% Done
 * 3) Operations with Decimals and Fractions
 * 4) /Exponents and Roots/
 * 5) /Order of Operations/
 * 6) /Factoring and Divisibility/
 * 7) Applications of Fractions and Decimals
 * 8) /Units/
 * 9) Rounding and Estimation
 * 10) Introduction to Statistics
 * 11) /Problem Solving/

Extra

 * 1) Review
 * 2) Practice Problems
 * 3) Solutions to Practice Problems
 * 4) Glossary

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Algebra/Chapter 1


text:
Algebra/Chapter 2

Chapter 2: An Introduction to Algebra

Sections

 * 1) Mathematical Expressions
 * 2) Mathematical Sentences
 * 3) /Sets/
 * 4) Properties Of Real Numbers
 * 5) /Mathematical Reasoning/
 * 6) /Logic and Proofs/
 * 7) /Concepts of Geometry/

Extra

 * 1) Review
 * 2) Practice Problems
 * 3) Solutions to Practice Problems
 * 4) Glossary

title: Algebra/Chapter 2


text:
Algebra/Chapter 3

Chapter 3: Solving Equations

 * 1) What are Equations?
 * 2) Inverses of Numbers
 * 3) Properties of Equality
 * 4) /Absolute Value Equations/
 * 5) /Word Problems/
 * 6) Ratios, Proportions, and Percentages
 * 7) Direct and Inverse Variation

title: Algebra/Chapter 3


text:
Algebra/Chapter 4

Chapter 4: Inequalities

 * 1) What are Inequalities?
 * 2) Interval notation and Graphing Intervals
 * 3) Solving Inequalities
 * 4) /Compound and Absolute Value Inequalities/

title: Algebra/Chapter 4


text:
Algebra/Chapter 1/The Number Line

1.1: The Number Line

The concept of number is the basis of all of modern mathematics. Numbers have many uses, mostly for counting and measuring things. This section will cover the concept of numbers, as well as the different types of numbers that are seen in mathematics, and their uses.

An Introduction to Numbers
Many of us take the concept of "number" for granted.

Numbers are an abstract concept that often used to count or measure things. Though, depending on where they're used, they may be used for many different purposes. Numbers are merely an idea in our head, things that we talk about and write about, but never actually see beyond that. To do this, we represent these numbers using symbols called numerals, such as "5" or "five". We could also represent the number by holding up 5 fingers, tap a table 5 times, or jump and down 5 times. These are only a few ways to represent the number 5.

Generally speaking, a number is an idea, and a numeral is a way that we can express that idea with. Numbers represent what is known as a quantity. This is how much of something there is in a group of objects, such as “six” lollipops, “seven” slices of pizza, “twenty-five” dollars, or “one hundred” millenniums. To represent these quantities, we can say 6 lollipops, 7 slices of pizza, 25 dollars, or 100 millenniums.

Number Systems
Numbers can be denoted in many ways. The collection of symbols and notation that we may use to represent these numbers is referred to as a number system. Number systems make use of digits or other symbols in a consistent manner. For example, in the earliest of civilizations, people had been believed to rely on the counting of their fingers, or a group of sticks and pebbles. It is not known for certain what the earliest number system may have been, but quite early, groups of symbols, |, ||, |||, ||||, etc. were used to represent numbers. These are still used today as tally marks, |, ||, |||, ||||, and |||| as 1, 2, 3, 4, and 5 respectively.

One of the oldest, and most useful, applications of number is time. We use time in order to keep track of the order of a sequence of events, to compare the duration of events or the intervals between them, as well as to quantify the rates at which things change. There have been numerous ways in order to keep track of time, from the prehistoric tribes who kept track of time through the rise and setting of the sun, to the markings on bones and trees.

During the late Stone Age, hunting and fishing societies had developed in the Nile Valley and in many parts of Africa. An interesting find from these ancient civilizations is a carved bone discovered in Lake Edward in the Democratic Republic of Congo. This bone, referred to as the Ishango Bone, had several notches in it, indicating a pattern. The exact purposes of this bone are unknown, but from a mathematical perspective, this may be viewed as a sort of tally system.

The bone has several separate markings. One set of markings has four groups of 11, 13, 17, and 19; another 11, 21, 19, 9; and the third has seven groups of 3, 6, 4, 8, 10, 5, 5.

The Hindu-Arabic Numeration System
The number system that we use today is referred to as the Hindu Arabic numeration system, which, as implied by its name, were created by the Hindus before the third century, this number system was introduced into the Western World 1000 years later by Italian mathematician Lenoardo Fibonacci. This number system would then be popularized by the Arabs, hence its name.

All numbers in this system consist of digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Numbers are written using these digits, placing them in certain positions. The Hindu mathematician who devised the system had stated that "from place to place each is ten times the preceding." For instance, the number three is written with the symbol "3", and the number two million is written using the numeral "2", followed by six zeroes.

Place Values
As you may know, we arrange the digits in a number by their place value. When numbers are composed of more than three digits, we sometimes use commas to separate the digits into three. These groups of three digits are referred to as periods. In the Hindu-Arabic number system, the positional value in the first period are the ones, tens, and hundreds.

Every other period in a number also consist of a one, ten, and hundred. When we read the numbers from right to left, notice that the value of each position is ten times the preceding one.

Order of Numbers
The order of numbers is the method we use in order to arrange numbers from "smallest" to "largest".

The Number Line
The number line is a line that shows all numbers, with the left side decreasing, and the right side increasing. The typical number line shows all integers (whole numbers), but it can consist of the numbers between 2 numbers, like half or 3.7.

As one may gather from the above, numbers come in many varieties. For now, the numbers we shall be discussing about are whole numbers, decimals, fractions, negative numbers, and zero.

The magnitude of a number is its distance from zero on the number line.

For the rest of the section, we will be covering all of the features of this representation of numbers one at a time.

Comparing Numbers
The dictionary definition of "comparing" two things is viewing them in relation to one another. For this course, it is good to be able to compare numbers, or be able to know when one quantity is equal to, less than, or equal to another. In mathematics, the symbols that we use for comparing numbers are ">", which means "greater than", "<", which means "less than", and "=", which means "equal to".

In the diagram below, we see that 7 is greater than 3 by 4. In simple terms, this can be expressed as "7-3 = 4".

In case you have forgetten which number goes where, just remember the following:

BIG > small small < BIG

The small end will always point to the smaller number, and the larger end will always point to the larger number.

The number line can also be utilized as a tool in order to compare numbers, and it is very easy. All that one needs to know in order to do this are two simple facts:

 * For any two numbers on the number line, the number farther right is greater.
 * For any two numbers on the number line, the number farther left is smaller.

Whole Numbers
The most familar number system that is used is known as the counting numbers, more technically refered to as the natural numbers. These are the numbers 1, 2, 3, 4, and so on. These numbers are either the numbers you and I used for learning how to count, as indicated in the name, (as in "there are six pencils on the table"), as well as for ordering things (as in "this is the fourth smallest ball in the basket"). For large numbers, commas are used to separate digits into groups of 3.

In this example, the digit 3 is in the thousands place, the digit 5 is in the hundreds place, 2 is in the tens place, and 2 is in the ones place.

$$3,522$$

Fractions and Decimals
We understand how to count whole numbers. But what happens when we are not working with whole parts? Most of the time, you will come across instance "uncountable" objects, or things that cannot be represented as a group of objects.

Whereas whole numbers are useful for counting things in a group, in most instances, we are interested in measuring things that do not have quantities as precise as "one" or "three-hundred" or even "one million". We can "count" things like holes in a piece of cloth, the number of slabs of wood used to make a fence, and the number of stones in a lake. We cannot however "count" things like the amount of water in a bucket or the weight of an object. Instead, special tools are used to determine these quantities, such as rulers for length, scales for weight, and stopwatches for time.

Decimals can be thought of as the numbers that are found between the tick marks found on the number line. These are often identified with a decimal separator symbol, typically ".", which is placed between the digits (Though some countries use a comma "," instead). The digits on the right of this symbol represents the part that is between the tick marks. For example, the number 12.5 is between the numbers 12 and 13. It is more than 12 but less than 13. The number 12.3 is also more than 12, but it is also less than 12.5.

Fractions are also numbers that can be found between the tick marks on the number line. They also represent the idea as decimals, values that include part of a whole, but they are a different way to express that idea.

Contrary to how you might have been taught, you do not pronounce a "," in a number as "and".

It should also be noted that in some countries, the "," and the "." switch places. So "1,523.99" in America might be written as "1.523,99" elsewhere.

Negative Numbers and Zero
Throughout the history of mathematics, there were certain ideas that even the greatest scholars had trouble accepting that we now take for granted today, some of which today's youth can easily grasp. Two prime examples of this were the concept of "negative" numbers, and the number zero.

Mathematic's origins were routed in counting things, like the number of sheep one owned. If you had two sheep, you'd count them as "one, two". Logically, that meant that one could not ask you to "lend them three sheep", since you'd have no more sheep to give up after the two you already had.

Negative Numbers are the numbers found on the left of the number line. In other words, these are the numbers that are less than zero. Imagining a number less than zero, can be difficult. If you have a plate of cookies, and everyone takes all of them until there are none less, it is hard to imagine being able to take any more cookies.

Zero is the value which denotes that there is no amount, for example, if a person has zero hats, that means that they have no hats. Zero is found between the positive and negative numbers, located halfway between -1 and 1. The number zero is neither positive of negative, but it is even.

Long ago, the concept of zero was non-existent. However, the concept was first brought about by the Babylonians, Ancient Indians, and several Central American tribes at different times, though the number primarily served as a placeholder rather than an actual number. Zero as a number is thought to have been introduced in India as far back as 628 AD in the Brāhmasphuṭasiddhānta, which was theorized by the mathematician Brahmagupta.

However, some countries still did not impliment the number zero, which meant that mathematics was significantly more difficult to do.

title: Algebra/Chapter 1/The Number Line


text:
Algebra/Chapter 1/Arithmetic

1.2: Operations in Arithmetic

Arithmetic is the process of performing certain operations on quantities. In this section, we will cover four of the arithmetic operations: addition, subtraction, multiplication, and division.

Groups of Numbers
For the purposes of understanding operations involving numbers, we will first discuss what we mean by an "operation" as well as the groups of numbers that they occur in.

In an operation, we take one or more numbers of interest, and perform a procedure on them to receive a new number, or the result of the operation. A group of two numbers may be combined together with a given operation to produce a third number.

The Four Basic Operations
The four operations of mathematics take two different values, and convert them to a new value. These include:

Division: ÷
These four operations are commonly referred to as arithmetic operations.

Defining Addition
$$\ 1 + 1 = 2$$

To define the number one is a rather difficult task, but we all have a good intuitive sense of what "oneness" is. Oneness is the property of having or thinking of a singular quantity. For example, think of when you have one dollar, one bushel of potatoes, or one light year. From here we can recursively (that is, relating to the last one) define the natural numbers by assigning a new name for each new number of ones that we have:

For example, $$\ 5 + 3 = (1 + 1 + 1 + 1 + 1) + (1 + 1 +1) = 8$$

In the above, notice that we represent the numbers 5 and 3 as a repeated addition of 1 in parentheses, which we subsequently added together.

Performing Addition
Addition is the mathematical operation which explains the total amount of objects which we put together in a collection. In addition, the numbers being added together are called addends or terms, and the end result is referred to as the sum.

$$\ 25 + 12 = 37 $$

Addition has several important properties. One of these properties is that the order you add the numbers will not effect the final result. Refer to the diagram of the apples above. Much like how we can add 3 apples to a group of 2 will result in 5 apples in total, we can add 2 apples to a group of 3 to get 5 apples in total.

 Addition Problems

Subtraction
Subtraction is the term we use to describe how we "take away" one or more numbers from another. The term is used in two types of situations. The first is to answer the question of "how much is less?", and the second is "how much is needed?"

It can likewise be defined as counting initial quantity of ones and removing some amount of them. For example:

$$\ 5 - 3 = (1+1+1+1+1) - (1+1+1) = 2 $$ means 5 ones remove 3 ones, leaving a result of 2 ones.

 Subtraction Problems

Multiplication
Multiplication is a shorthand (that is, a faster way of writing something) for repeated addition. For example:

$$\ 5\cdot3 = 15 $$ What this means is to add up '3' 5 times; or add up '5' 3 times. $$\ 5\cdot3 = 3 + 3 + 3 + 3 + 3 = 15 $$

 Multiplication Problems  Note that in some regions and cases, it is better to use the cross symbol ($$ \times $$) or the letter "x" instead of the dot. Most regions use the dot instead of the cross symbol because the cross symbol looks like the letter "x".

Division
Division is the opposite of multiplication.

$$\ \frac{6}{3} = 2 $$

This example asks if six is 1+1+1+1+1+1, and three is 1+1+1, then how many sets of three can we break six into? The answer is of course 2, because

$$\ 6 = 1+1+1+1+1+1 = (1+1+1)+(1+1+1) = 3 + 3 $$; two sets of three.

 Division Problems

Division is the first operation where a problem arises. In all the previously defined operations (addition, subtraction, and multiplication) we could perform the operation on any pair of numbers we chose. However, with division we cannot divide by zero. Much will be said about this fact throughout the course of this book, and even through your studies in all of mathematics.

Converting between Fractions and Decimals
While fractions and decimals both essentially represent the same thing, one may be better suited for a problem than the other.

Quiz
Perform the following operations specified below.  {Example: 3+2=5}

{ 3+7={ 10_3 }
 * type="{}"}

{ 100-1={ 99_3 }
 * type="{}"}

{ 33+10={ 43_3 }
 * type="{}"}

{ 19-3={ 16_3 }
 * type="{}"}

{ 2+67={ 69_3 }
 * type="{}"}

{ 2*7={ 14_3 }
 * type="{}"}

{ 10*5={ 50_3 }
 * type="{}"}

{ 30+50={ 80_3 }
 * type="{}"}

{ 41-10={ 31_3 }
 * type="{}"}

{ 15/5={ 3_3 }
 * type="{}"}

{ 31+46={ 77_3 }
 * type="{}"}

{ 1/2 of 80 = { 40_3 }
 * type="{}"}

{ 67-40={ 27_3 }
 * type="{}"}

{ 8+{ 5_3 }+4=17
 * type="{}"}

{ 100/10={ 10_3 }
 * type="{}"}

{ 58+32={ 90_3 }
 * type="{}"}

{ 45+16={ 61_3 }
 * type="{}"}

{ 1/4 of 16 = { 4_3 }
 * type="{}"}

{ 60+{ 29_3 }=89
 * type="{}"}

{ { 12_3 }+25=37
 * type="{}"}

{ 54-15={ 39_3 }
 * type="{}"}

{ 60/5={ 12_3 }
 * type="{}"}

{ 92-85={ 7_3 }
 * type="{}"}

{ { 100_3 }-50=50
 * type="{}"}

{ 66-38={ 28_3 }
 * type="{}"}

title: Algebra/Chapter 1/Arithmetic


text:
Algebra/Chapter 1/Exponents and Roots

1.4: Exponents and Roots

Exponentiation
What are exponents?

Exponents are a shorthand used for repeated multiplication of the same number. Remember that when you were first introduced to multiplication it was as a shorthand for repeated addition. For example, you learned that: 4 × 5 = 5 + 5 + 5 + 5. The expression "4 × " told us how many times we needed to add. Exponents are the same type of shorthand for multiplication. Exponents are written in superscript (that is, a smaller number written above) after a regular-sized number.

For example: 23 = 2 × 2 × 2 = 8. The number in larger font is called the base. The number in superscript is called the exponent. The exponent tells us how many times the base is multiplied by itself. In this example, 2 is called the base and 3 is called the exponent.

The expression 23 is read aloud as "2 raised to the third power", or simply "2 cubed".

Here are some other examples:

6 × 6 = 62 (This would read aloud as "six times six is six raised to the second power" or more simply "six times six is six squared".) 7 × 7 × 7 × 7 = 74 (This would read aloud as "seven times seven times seven times seven equals seven raised to the fourth power". There are no alternate expression for  raised to the fourth power.  It is only the second and third powers that usually get abbreviated because they come up more often.  When it is clear what is being talked about, people often drop the words "raised" and "power" and might simply say "seven to the fourth".)

There are two other powers that play an important role in understanding powers. One is

 * $$2^0$$

and the other is

 * $$2^1$$

Any number raised to the power of zero is equal to 1. For example: $$6^0 = 1$$

Any number raised to the power of one is itself. For example: $$15^1 = 15$$

In general, an exponent of a number to power of n
 * a × a × a ... = $$a^n $$

The base is a and it is multiplied by itself n times.

 Exponentiation Problems

Roots
What are Roots?

Roots are the opposite for exponents. It's easy, although perhaps long, to compute exponents given a root. For instance 7*7*7*7 = 49*49 = 2401. So, we know the fourth root of 2401 is 7, and the square root of 2401 is 49. What is the third root of 2401? This article gives a formula for determining the answer, while this article gives a detailed explanation of roots.

Finding the value for a particular root is difficult. This is because exponentiation is a different kind of function than addition, subtraction, multiplication, and division. When we graph functions we will see that polynomials that use exponentiation use curves instead of lines. Using algebra we will see that not all of these polynomials are functions, that knowing when a polynomial is a relationship or a function can allow us to make certain types of assumptions, and we can use these assumptions to build mental models for topics that would otherwise be impossible to understand.

For now we will deal with roots by turning them back into exponents.

The positive nth root of $$x$$ is represented as $$\sqrt[n]{x} = r$$. We get rid of the root by raising our answer to the nth power, $$r^n = x.\!\,$$  Problems Using Roots

title: Algebra/Chapter 1/Exponents and Roots


text:
Algebra/Chapter 1/Order of Operations

1.5: Order of Operations

The Order of Operations is used when doing expressions with more than one operation (e.g., ×, +, -). These are rules so you only get one answer all the time.

Example: When faced with $$4+2 \times 3$$, how do you proceed?

There are two ways:

$$4 + 2 \times 3 = (4 + 2) \times 3$$

$$4 + 2 \times 3 = 6 \times 3$$

$$4 + 2 \times 3= 18$$ or

$$4 + 2 \times 3 = 4 + (2 \times 3)$$

$$4 + 2 \times 3 = 4 + 6$$

$$4 + 2 \times 3 = 10$$

This is confusing, so which is correct? (Parentheses, "(" and ")" are used to show what to do first)

In order to communicate using mathematical expressions we must agree on an order of operations so that each expression has only one value.

For the above example all mathematicians agree the correct answer is 10.

You're probably wondering what this order is.

The Standard Order of Operations
Evaluate expressions in this order.

 * Parentheses or Brackets (evaluate what's inside them)
 * Exponents
 * Multiplication and/or division from left to right
 * Addition and/or subtraction from left to right

An Easy Way of Remembering
Use this memory tool to help remember the order! Please Excuse My Dear Annoying Sister It is also commonly called by its acronym, PEMDAS.

An alternative form of this is; Brackets Indices Division or Multiplication Addition or Subtraction (BIDMAS).

Yet another way of remembering this is Brackets Orders Division Multiplication Addition Subtraction (BODMAS)

or Bring Our Dear Mother Along Saturday

Practice Problems
Note: the expressions in the following quiz, use an asterisk (*) to indicate multiplication ($$\times$$) between adjacent factors. This use of the asterisk is nearly ubiquitous with the various computer languages, as the times symbol is not an historically available keyboard character. Hand written expressions commonly use a small vertically centered dot (·) to indicate multiplication. Where unambiguous, multiplication is implied between factors and a symbol is extraneous.

Quiz
Evaluate the following expressions

{Evaluate the numerical expression $$2+4*3=$${ 14_2 }
 * type="{}"}

{Evaluate the numerical expression $$2*4+3=$${ 11_2 }
 * type="{}"}

{Evaluate the numerical expression $$(2+4)*3=$${ 18_2 }
 * type="{}"}

{Evaluate the numerical expression $$9^2 + 1 -7*(8+4)/2=$${ 40_2 }
 * type="{}"}

title: Algebra/Chapter 1/Order of Operations


text:
Algebra/Chapter 1/Statistics

1.10: Introduction to Statistics

Measures of Center and Spread
The following three numbers represent 3 different ways to think about the average value of your set.

Mean - This is what we usually think of as the "average" of a data set. The mean can be found by summing all the values in the data set and dividing by the size of the data set (that is number of elements in the set). In mathematical notation,

For example: Suppose 1, 2, 4, 6, 8, 9 is our data set then the sum is 1 + 2 + 4 + 6 + 8 + 9 = 30 and there are 6 elements in the data set, so the mean is 30/6 = 5.

The mean, while a very useful statistic, has its flaws. Notably, its value may be heavily influenced by outliers - numbers in a data set which are significantly higher or lower than the majority of the data. It is often preferable to use the median instead to describe such data sets.

Median - This is the middle of our data set. To find the median you must first put your data values in numerical order (say, from smallest to largest). If you have an odd number of elements in your data set there will be exactly one number in the middle, this number is the median. If you have an even number of elements in your data set then the median is the average of the middle two numbers. et For example. If our data set was 2, 2, 3, 4, 4, 5, 6, 7, 8, 9, 12, 13, 16, 22 is our data s data set. Since it has an even number of elements, we have to take the mean of the middle two, in this case 6 and 7, so the median is 6.5.

Mode - Mode refers to how many times a number or numbers occur in a data set. Since mean, median, and mode often are confused with each other, an easy way to remember mode is 'most often'. The first two letters in mode are 'm' and 'o', imagine this stands for 'most often' to help you remember. In the case that two or more different values are tied for the most number of repeats then that data set is said to have multiple modes. If your asked to find the mode of a data set with multiple modes, then all of the modes should be listed. If no element of the data repeats, then there is no mode.

For example. Suppose 1, 2, 2, 2, 3, 3, 4, 5, 5, 5, 7 is our data set, then the mode would be both 2 and 5. They both occur three times and three is the maximum number of repeats in our data set.

The following quantity tell us how spread out our data set is.

Range - The difference between the largest and smallest numbers in our data set. Notice this means the range is never negative.

Examples
Mean

Let's look at the following data set:

Data Values: 10, 13, 4, 7, 9              so n = 5

Now add the values together:

10 + 13 + 4 + 7 + 9 = 43

43 / 5 = 8.6

Mean = 8.6

Median

Case 1:

Data Values: 10, 13, 4, 7, 8              so n = 5

Numerical Order: 4, 7, 8, 13, 10

Since 8 is the middle number,

Median = 8

Case 2:

Data Values: 10, 13, 4, 7, 8, 10          so n = 6

Numerical Order: 4, 7, 8, 10, 10, 13

Middle Numbers: 8 and 10

Find Mean: 8 + 10 = 18

18 / 2 = 9 Median = 9

Mode

Data Values: 10, 13, 4, 7, 8, 10

10 is in the data set twice.

Mode = 10

Data Values: 4, 9, 13, 18, 4, 2, 9, 4, 13, 8, 9

4 and 9 both have three data values.

Mode = 4, 9

Range

Data Values: 10, 13, 4, 7, 8

Numerical Order: 4, 7, 8, 10, 13

Difference of last and first: 13 - 4 = 9

Range = 9

Quiz
 {Find the mean, median, mode, and range of the following data sets:}

{5, 8, 12, 4, 8, 9, 11, 2}

{ Mean = { 7.375_6 }
 * type="{}"}

{ Median = { 8_6 }
 * type="{}"}

{ Mode = { 8_6 }
 * type="{}"}

{ Range = { 10_6 }
 * type="{}"}

{24, 26, 37, 24, 16, 44, 26, 34, 24}

{ Mean = { 28 1/3_6 }
 * type="{}"}

{ Median = { 26_6 }
 * type="{}"}

{ Mode = { 24_6 }
 * type="{}"}

{ Range = { 28_6 }
 * type="{}"}

{15, 48, 89, 74, 25, 36, 57, 51, 17, 22}

{ Mean = { 43.4_6 }
 * type="{}"}

{ Median = { 42_6 }
 * type="{}"}

{How many modes? { 0|10_2 }
 * type="{}"}

{ Range = { 74_6 }
 * type="{}"}

{2, 6, 8, 7, 8, 2, 2, 9, 10}

{ Mean = { 6_6 }
 * type="{}"}

{ Median = { 7_6 }
 * type="{}"}

{ Mode = { 2_6 }
 * type="{}"}

{ Range = { 8_6 }
 * type="{}"}

title: Algebra/Chapter 1/Statistics


text:
Algebra/Chapter 2/Variables

2.1: Mathematical Expressions

In this section, we review the definitions and applications of variables and expressions. In case this is not review, we will attempt to go over the ideas carefully enough so that this section may serve as a first introduction.

Algebra
Throughout the rest of the book, we will be talking about unknown values.

Numerical and Algebraic Expressions
An expression is a collection of symbols you could put into a calculator and get a number. To be precise, an expression is a well-formed formula, but don't worry about the formal details. Expressions are things we can evaluate, and I like to call them "numbers in fancy clothing."

$$ \begin{align} \text{4} + 2 \\ \end{align} $$

This is an expression, since we can simply add $$ 4 + 2 $$ and get $$ 6$$.

$$ \begin{align} \text{(}4+10)/8 \\ \end{align} $$

This is also an expression, since it can be evaluated to $$14/8 = 7/4$$.

$$ \begin{align} \text{(}*3 + 2 \\ \end{align} $$

This is not an expression. Note that $$3$$ is not being multiplied by anything, and there is no right parenthesis to match left one.

$$ \begin{align} \sqrt{-4} \end{align} $$

While $$\sqrt{-4}$$ cannot be evaluated, it is still an expression since it follows the rules of the order of operations.

All of the expressions in Example 2.1 above, along with the ones that we have worked with in Chapter 1 are referred to as numerical expressions. These are expressions that involve a set of numbers being placed together utilizing addition, subtraction, multiplication, and division. In these scenarios, we knew about the number values to work with. However, when working with mathematics or science problems it is frequently necessary to talk about numbers whose value you do not know immediately.

For example, suppose you are asked how many 1 × 1 squares fit inside a rectangle that measures 9 squares by 10 squares? In this case the number we do not know is "the number of squares." Some people answer correctly right away by saying "The number of squares that fit inside the rectangle is 90." But how did they find this? They could say "The number of squares in the rectangle is equal to the number of squares of its length times the number of squares of its width. The rectangle is 9 squares long and 10 squares wide, so it contains 90 squares." This is a lot of words. We could express ourselves mathematically by writing:

$$ \begin{align} \text{Area} & = \text{Length} * \text{Width} \\ \text{Length} & = 9 \text{ and } \text{Width} = 10 \\ \text{Area} & = 9 * 10 \\ \text{Area} & = 90 \end{align} $$

For mathematicians, this is still a lot of writing. Mathematicians usually choose to name each unknown with just a single letter, mostly out of a need for conciseness (that is, shortness), since writing one letter is easier than writing a word. A variable is the letter (or symbol) we are using to represent an unknown number. We could rewrite the calculations above as:

$$ \begin{align} \text{A} &= L * W \\ \text{L} & = 9 \mbox{ and } W = 10 \\ \text{A} & = 9 * 10 \\ \text{A} & = 90 \end{align} $$

Now that you're familiar with what expressions look like, let's consider a practical example.

Let's start with figuring out how much I spend on coffee in a week.
 * Let's call the total number of cups of coffee I drink in a week T. We can give a name to how much money I spend, lets call it C (for cost).  C is an expression, but we need a more useful one involving T. Every cup of coffee costs $2.00, so:
 * 0 cups of coffee costs $0, as 0×2 is 0.
 * 1 cup of coffee costs $2, as 1×2 is 2.
 * 2 cups of coffee costs $4, as 2×2 is 4.
 * T cups of coffee costs ... $2T.

So, 2T is an expression representing the cost of T cups of coffee.

This is the first step in figuring out how much I spend on coffee in a week. To get further we need to know the value of T: how many cups of coffee I drink in a week. If I drink 4 cups a day, and there are 7 days in a week, this must mean that
 * T = 4 × 7 = 28.

By replacing T with the number we calculated we can see that
 * C is the same as 2.00 × 28 = 56.00.

So I spend $56.00 a week on coffee! Maybe I should cut back!

What about how much I spend in a year? Well, we figure this out in the same way. Let's first think about a year, if we keep our names the same we still have that
 * C is the same as 2.00 × T.

But now there are 365 days in a year, so the total number of cups of coffee (that is the number T) changes. We can calculate it just as we did before, drinking 4 cups a day leads to:
 * T = 365 × 4 = 1,460.

Now that we know the value of T we can see that:
 * C is the same as 2.00 × 1,460 = 2,920.00.

Now I can see that I spend $2,920.00 a year on coffee. Yes, I need to cut back!!

Notice that solution of these two problems is basically the same, the only things that changes is the values of C and T. This is why they are called variables, because the exact numbers varied. Some of the numbers we used in this calculation do not change. For example, the number of days in a week is always 7. We may still use a letter to represent the number of days in a week if we want, but since the number is not going to change, we simply leave it as 7. Letters or symbols that represent specific unchanging numbers are called constants.

In practical situations what you think of as a constant sometimes depends on how you think about the problem. A careful reader may point out that there are not always 365 days in a year: on leap years there are 366. Since I was working out how much I spent for the year 2010, the number of days in the year is constant and equal to 365. If I wanted to make a table of how much I spent on coffee in the last 10 years, then it might be better to use a variable to represent the number of days in a year.

Another example, which might be familiar from physics, is the amount of acceleration of a falling object due to gravity. For most problems this acceleration is treated as a constant g = 9.8 m/s2. However, for problems involving objects that aren't on Earth this may be a bad approximation. The value we used for g in equations concerning objects on other planets may not be considered constants. The problems here are not going to ask us to worry about when it is OK to treat the acceleration due to gravity as a constant, as that is much more appropriate for a physics course. For us, constants are going to be those numbers fixed in the problem and that do not change like the price of coffee in the example above or in some cases some very well known fixed numbers may come up (such as the number of cards in a deck, the number of days in a week, etc.).

Variables are typically written using letters, such as x, t, or C. For cultural reasons x is an extremely common choice for the name of a variable. But when naming a variable or constant yourself it is best to choose something connected to the problem such as C for cost, T for total, etc. This makes it much easier to make sense of the equations you end up looking at. Constants are typically written as the numbers themselves, such as 2, -5, and 0.75, or in some cases may be represented by letters, such as g (from above) and π.

As you can imagine, equations that come up are often more complicated than the equation in the preceding example. We are going to need some vocabulary for the different parts of the equations we encounter. For example, if I let G be the amount of gas I spend driving to the café each day, then the expression for how much I spend C might look like:
 * 2.00 × T + G × D

Now C is the sum of two things called terms. In this case there are two terms in the expression, namely 2.00 × T and G × D. A term of a sum is just one of the pieces we are adding together. In the expression 2.00 × T + G × D − 7 there are three terms. Two of the terms are 2.00 × T and G × D. There are two possibilities for handling − 7. The first possibility is to consider subtraction the same as addition and consider 7 to be the third term. The second possibility is that we think of subtracting 7 as adding −7. In this case we might say that the terms are 2.00 × T, G × D, and −7. To be honest, it doesn't really matter much how we choose to think of things, but we should try to be consistent. Since we used the word "sum" in the definition of term above, we shall try to consistently use the second possibility to describe terms that are subtracted instead of added.

Coefficients
There are many ways to indicate that two numbers should be multiplied. You are probably most familiar with using the symbol ×, in equations such as 2 × 2 = 4. But because mathematics developed in many places, there are other symbols that sometimes get used and this becomes particularly important in algebra. Why have more than one notation? Believe it or not, for convenience! For cultural reasons the most common letter to choose for a variable is x, but now things can look confusing when we try to write out expressions like x × 2. It is just an unfortunate fact of life that our variable and our multiplication symbol look a lot alike. Add poor penmanship to this and you're asking for trouble! There are two common ways to deal with this. The first is to introduce another symbol for multiplication, namely a dot written in the middle of the line. For example instead of writing 2 × 2 = 4 one can write 2 · 2 = 4. Another even more common strategy is to do away with writing anything at all! Suppose I want to multiply x by 2. Since it can lead to confusion I do not want to write 2 × x. I could write, with  our new dot notation 2 · x, or I could just be concise and decide you'll know what it means if I write 2x. That's right, I skipped writing any symbol for multiplication at all! This is known as implied multiplication, because I never really said I was multiplying, I just implied it. This is by far the most common way to express multiplying in algebra.

At first this may seem a crazy way to do things, but it works particularly well with our intuition about units. We are taught from elementary school that if I have 1 apple and someone gives me 1 apple then I have 2 apples. In the same way it seems very natural to write 1x + 1x = 2x. If I have 1 x and someone gives me 1 x then I will just have 2 x ' s. Because variables have such a strong similarity to our units in simple examples like this, it has been culture to never place the variable before the number in implied multiplication. While it may be in some technical sense correct, to write x2 for the product of 2 and x, people may not understand what it means. So always place explicit numbers before the variables when using implied multiplication.

Implied multiplication is also frequently used between two different variables, or even two whole expressions (provided you use parentheses). So we may encounter expressions like xy for the product of x and y, or x(a + b) for the product of x and a + b. Because implied multiplication is so common it gives us even more reason to use single letters for variables. While it might have been nice to use a variable YC for the yearly cost for my coffee, if I were not very careful to explain what I meant some readers may think this represents some variable Y multiplied by another variable C. On the other hand in complicated situations with lots of variables sometimes it is worth risking confusion to choose variable names that make sense.

Finally, you can use implied multiplication between two numbers. Just try it, if we wanted to write 2 × 2 = 4 using implied multiplication we would end up writing 22 = 4, but twenty-two is not four! Instead, we write one or both constants in parentheses: 2(2) = 4 or (2)(2) = 4. Both forms are correct. Follow what your teacher says when in doubt.

Evaluating Expressions
We treat variables just like numbers — numbers that maybe we do not know at the beginning, but numbers nonetheless. When we know what the numbers associated with a variable are we can figure out what some expression we have written before equals. Suppose we are asked to find the value of x − 5 when x = 7. To do this we substitute 7 for x. This means we rewrite the expression, except everywhere we would write x we write 7. So we get 7 − 5, and now we can use simple arithmetic to figure out that the expression equals 2. If you look back to discussion analyzing my coffee drinking habits, you'll see see we substituted a few times already. Let's look at some more examples.

We will do this in two steps. First we will substitute 2 for x to get:
 * 2 · y − 9.

Now we will substitute 3 for y to get:
 * 2 · 3 − 9 = 6 − 9 = -3.

In the last line, since we had gotten to a problem of arithmetic we simply used the rules of arithmetic we are already familiar with. One time to be a little bit careful is when implied multiplication is involved. Consider the following example. We needed to use our precedence rules to work the simple arithmetic in the right order in order to figure out that the expression equals 2.

Solution. Substituting 4 for x we get:
 * 2 · 4 + 2 = 8 + 2 = 10.

Notice a very subtle change when I rewrote the expression. Specifically I inserted a multiplication symbol where there was an implied multiplication. Imagine that I hadn't: the last line would have started 24 + 2, which is not what we meant! We wanted to multiply x by 2, since x = 4, that means we wanted multiply 4 by 2. The number 24 shouldn't be part of our calculation.

It is very important when evaluating expressions to follow the correct order of operations (Don't forget "Please excuse my dear aunt Sally"). For example

Solution. Writing first by substituting for x, we have
 * 3 · 22 · (2z+k)

Now substiting for z
 * 3 · 22 · (2 · 1/2 + k)

Finally we substitute for k to get:
 * 3 · 22 · (2 · 1/2 + 1).

First we need to figure out the value inside the parentheses, that is we need to calculate 2 · 1/2 + 1. The correct order to do this in is to first multiply, then add. That is 2 · 1/2 + 1 = 1 + 1 = 2.

Now that we have figured out what is inside the parentheses, the problem is now to calculate:
 * 3 · 22 · (2).

We do the exponentiation first to get
 * 3 · 4 · (2).

Now we simply multiply, and our answer is 24.

Formulas
<!---==Parentheses== Let's look at when it is OK to add or remove parentheses. The purpose of parentheses is to establish precedence. Precedence tells you which operation goes first. The operation rules for precedence say to evaluate the parentheses first (PEMDAS!). But, what can you do with $$\frac{x + 1}{2} = \frac{1}{2}$$ ? $$x + 1$$ doesn't have just one value; it has as many values as we choose to assign to $$x$$. This is where the distributive property shows its power. It allows us to rearrange the operations while maintaining precedence.

$$ \frac{x + 1}{2} = \frac{1}{2} * (x+ 1) = \frac{1}{2}x + \frac{1}{2} $$ So we can change our equation to $$ \frac{1}{2}x + \frac{1}{2} = \frac{1}{2} $$ Add $$\frac{-1}{2}$$ to both sides $$ \frac{1}{2}x + \frac{1}{2} - \frac{1}{2} = \frac{1}{2}- \frac{1}{2} $$ $$ \frac{1}{2}x = 0 $$ And multiply both sides by the inverse of $$ \frac{1}{2}$$ $$ \frac{1}{2}x * 2 = 0 * 2 $$ $$ x = 0  $$ Parentheses allow us to ensure that we treat expressions that have variables as if they were a value. For instance, if we want to know for which values the expression $$ \frac{2}{x + 1} = 3 $$ is true we need to use the properties of real numbers to place the variable $$x$$ by itself on one side of the equals sign. To do this we need to get $$x+1$$ out of the denominator of the fraction. We can do this by multiplying both sides of the equation by $$(x + 1)$$. We don't know what the value of $$(x + 1)$$ is, but it will always be the same thing on both sides of the equation so it doesn't change the notion of equality.

$$ (x + 1) * \frac{2}{x+ 1} = 3 * (x+ 1) $$

We use the inverse property to re-write $$\frac{2}{x+1}$$ as multiplication. $$ (x + 1) * 2 * \frac{1}{x+ 1} = 3 * (x+ 1) $$

And the associative property to re-write the multiplication. $$ 2* (x + 1) * \frac{1}{x+ 1} = 3 * (x + 1) $$

And the identity property to re-write $$(x + 1) * \frac{1}{x+ 1}$$ $$ 2 * (1) = 3 * (x+ 1) $$

Since 2 * (1) has no variables we can evaluate it. We use the distributive property to re-write 3 * (x+ 1) $$ 2 = 3*x+ 3*1 $$ $$ 2 = 3*x+ 3 $$

We subtract 3 from both sides of the equation.

$$ 2 - 3 = 3*x+ 3 -3 $$ $$ -1 = 3*x $$

And multiply both sides by $$\frac{1}{3}$$. $$ -1*\frac{1}{3}= 3*x*\frac{1}{3} $$ $$ -\frac{1}{3}= x $$

Using parentheses and the properties of real numbers and equality we were able to get x alone to determine the only number for which our initial statement is true. id:Subjek:Matematika/Materi:Aljabar

title: Algebra/Chapter 2/Variables


text:
Algebra/Chapter 2/Different Types of Relationships

2.2: Mathematical Sentences

In this section, we introduce mathematical sentences and the ways that expressions can be related.

The Language of Mathematics
People usually have trouble understanding mathematical ideas. This is not necessarily because math is difficult, but it is usually because mathematics is often presented as a foreign language.

In much mathematical literature, you will come across statements such as: "Let X, Y, and Z be integers". This kind of language is customary, and will very often appear in mathematical literature. When we are told to "let" things be a certain way, you are essentially being asked to imagine or assume that those things are that way.

Previously, we've been talking about expressions. These were quantities of interest that were being represented by symbols and operations. Algebraic expressions are also quantities, just with several unknown parts. Unfortunately, on their own, expressions do not communicate anything, as they are only a single quantity. Expressions in mathematics can be thought of as being analogous to the nouns used in a language, in which both are used to refer to objects of interest.

On its own, the word "cat" does not communicate anything, but telling someone that "a cat is a type of mammal" would communicate a complete thought to them. Likewise, the expression "x" does not communicate anything. Telling someone that "x is equal to 1" would also communicate a complete thought, in this case communicating what the exact value of the unknown value x is. Much like how a correct arrangement of words is referred to as a "setence", an arrangemetn mathematical symbols that expresses a complete thought is referred to as a sentence.

The diagram below provides the general analogy of mathematics as a language.

Relationships
Perhaps the most common kind of mathematical sentence you will see in all of mathematics are ones that describe how quantities are related. Intuitively, we refer to these sentences as relationships. Up to this point, you have encountered relationships all throughout, such as when we said that quantities were "equal", "greater than", "less than", etc. Here, we will cover what these relationships mean in a broader sense.

Numbers have many different names. In our language analogy, we can say that numbers have "synonyms" like nouns do.

$$6 \ \ \ \ \ \ \ \ 1 + 1 + 1 + 1 + 1 + 1 \ \ \ \ \ \ \ \ 2 + 4 \ \ \ \ \ \ \ \ \frac{18}{3} \ \ \ \ \ \ \ \ (7-2) + 1$$

Even though all of these expressions look different, they all refer to the same number. The idea that all numbers can be expressed in so many ways is an extremly crucial one to all of mathematics, as we will see later in the section.

In mathematics, when an expression A refers to a quantity B, or when expressions A and B refer to the same value, we say that they are equal. We show this relationship with an equal sign (=).

$$ A = B $$

We can extend this definition even futher with more than two quantities or expressions. If we know that A, B, and C all represent the same quantity, we can express this as follows:

$$ A = B = C$$

The equality of the expressions can be expressed as:

$$6 = 1 + 1 + 1 + 1 + 1 + 1 = 2 + 4 = \frac{18}{3} = (7-2) + 1$$


text:
Algebra/Chapter 2/Sets

2.3: Sets

In this section we mainly set up some useful notation. While the ideas here are not very central to the study of algebra, they do come up from time to time, so pay attention! It is exactly because these ideas don't reoccur on every page that they can be confusing when they suddenly come up later on. So be prepared to revisit this section as necessary to refresh your memory.

Sets and the Number Line
A set is a collection of things. They are also an example of an expression, as sets also describe mathematical objects of interest, in this case a group of objects.

Examples of sets might be the collection of letters used in the English alphabet, or the set of books written by John Steinbeck. For us, the sets we will discuss will usually be collections of numbers because these are the sets that are important in algebra. Each of the things in a set is called an element of the set. The number of elements in a set could be finite or could be infinite. The only requirement is that the elements of a set should be described explicitly in some way either now or in the future (after we solve some problem). In this book we will mostly try to use capital letters as the symbols for sets whereas lower case letters are often (but not always!) used for variables.

There may be no elements in a set; such a set is called an empty set or a null set. For the empty set, we use the symbol ∅ to represent it.

A set can be written by putting braces, that is { and }, around a list of the elements of the set, with each element being separated by a comma. For example, a set S containing natural or whole numbers from 1 to 10 could be shown as follows:

$$S=\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\,$$

It is not always possible to list out all the numbers in a set. In these cases we rely on English to describe the set. That's right! Words are an important part of math too. The last and probably most common notation involves using variables and algebraic expressions, together with a description of what values the variables may take. For example, to describe the set of numbers that are a perfect square, we might write:
 * $$\{n^2 \mid n \text{ is a whole number.}\}$$

or we can even use other sets in the description like:
 * $$\Big\{n^2 \mid n\in \{0, 1, 2, 3, \ldots\}\Big\}$$

Sometimes we want to explicitly make clear that a particular number (or thing) is in a set. Keeping S from the example above, we know that 2 is an element of S, rather than writing this out in English, sometimes people use the shorthand 2 ∈ S. The symbol ∈ is chosen because it looks like an E, and E is the first letter of the word "element". If we want to express that something is not part of a set we use the symbol ∉. So to continue our example we know that 11 ∉ S.

Let's take a look at a practice problem.

We now introduce the basic ideas that come up when you have two or more sets.

Subsets and Super sets
Sometimes every element in one set is contained in another set. For example, let S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and T = {2, 4, 6, 8, 10}. Clearly T is just the even numbers between 1 and 10, and every number in T is already in S. In this example we would say that T is a subset of S. Instead of saying which set is the smaller one, we could instead say which set is the bigger one by calling it a superset. That is, we could say S is a superset of T. As one deals with sets more and more, it becomes increasingly tempting to say that T is smaller than S (or maybe less than S). In fact, we already have! Because the relationship of one set being contained in another is so similar to relationship of one number being less than another it is natural to introduce a symbol very similar to inequality symbol <. To avoid confusing sets with numbers we don't want to use exactly the same symbol, so we will round out the point a bit and use the symbol ⊂. So instead of writing out in words "T is a subset of S", we could write T ⊆ S. Just like inequality we can flip the symbol around. We just have to make sure the rounded side points to the smaller thing. That is, for our example we could write S ⊃ T.

What if two sets have exactly the same elements? In this case we say that the two sets are equal. So if someone else came along and said let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then we say that S and U are equal. This time we don't worry about confusing sets and numbers and we will stick with the symbol = to denote when two sets are equal. So we can write S = T. Are there relationships corresponding to ≤ and ≥? Yes, the are ⊆ and ⊇ and they work like you might expect. Here is a table that explains how each of these symbols work.

The empty set is a subset of all sets.

Venn Diagrams
A Venn Diagram shows the logical relation between sets using simple closed curves such as circles drawn a plane.

Unions and Intersections
There are two other things to do with sets. Given sets in S and T we may want to talk about all the elements that are in either S or T. Since a set is just a collection of things, and "the elements that are in either S or T" is a collection, you can think of this as defining a new set called the union of S and T. We write the union of S and T by S ∪ T. We use the symbol ∪ because it looks like a u, and u is the first letter in word "union". Let's do an example.

An important thing to notice in this example is that S ∪ T doesn't contain two 6's. The union contains all the elements in either set, but 6 is still just one thing that happens to be in both sets.

Give sets S and T, instead of thinking about things that are in either set, it is sometimes handy to think about things that are in both sets. Again we think of the collection of "the elements in both S and T, this set is called the intersection of S and T. We write the intersection of S and T by S ∩ T.  We don't use the symbol ∩ because it looks like an i.  It doesn't. Somehow, an i between two symbols just wouldn't look as good, so we want to pick something else.  This symbol is just the upside down symbol of the symbol for union.  Let's consider what the intersection looks like in the problem above.

The number 6 is the only number in both sets, so it is the only element of the intersection.

Subscripts
In a sentence like "let x and y be numbers", we can presume the possibility that $$x=y$$. If we wish to exclude this possibility, we state it explicitly. We say "let x and y be distinct numbers". We can keep going on with this pattern, talking about more than two variables, for example in the sentence "let x, y, and z be numbers". But there are only twenty-six letters in the alphabet. With such limited options to choose from, we would obviously run out of letters in enumerating variables. Hence we use a new notation with subscripts, tiny numbers written to the side of a number. We can use this new notation as follows:

$$Let\ x_1,\ x_2\ be\ numbers.$$ $$Let\ x_1,\ x_2,\ x_3\ be\ numbers.$$ $$Let\ x_1,\ x_2,\ x_3,\ x_4\ be\ numbers.$$

The most general case of this sentence is of course:

$$Let\ x_1,......,x_n\ be\ numbers.$$

Where n represents the amount of objects in a set.

Practice Problems
 Problem 2.38 (Element or Not?) Determine if the number 10 is an element in the following sets.

a. $$\{4, 5, 6,...,15\}$$ b. $$\{1, 2, 3, 4, 5,...\}$$ c. $$\{1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8},...\}$$ d. $$\{x|x\ is\ a\ whole\ number\ greater\ than\ 11\}$$ e. $$\{x|x\ is\ a\ whole\ even\ number\}$$ f. The set $$C$$ made up of composite numbers

 Problem 2.39 (Roster Notation) Each of the sets below are defined using roster notation.

1. $$\{1, 4, 9, 16, 25...\}$$ 2. $$\{0, 4, 8,...,96, 100\}$$ 3. $$\{3, 9, 15, 21, 27...\}$$ 4. $$\{..., -\pi^4, -\pi^3, -\pi^2, -\pi, 1\}$$

a. Determine four other elements that may appear in the sets above. b. Use set builder notation to describe the sets above.

 Problem 2.40 (Set of Sets) Give three examples of sets whose elements are sets.

 Problem 2.41 (Set of Sets of Sets) Give an example of a set whose elements are sets of sets.

 Problem 2.42 (Sorting Automobiles) Construct a Venn Diagram which illustrates the possible unions and intersections of the following sets relative to the universal set consisting of automobiles made in the United States.

$$F: Four door, S: Sunroof, P: Power steering$$

 Problem 2.43 (Statements about Sets) Determine if the following statements are true or false.

 Problem 2.44 (Working with Sets I) Let $$U = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}$$, $$M = \{0, 2, 4, 6, 8\}$$, $$N = \{1, 3, 5, 7, 9, 11, 13\}$$, $$Q = \{0, 2, 4, 6, 8, 10, 12\}$$, and $$R = \{0, 1, 2, 3, 4\}$$.

Use these sets to find the following.

 Problem 2.45 (Working with Sets II) Let $$U = \{copper, sodium, nitrogen, potassium, uranium, oxygen, zinc\}$$, $$A = \{copper, sodium, zinc\}$$, $$B = \{sodium, nitrogen, potassium\}$$, $$C = \{oxygen\}$$.

Use these sets to find the following.

 Problem 2.46 (Working with Sets III) If $$U = \{x|0<x<12\}$$, $$M = \{x|1<x<9\}$$, and $$N = \{x|0<x<5\}$$.

Use these sets to find the following.

 Problem 2.47 (Working with Sets IV) Suppose $$A$$, $$B$$, and $$C$$ are subsets of the universal set $$U$$.

Using Venn Diagrams, shade the areas that represent the following.

 Problem 2.48 (Working with Subscripts) For a whole number $$j$$, $$x_j = (-1)^j$$. Find the value of $$x_0$$, $$x_1$$, and $$x_{183}$$.

 Problem 2.49 (List of Numbers) Refer to the set of numbers below, and use it to answer the following questions.

$$x = \{5, 11, 14, 9, 3, 25, 16, 8, 1, 11, 68, 63, 43, 99, 35, 100\}$$

a. In the set, which number is represented by $$x_3$$? b. In the set, which number is represented by $$x_{10}$$? c. What symbol(s) can be used to represent the number 25 in the list? d. What symbol(s) can be used to represent the number 11 in the list? e. What number represents $$n$$ in $$x_n$$? f. What value does $$x_n$$ take?

 Problem 2.50 (Average) Use subscript notation to write an expression which represents the average of $$n$$ numbers.

title: Algebra/Chapter 2/Sets


text:
Algebra/Chapter 2/Real Numbers

2.4: Properties of Real Numbers

<!---==Subsets of the Real numbers== In this section, we give names to some of the important classes of number.

The first important set of numbers is probably the first set of numbers we are really introduced to, namely the Natural Numbers, which we will call $$\mathbb{N}$$. The natural numbers are:
 * $$\mathbb{N}=\{1, 2, 3, 4, \ldots\}$$.

The next set is just a little bigger, and includes the next number we usually learn in elementary school. The natural numbers, together with the number 0 will be called the Whole Numbers, and denoted by $$\mathbb{W}$$. The whole numbers are:
 * $$\mathbb{W}=\{0, 1, 2, 3, 4, \ldots\}$$.

Of course we are missing the negative numbers. The set of whole numbers together with all of the negative numbers is called the Integers denoted by $$\mathbb{Z}$$. (You might ask why a letter that looks like Z. The reason is because it comes from the German word for number, Zahlen.  English speakers are not the only ones to make important contributions to mathematics! Today, Z is the letter used almost universally.)  The integers are:
 * $$\mathbb{Z}=\{\ldots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \ldots\}.$$

Next, as you might guess we need a set of numbers that includes fractions. The set of all numbers that can be written as a fraction is written is called the Rational Numbers and is denoted by $$\mathbb{Q}$$. You might ask why a letter that looks like Q? Well, first mathematicians save the letter R for real numbers (described below) and F for a general number field (a concept a bit beyond this book). But since a quotient is another word for fraction, and we are not using Q for anything else, it seems the sensible choice. The rational numbers are:
 * $$\mathbb{Q}=\left\{\frac{a}{b} \,\Big|\, a,b\in\mathbb{Z} \text{ and } b\neq 0 \right\}$$

What about just the decimal numbers, we spent a long time working with them. As mentioned in the section on variables the set of all numbers decimal numbers (including those whose that continue indefinitely after the decimal point) is known as the Real Numbers and is written with the symbol $$\mathbb{R}$$. In this case we will not attempt to give a formula that describes the set, and instead just rely on its English description. But we should give a few examples of real numbers.

It may seem difficult to believe, but not every number can be written has a fraction. As we will see later, one such number is $$\sqrt{2}=1.41421356237\ldots$$, but this is far from the only example. Indeed every integer can be written as a decimal just by adding a decimal point and infinitely many zeros to it. For example, 0 = 0.000… and -3 = -3.000… and we have made those to integers into decimal numbers. What about fractions? Yes every fraction can be written as a decimal simply using long division. We can also add, subtract, multiply and divide any two real numbers to get another real number (as long as we don't divide by 0). Unfortunately it gets to be very difficult to describe why this is. The algorithms we learn in school for adding, subtracting, multiplying and dividing real numbers all being with the decimal place furthest to the right. If the decimal goes on forever, it is awfully hard to find the decimal furthest to the right. So what do we do? For the moment the answer has to be "not worry about it too much". Later, after we have mastered a few more mathematics courses we will be ready to tackle the task of making sense of the arithmetic of real numbers. In the mean time your intuition about decimal numbers will probably not lead you astray. And where ever possible we will stick to fractions, or expressions like $$\sqrt{2}$$, rather than having to deal with infinitely long decimals.

Notice, the above list of numbers is increasing. That is, $$\mathbb{N}\subset \mathbb{W}\subset \mathbb{Z}\subset \mathbb{Q}\subset\mathbb{R}$$.

There is one last set of numbers it we should name. The Irrational Numbers is the set of all real numbers which are not rational numbers, we shall denote this set by $$\mathbb{I}$$, though other books may choose other names. To give a formula, we write:
 * $$\mathbb{I}=\{x\in\R \mid x\not\in\mathbb{Q}\}.$$

We have never quite given the definition of a set this way. We added emphasis that the numbers x needed to come from the set of real numbers. This is also a common way to denote a set, though we may not use it much. We should point out there are numbers that are irrational. The most famous example is $$\textstyle \sqrt{2}$$, but there are many many more. In fact the square root of any number which is not a perfect square will be an irrational number. As will the cube root of anything which isn't a perfect cube, etc.

We should point out that
 * $$\mathbb{R}=\mathbb{Q}\cup\mathbb{I}$$

Why? Well because of the definition of $$\mathbb{I}$$. Maybe your thinking "wait, what does this mean again?" Remember that two sets are equal if they have the same elements. So we really should explain why everything in $$\mathbb{Q}\cup\mathbb{I}$$ is in $$\mathbb{R}$$ and why everything in $$\mathbb{R}$$ is in $$\mathbb{Q}\cup\mathbb{I}$$. But we will leave this as an exercise to the ambitious reader.

Practice Problems
 {Name the smallest of the sets given above to which the following numbers belong to.}

{0 -$$\mathbb{N}$$ +$$\mathbb{W}$$ -$$\mathbb{Z}$$ -$$\mathbb{Q}$$ -$$\mathbb{I}$$ -$$\mathbb{R}$$
 * type=""}

{$$\sqrt{21} \,$$ -$$\mathbb{N}$$ -$$\mathbb{W}$$ -$$\mathbb{Z}$$ -$$\mathbb{Q}$$ +$$\mathbb{I}$$ -$$\mathbb{R}$$
 * type=""}

{0.0110211 -$$\mathbb{N}$$ -$$\mathbb{W}$$ -$$\mathbb{Z}$$ +$$\mathbb{Q}$$ -$$\mathbb{I}$$ -$$\mathbb{R}$$
 * type=""}

{$$\sqrt{49}$$ +$$\mathbb{N}$$ -$$\mathbb{W}$$ -$$\mathbb{Z}$$ -$$\mathbb{Q}$$ -$$\mathbb{I}$$ -$$\mathbb{R}$$
 * type=""}

{-32 -$$\mathbb{N}$$ -$$\mathbb{W}$$ +$$\mathbb{Z}$$ -$$\mathbb{Q}$$ -$$\mathbb{I}$$ -$$\mathbb{R}$$
 * type=""}

{$$\textstyle\sqrt{\frac{1}{4}}$$ -$$\mathbb{N}$$ -$$\mathbb{W}$$ -$$\mathbb{Z}$$ +$$\mathbb{Q}$$ -$$\mathbb{I}$$ -$$\mathbb{R}$$ --->
 * type=""}

Types of Numbers
We have already talked about the different types of numbers in Chapter 1. However, in this section, we will be using more sophisticated language to refer to them, and take a look at each of their unique properties.

In mathematics there are names for many different types of numbers and you've encountered lots of these types already and some of these types contain the others. For instance we can start with the whole numbers such as 0, 1, 2, 3, etc. Using subtraction we can build negative numbers by subtracting a bigger number from a smaller giving us an answer in the set {... -3, -2, -1, 0}.

Using division we can identify fractions between 0 and 1 by dividing a smaller number by a bigger e.g. {1/2, 2/3, 3/4, ...} or {-1/-2, -2/-3, -3/-4, ....} We can also identify negative fractions between -1 and 0 by dividing a negative number by a positive or a positive number by a negative {-1/2, -2/3, -3/4, ...} or {1/-2,  2/-3, 3/-4, ...}. Every whole number can be written as a fraction, such as $$\textstyle 2 = \frac{2}{1}$$. The rational numbers are exactly those numbers which can be written as fractions.

Rational numbers are a subset of numbers we call real numbers. Some calculators allow you to differentiate between rational numbers and real numbers by representing the rational number as a fraction. If you use decimal notation the decimals in your rational number may go on forever, for example $$\textstyle \frac{1}{3}=0.333\ldots$$. The real numbers include all of the types of numbers mentioned before (whole numbers, negative numbers, fractions, etc.) and others that require special operations such as roots to represent. These other numbers may not have any recognizable pattern to their digits, such as $$\sqrt{2}=1.41421356237\ldots$$. But, at the end of the day, the real numbers act just like the rational numbers that you're already familiar with. For those readers that are geometrically inclined, one may think of the real numbers as a line (or ruler), where every point on the line corresponds to exactly one number, as in the picture below.

Practice Problems
 Problem 2.52 (Relating Types of Numbers) Create a Venn Diagram which shows how each of the number types above are related to each other.

 Problem 2.53 (Classifying Numbers) Identify the set(s) of numbers each number belongs to.

Properties Of Real Numbers
We begin this section with a review of the fundamental properties of arithmetic. It may seem unusual to give so much emphasis to the few properties listed below, but there is a good reason. Roughly speaking, all of algebra follows from the 5 properties listed in the table below. In the table below, a, b and c can be any number unless stated otherwise. So let's take a look:

But what does all this mean? The commutative property is that you can exchange two numbers and still get the same answer. The associative property is that you can change the grouping (i.e., change the position of the parenthesis) and still get the same answer. The identity property is that there is a certain number that when operated with a number doesn't change it. The inverse property is something that results to the identity number. The distributive property means that you can distribute the operation. Out of all of those properties, the distributive property is the one you'll probably use the most, because it is the only one that mentions both addition and multiplication at the same time. To give an example: these properties even imply fundamental things such as: "multiplication is repeated addition". This book is not going to prove many things, but it would be useful for us to take a look at how this works.

We apply the distributive property for a = 7, b = 1 and c = 1.

 * + = 7 + 7

Though it may seem obvious, this is identity property for multiplication listed above. Now let's try to do the same thing with.

Just like before, this is just the fact that 3 = 1 + 1 + 1 together with substitution.

Once again, we apply the distributive property. Note that we can apply it to expressions with more than two numbers being added in parentheses. The proof is below. While 7 · (1 + 1 + 1) = 7 · 1 + 7 · 1 + 7 · 1 is not covered by the distributive property alone, this problem is solved by grouping the last two 1s with parentheses. Rather than writing this as 7 · (1 + 1 + 1), we could write it as 7 · (1 + (1 + 1)), then used the distributive property with a = 7, b = 1 and c = (1 + 1). Then: 7 · (1 + (1 + 1)) = 7 · 1 + 7 · (1 + 1). Now we apply the distributive property just to the second (taking a = 7, b = 1, and c = 1. Then (looking just at the second term) we have 7 · (1 + 1) = 7 · 1 + 7 · 1.  Finally we can substitute this expression for the second term back into the equation to get: 7 · (1 + 1 + 1) = 7 · 1 + 7 · 1 + 7 · 1.

This looks like a lot of mindless parenthesis juggling, but the point is that the distributive property applies to arbitrarily long sums and products. It is also true that

Or we could make it even longer! We could have as many terms in the sum as we like; as long as "" appears in front of each term on the right hand side we will have a true statement. We will use this fact without justification (that is, without proof). Let's remind ourselves what these properties tell us about arithmetic. Commutativity and Associativity together imply that it doesn't matter what order we add things up in. Let's see why. Associativity says that a + (b + c) = (a + b) + c. This should be thought of as a statement about the sum a + b + c. Why? Because usually addition is just defined between two things, so someone writes down something like a + b + c some people may first add b and c first then add in a, and other people might add a and b first and then add in c. This property says (using a formula) that it doesn't matter which way you do it. What about those people who add a and c together first? Well, that is where commutativity comes in. It tells us that we don't have add things up in exactly the order people write things down. You can switch things around and still get the same answer. Let's do one more example of using these properties to "juggle parentheses" to see that commutativity says you really can add a and c first and get the same answer.

Commutativity and associativity tell you that it doesn't matter in which order you add up a + b + c. You will get the same answer regardless of order. The rule holds even if there are more than three terms: There may be 4, 12, or several thousand. These properties would still tell us that it doesn't matter how we add things up.

The same properties for multiplication tell us it doesn't matter in what order we multiply things together. We are free to change the order to anything that we find easier. Does it ever really make things easier? Sure! For example if you were asked to calculate, then I would personally think it would be easier to calculate

The identity and inverse properties really capture what it means to say that "addition and subtraction are opposites" and "multiplication and division are opposites, as long as it isn't zero that we multiply by." We shall leave it as an exercise to the interested reader to think about why this is.

You can often simplify expressions using the Distributive Property. This is one of the reasons it is so important. For example, consider the expression 2(x − 7) + 14. What happens if we use the distributive property on the first term in this expression? Let's work it out. According to the Distributive Property
 * 2(x − 7) = 2x − = 2x − 14

Plugging this into the expression above we get 2(x − 7) + 14 = 2x − 14 + 14 = 2x. Clearly 2x is a lot easier to evaluate than 2(x − 7) + 14!

Commutative properties of Division
Division is not commutative. That means usually a ÷ b is not equal to b ÷ a, and can be demonstrated simply by example.

$$ \frac 1 2 \ne \frac 2 1 $$

While division itself is not commutative, there are two special cases where the answer is the same if you reverse the order of operation. These cases occur when the answer (quotient) is 1 or when the answer is -1:

$$ a \div b = b \div a \iff \mbox{(rewrite as fractions)} $$

$$ \frac a b = \frac b a \iff \mbox{(multiply both sides by} \ ab) $$

$$ a^2 = b^2 \iff \mbox{(take both square roots)} $$

$$ a = \sqrt{b^2} \quad\mbox{ or }\quad a = -\sqrt{b^2} $$

$$ a = b \quad\mbox{ or }\quad a = -b $$

$$ a \div b = 1 \quad\mbox{ or }\quad a \div b = -1 $$

Basic Laws In Algebra
There are several basic laws in algebra. Understanding these will help you to manipulate and solve equations, and to understand algebraic relationships.

1. Commutative Law
In general, the order of the items can be changed without affecting the results. For addition, $$A + B = B + A$$ indicates that changing the order of the items added does not affect the sum. For multiplication, $$X Y = Y X$$ indicating that the changing of the order of the items multiplied does not affect the product.

Note that the commutative law does not apply to subtraction or division.

2. Associative Law
In general, the grouping of the items can be changed without affecting the results. (Seems to be an extension of the commutative law).

For addition, $$A + (B + C) = (A + B) + C$$ indicates that changing the grouping of the items added does not affect the sum.

For multiplication, $$X ( Y Z ) = ( X  Y ) Z$$ indicates that changing the grouping of the items multiplied does not affect the product.

As with the commutative law, the associative law does not apply to subtraction or division.

3. Distributive Law
Indicates that common terms can be factored, or that factors can be distributed. (A + B) X = (A X) + (B X)  (The "X" terms on the right are combined into a factor on the left side; the factor "X" on the left is distributed on the right side).

Consider the substitution of X = (Y + Z) into the above equation  yields (A + B) (Y + Z) = A (Y + Z) + B (Y + Z). Apply the distributive law to each term on the right yields A Y + A Z + B Y + B Z. We can skip the intermediate step if we multiply the terms identified by “F O I L” in the following expression (A + B) (Y + Z) =

4. Law of Identity
For addition and subtraction the law of identity indicates that the addition and subtraction of a given term or quantity results in the zero, 0, the identity element for addition and subtraction. Alternately, adding the identity element results in no change to the original value or quantity.

$$A - A = 0$$

Adding A to both sides of the first equation we get (A - A) + A = 0 + A. Re-arranging or substituting gives 0 + A = A

 * Note the special case(s) where A = A + 0 = A + 0 + 0

For multiplication and division the law of identity indicates that the multiplication and division of a given term or quantity results in "one," 1, the identity element for multiplication and division. Alternately, multipling or dividing by the identity element results in no change to the original value or quantity.

$$ 1 = \frac{Y}{Y}$$, or $$ 1 = (\frac{Y}{1}) (\frac{1}{Y})$$

 * Note that dividing 1 by a term or quantity gives the reciprocal of the term or quantity. Multiplying by the reciprocal is the same as dividing by the term or quantity. In the above equation on the right (Y / 1), and (1 / Y) are reciprocals of each other
 * Note the special case where $$1 = \frac{1}{1}$$, Multiplying this equation by “1” gives $$1(1) = (1)( \frac{1}{1} )$$, and then dividing by one gives $$\frac{1(1)}{1} = (1)(\frac{1}{1} ) = $$.
 * Simplify this by substititing the first special case equation to get $$1 = 1(1)$$, and $$1 = 1(1)(1) $$, . . .

By multiplying both sides of the first equation by “Y” we get $$ (Y)(1) = (Y)(\frac{Y}{Y})$$, which simplifies and becomes  (Y) = (1) Y.

Practice Problems
 Problem 2.54 (Determining Properties of Real Numbers) Determine if the following statements are always, sometimes, or never true. If the statement is always true, explain your reasoning. If the statement is not always true, provide a counterexample.

$$a.\ An\ integer\ is\ a\ whole\ number.$$ $$b.\ If\ a\ number\ is\ whole\ it\ is\ a\ natural\ number.$$ $$c.\ If\ a\ number\ contains\ a\ decimal\ it\ is\ an\ integer.$$ $$d.\ If\ a\ number\ is\ nautural,\ then\ it\ is\ a\ real\ number.$$ $$e.\ The\ product\ of\ two\ irrational\ numbers\ is\ an\ irrational\ number.$$

 Problem 2.55 (Identifying Properties of Real Numbers) Identify the following properties being expressed.

$$a.\ 4(3x + 4) = 12x + 16$$ $$b.\ 6 + 0 = 6$$ $$c.\ (2 + 7) + 5 = (2 + 5) + 7$$ $$d.\ (3/4)(4/3) = 1$$ $$e.\ To\ divide\ 3072\ by\ 512,\ you\ can\ divide\ 3072\ by\ 16,\ again\ by\ 8,\ and\ again\ by\ 4.$$

 Problem 2.56 (Product Pattern) Use the Associative Law to explain why the products in each rule are equal.

 Problem 2.57 (Gauss's Trick) In the late 1700s, the kindergarten class of the mathematician Carl Fredrich Gauss was asked to find the sum of all of the natural numbers between 1 and 100. While most of the class had struggled with this seemingly impossible task, Gauss was able to determine the solution to this problem rather quickly. How was he able to do this?

 Problem 2.58 (Manipulating Gauss's Trick) We can use techniques similar to the one we used in Problem 2.55 to find the sum of several numbers. Can you find the following?

a. $$1 + 2 + 3 + 4 + ... + 201$$ b. $$2 + 4 + 6 + 8 + ... + 200$$ c. $$101 + 102 + 103 + ... + 998 + 999 + 1000$$ d. $$9 + 12 + 15 + ... + 54 + 57 + 60$$

 Problem 2.59 (Inverses of Numbers) The Additive Inverse Property states that if you add a number and its opposite, or its additive inverse together, you get zero. Likewise, the Multiplicative Inverse Property states that if you multiple a number by its recipricol, or its multiplicative inverse together, you get one. Find the additive and multiplicative inverses of the following numbers.

$$a.\ -6$$ $$b.\ 4\frac{2}{3}$$ $$c.\ -0.33$$ $$d.\ 2 + \sqrt{5}$$

 Problem 2.60 (Using the Distributive Property) Use the Distributive Property to simplify these expressions.

$$a.\ 2(14x - 26)$$ $$b.\ (2/3)(3x + 9)$$ $$c.\ 3(12x + 4y)$$ $$d.\ 2(5x - 6) + 3(3x + 2)$$ $$e.\ (4x + 7)(2x - 3)$$ $$f.\ (x + 1)(x + 2)(x + 3)$$

 Problem 2.61 (Distribution with Three Terms) What is the coefficient of y in the expansion of the expression below?

$$(5x+2y-4)(2x+7y+3)$$

 Problem 2.62 (Rewriting Expressions) Evaluate the following expression without using a calculator:

$$\frac{2013*2014 - 2013*1992}{2014-1992}$$

 Problem 2.63 (Multiplication and the Distributive Law) Point out in what sense the usual arrangement of the multiplication of 365 and 392 is an instance of the Distributive Law.

 Problem 2.64 (Suare of Sum/Difference) For two numbers $$a$$ and $$b$$, find the following:

$$a.\ (a + b)^2$$ $$b.\ (a - b)^2$$

 Problem 2.65 (Tricky Products) Evaluate the following expressions without using a calculator:

$$a.\ (101)^2$$ $$b.\ (95)^2$$ $$c.\ (998)(999)$$ $$d.\ (63)(57)$$ $$e.\ (71)^2$$

 Problem 2.66 (Secret of 1001) A boy claims that he can figure out the product of any three digit number and 1001. A student in his arithmetic class challenges him to find the product of 1001 and 865, and he gets the correct answer immediately. Compute the answer, and determine the boy's secret.

 Problem 2.67 (ABCD) Prove that the following expression can be written as a product between $$a-d$$ and $$b+c$$

$$ab - cd + ac - bd$$

 Problem 2.68 (Density Property of Real Numbers) The Density Property of Real Numbers states that between any two real numbers, there is another real number. Use this property to prove that there are infinitly many real numbers between 0 and 1.

Closure
Closure is a property that is defined for a set of real numbers and an operation. This Wikipedia article gives a description of the closure property with examples from various areas in math. As an Algebra student being aware of the closure property can help you solve a problem. For instance a problem might state "The sum of two whole numbers is 24." With practice you will be able to see that the possible set of numbers will be either all odd (e.g. (1,23),(3,21), ... etc.) or all even (e.g. (2,22), (4,20), ... etc.). The problem might not explicitly state the idea of whole numbers. It might state that two sides of a square sum to 24. If you remember working a problem like this before you know that the sides of a square need to be equal and you divide by two. The author of the problem might want to be trickier and say that two sides of an equilateral triangle sum to 24 and then ask for the perimeter of the triangle. In this case you might want to write the equation $$3x = p$$ for the perimeter of an equilateral triangle. This might make it easier for you to see that again you just need to divide 24 by 2 to find the length of one side and plug it into the equation.

Practice Problems
 Problem 2.69 (Closure of Operators) Complete the following table which represents the closure properties of operations with different types of numbers. Use a check mark to represent closure, and a cross to represent no closure.

 Problem 2.70 (Closure of a Set) Two letters from the set $$\{a, b, c, d, e\}$$ are chosen and multiplied together. The results after doing this are as follows:

Is the set closed under multiplication?

Order and Absolute Value
The absolute value (or modulus) of a real number $$a$$, denoted by $$|a|$$ refers to its distance from zero on the real number line. This value is always taken to be nonnegative. For example, the illustration on the left shows the following: $$|-5| = 5 \ |3| = 3$$ The absolute value of -5 is 5 because it is 5 away from zero, and the absolute value of 3 is 3 because it is 3 away from zero. The absolute value of a positive number or zero is always itself. Conversely, the absolute value of a negative number is its opposite.

Likewise, the distance between two numbers on the number line can be thought of as the absolute value of the difference between them.

Practice Problems
 Problem 2.71 (Ordering Numbers I) Order the following set of numbers from: (a) least to greatest (b) greatest to least. $$2.1, -4,\ \frac{1}{2},\ \pi,\ 3.99,\ -\frac{3}{4},\ -0.25,\ \frac{\pi}{3}$$

 Problem 2.72 (Ordering Numbers II) Order the absolute values of the numbers from Problem 2.68 from: (a) least to greatest (b) greatest to least.

 Problem 2.73 (Absolute Value Expressions) Evaluate the following expressions that involve absolute values.

 Problem 2.74 (Absolute Ratio) Simplify the following expression given that $$x<0.$$

$$\frac{|x|}{x}$$

 Problem 2.75 (Range of Values I) If $$24<x<39$$, what is the value of the following expression?

$$|x - 24| + |x - 39|$$

 Problem 2.76 (Range of Values II) If $$-12\leq	x<12$$, what is the value of the following expression?

$$|x - 14| + |x - 12| + |x + 12| + |x + 14|$$

 Problem 2.77 (Range of Values III) If $$-19\leq x \leq y \leq 4$$, what is the value of the following expression?

$$|x + 19| + |x - y| + |y - 4|$$

 Problem 2.78 (Least Possible Absolute Value) If n is an integer, what is the smallest possible value of the following expression?

$$|123 - 5n|$$

 Problem 2.79 (The Triangle Inequality) For any triangle, the sum of the lengths of any two of its sides must be greater than or equal to the length of the third. This relation is represented as follows:

$$|a + b| \leq |a| + |b|$$

a. Use this relation to determine if a triangle with the side lengths of 6, 9, and 14 exists. b. Use this relation to determine if a triangle with the side lengths of 5, 10, and 15 exists. c. Outside of geometric applications, the above inequality also states that the absolute value of the sum of two numbers a and b are less than or equal to the sum of the absolute value of a and the absolute value of b. Prove that this relation is true.

Lesson Review
All numbers that we will be working with for the majority of Algebra are called Real Numbers. They consist of Rational and Irrational Numbers. Irrational Numbers are numbers that have infinite, non-repeating decimals, such as pi. Rational Numbers are all numbers that can be expressed as a fraction of integers, which include Natural Numbers, Whole Numbers, Integers, and Rational Numbers. For all Real Numbers, there are a few properties of addition and multiplication: Commutative, Associative, Identity, Inverse, and Distribution. The Distribution will come in handy for the rest of the course.

Lesson Quiz
 {Identify the set(s) of numbers each number is in. If the number does not belong to any set, leave all boxes unchecked}

{$$\sqrt{7}$$ -Natural -Whole -Integer -Rational +Irrational +Real
 * coef="2"}

{$$\sqrt{-25}$$ -Natural -Whole -Integer -Rational -Irrational -Real
 * coef="2"}

{$$-\sqrt{36}$$ -Natural -Whole +Integer +Rational -Irrational +Real
 * coef="3"}

{$$(-4)^2$$ +Natural +Whole +Integer +Rational -Irrational +Real
 * coef="5"}

{Identify the property being expressed.}

{$$2 \cdot\! 3 = 3 \cdot\! 2$$ { Commutative (i) _15 } Property of { Multiplication (i) _15 }
 * type="{}" coef="2"}

{$$x\left(\frac{1}{x}\right) = 1$$ { Inverse (i) _15 } Property of { Multiplication (i) _15 }
 * type="{}" coef="2"}

{$$x + (-x) = 0 \,$$ { Inverse (i) _15 } Property of { Addition (i) _15 }
 * type="{}" coef="2"}

{Perform the Distributive Property of Multiplication to simplify each of these expressions.}

{ $$3(2x + 7)=$${ 6x + 21 (i)|6x+21 (i) _12 }
 * type="{}"}

{ $$15(6x - 22)=$${ 90x - 330 (i)|90x-330 (i)|30(3x - 11) (i)|30(3x-11) _12 }
 * type="{}"}

{ $$3(20x + 42y) - 2(7x + 20y)=$${ 46x + 86y (i)|46x+86y (i)|2(23x + 43y) (i)|2(23x+43y) (i) _12 }
 * type="{}"}

{Challenge Questions. Note: Answer the "Why" questions on paper.}

{When two rational numbers are multiplied, does it always result in a rational number? +yes -no
 * type="" coef="1.5"}

{Why?
 * type="{}" coef="5"}

{When two irrational numbers are multiplied, does it always result in an irrational number? -yes +no
 * type="" coef="1.5"}

{Why?
 * type="{}" coef="5"}

{When two irrational numbers are added, does it always result in an irrational number? -yes +no
 * type="" coef="1.5"}

{Why?
 * type="{}" coef="5"}

{When the square root of an irrational number is taken, does it have to be irrational? +yes -no
 * type="" coef="1.5"}

{Why?
 * type="{}" coef="5"}

{If $$x(x+1)$$ is irrational, does x have to be irrational? +yes -no
 * type="" coef="1.5"}

{Why?
 * type="{}" coef="5"}
 * /Answers to "Why" questions/

title: Algebra/Chapter 2/Real Numbers


text:
Algebra/Chapter 2/Mathematical Reasoning

2.5: Mathematical Reasoning


text:
Algebra/Chapter 2/Logic and Proofs

2.6: Logic and Proofs

Practice Problems
 Problem 2.80 (Using Properties of Numbers) Justify each step, using the properties of communativity and associativity in proving the following identities.

$$a.\ (a + b) + (c + d) = (a + d) + (b + c)$$ $$b.\ (a + b) + (c + d) = (a + c) + (b + d)$$ $$c.\ (a - b) + (c - d) = (a + c) + (-b - c)$$ $$d.\ (a - b) + (c - d) = (a + d) - (b + c)$$ $$e.\ (a - b) + (c - d) = (a - d) + (c - b)$$ $$f.\ (a - b) + (c - d) = -(b + d) + (a + c)$$ $$g.\ ((a + b) + c) + d = (a + c) + (b + d)$$ $$h.\ (a - b) - (c - d) = (a - c) + (d - b)$$

 Problem 2.81 (Using Properties of Numbers) Determine if the following statements are true or false. Justify your conclusions.

a. If $$a$$, $$b$$, and $$c$$ are integers, then the number $$ab + bc$$ is an even number. b. If $$b$$ and $$c$$ are odd integers, and $$a$$ is an integer, then the number $$ab + bc$$ is an even number.

 Problem 2.82 (Using Properties of Numbers) We define an integer $$a$$ to be of

 * Type I if $$a=4n$$ for some integer $$n$$
 * Type II if $$a=4n + 1$$ for some integer $$n$$
 * Type III if $$a=4n + 2$$ for some integer $$n$$
 * Type IV if $$a=4n + 3$$ for some integer $$n$$

a. Provide at least two examples of each of the four types of integers above. b. Is it true that if $$a$$ is even, then it is of type I or III? Justify your answer. c. Is it true that if $$a*b$$ is of type I, whenever $$a$$ or $$b$$ are of type III? Justify your answer.

 Problem 2.83 (Using Properties of Numbers) For all real numbers $$x$$ and positive integers $$n$$, show that:

$$(1-x)(1+x+x^2+...+x^{n-1}+x^n)=1-x^{n+1}$$

title: Algebra/Chapter 2/Logic and Proofs


text:
Algebra/Chapter 3/Equations

3.1: What are Equations?

Balancing variables
Mathematics began with the concept of counting, but it grew because of the idea of equivalence. Tally marks or counters are equivalent to a value depending on the concept being expressed. For instance a shepherd could use a bag of rocks to make sure that he brought home the same number of sheep he left with. The idea of equivalence as shown with an equal sign allows us to state "is the same as".

An expression is a statement that is well formed. The neolithic shepherd used his bag of rocks to create expressions for managing his herd. For instance, the shepherd hoped that the size of his flock remained constant between the time he took his flock out to graze and the time that he returned them home. When he used the bag of rocks the shepherd used two expressions: # Sheep and # Rocks to create the equation # Sheep = # Rocks. Mathematics describes the operations the shepherd performed on the expressions in order to keep the equation in balance. For instance, when the shepherd used his bag of rocks to count his flock at night he might put his rocks in a bowl next to the gate on his pen. Each time a sheep went into the pen the shepherd would move a rock from the bag into the bowl. If rocks were left over in the bag, then the shepherd knew he needed to return to the field to find his lost sheep.

Expressions
An expression is the arrangement of mathematical symbols denoting values and operations, while the equation is a form of an expression that indicates that two values are equal. While equations are generally expressions, you will discover that some expressions are not equations. Expressions in the form of functions or inequalities will be more apparent later in the book.

When you simplify an expression, you are performing simple arithmetic steps until you receive the simplest answer, which is usually an integer or fraction.

Equations

An equation is just like a balance. A balance is a machine that compares whether two quantities are the same or not the same. In mathematics, two quantities balanced translates to "having the same value." An equation is true if the two sides are balanced and false if the two sides are not balanced. As 5=5. This is a very simple equation. We can use variables too(almost used in mathematics)e.g. x=5. This is same as while using a balancing machine, we use some quantity on one side and some matter on the other which is to be measured. As an example we know that 5 is equal to 5 and x is equal to x(whatever its value is). The simplest example of a false equation is that 5 is not equal to 20.

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Remember from the previous section, a variable is a symbol, usually a letter, that stands for a number. We'll use the letters $$x$$ and $$y$$ most often. If $$x=5$$ is true, then any time I use $$x$$, you can replace $$x$$ by 5. We're saying $$x$$ and 5 have the same value, but different appearance. In the equation $$x=5$$, the equal sign is like the balance. If $$x=5$$ is true, we know that whenever we see an $$x$$ we can substitute the number 5 if that helps us understand. Even more important, we can check if a given number substituted for a variable in an equation makes that equation true.

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Frequently we don't know what the value of a variable is that would make the equation true, and we need to find out! Finding out the values that make an equation true is called Solving an Equation, which we will do without guessing in the next section.

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Conceptual Questions
 Problem 3.1 (Explaining Equations) In no more than one paragraph, explain in your own words what an "equation" is. In your definition, discuss an equation's purpose, as well as what a solution of an equation means.

 Problem 3.2 (Anatomy of an Equation) Given the statement x + 6 = 12, answer the following.

a. Is this statement true or false? b. Does x=5 make the statement true? c. Does x=6 make the statement true?

 Problem 3.3 (Identifying Equations) Determine which of the following are equations.

a. x + 4 b. a - 64 = 45 c. 7 < 8 d. a + 6 > 9 e. 1 + 7 = 8

 Problem 3.4 (Special Solution Sets) Make up an equation whose solution is the null set, and explain why this is the solution set. After this, make up an equation whose solution is the set of all real numbers, and explain why this is the solution set.

 Problem 3.5 (Equivalent Equations) Are the statements 9x + 5 = 4 and 4 = 9x + 5 the same equation? Explain your reasoning.

Exercises
 Problem 3.6 (Checking Solutions) Check if the given value(s) is/are solutions to the following equations.

 Problem 3.7 (Equation Diagrams) Write the equation that best represents the following diagrams.

 Problem 3.8 (Scales) Look at the diagrams below. How many circles would you need to balance the third scale?

 Problem 3.9 (Scales) Which is heavier? A red triangle or a blue square?

title: Algebra/Chapter 3/Equations