# Tagged Questions

Questions on basic arithmetic, e.g. addition, subtraction, multiplication, division, powers, roots, etc.

2answers
98 views

### Stuck on simple proving

Prove that $\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\cdots-\frac{1}{2009}+\frac{1}{2010}<\frac{3}{8}$ Oh my, I feel embarrased for not knowing how to solve such an elementary ...
4answers
185 views

### Divide inside a Radical

It has been so long since I have done division inside of radicals that I totally forget the "special rule" for for doing it. -_- For example, say I wanted to divide the ...
2answers
206 views

### Primes in nonstandard models of PA

What is known about prime numbers in nonstandard models of PA? Restricted to true natural numbers the sets are identical, but does there always exist nonstandard primes? Can we explicitly define one ...
3answers
287 views

### How to compute these GCD's?

Please suggest me how to compute the GCD of theese really big numbers : GCD of $2^{120547564397}-1$ and $2^{356946681940}-1$ GCD of $2^n-1$ and $n!$ where $n=3^{19}$ Thanks to Bill Dubuque's ...
5answers
285 views

### Division with 4 digit number in denominator

I've got a question in my task sheet. The question is as follows. $$\frac{43\cdot93\cdot47\cdot97}{3007}=X$$ Find the exact value of $X$. I've tried a lot, but couldn't find easier way to do it ...
2answers
238 views

### What algorithm can I use to add negadecimal numbers?

I am trying to figure out how to add negadecimal numbers by hand. I can add normal decimal numbers using an algorithm I learned in kindergarten: start with the least significant digits, add them, ...
4answers
121 views

### How to factor 5671?

The other day I wanted to factor 5671 in my head. (It turns out to be $53\cdot107$, but I did not know this at the time.) I quickly ruled out the easy divisors, 2, 3, 5, 7, 11, and 13. At this point ...
5answers
170 views

### What is the square root of $i^4$?

What is the $\sqrt{i^4}$? $i^4$ = $(i^2)^2$ So is $\sqrt{i^4}$ = $\sqrt{(i^2)^2}$ = $i^2$ = $-1$? Or is $\sqrt{i^4}$ = $\sqrt{1}$ = $1$? When I plug it into my TI-89 Titanium, I get $1$. Edit: I ...
1answer
60 views

### Confused about exponents and imaginary/real answers

I am confused about some exponent behavior. $$(-2)^{7.6} = (-2)^{\frac{76}{10}} = ((-2)^{76})^{\frac{1}{10}} = ((-2)^{\frac{1}{10}})^{76}$$ Is there something wrong in this logic? When I plug the ...
1answer
2k views

### rules for rounding (positive and negative numbers)

I'm looking for clear mathematical rules on rounding a number to n decimal places. Everything seems perfectly clear for positive numbers. Here is for example what I found on math.about.com : Rule ...
2answers
47 views

### How do I append an integer to the left of another integer?

For example: . is my append operator f(x,y) = |x| . |y| f(1,45) = 145 f(233,10) = 23310 f(8,2) = 82 f(0,1) = 1 This is a trivially easy problem to ...
2answers
224 views

### Is there a log-space algorithm for divisibility?

Is there an algorithm to test divisibility in space $O(\log n)$, or even in space $O(\log(n)^k)$ for some $k$? Given a pair of integers $(a, b)$, the algorithm should return TRUE if $b$ is divisible ...
3answers
290 views

### A “fast” way to find the sum of the sequence $5,5.5,5.55,5.555,5.5555,\ldots$ (20 terms)

My initial approach is diving the whole sum by $9$ and taking the common $5$ out which gives $$\frac{5}{9}[(10-1)+(10-0.1)+(10-0.01)+\cdots + (10-10^{-19})]$$ after some algebra this could be reduced ...
1answer
757 views

### Chinese Remainder theorem with non-pairwise coprime moduli

Let $n_1,...,n_k \in \mathbb{N}$ and let $a_1,...,a_k \in \mathbb{Z}$. How to prove the following version of the Chinese remainder theorem (see here): There exists a $x \in \mathbb{Z}$ satisfying ...
2answers
856 views

### How can I calculate non-integer exponents?

I can calculate the result of $x^y$ provided that $y \in\mathbb{N}, x \neq 0$ using a simple recursive function:  f(x,y) = \begin {cases} 1 & y = 0 \\ (x)f(x, y-1) & y > 0 \end ...
2answers
3k views

### How to calculate the number of decimal digits for a binary number?

I was going to ask this on Stack Overflow, but finally decided this was more math than programming. I may still turn out to be wrong about that, but... Given a number represented in binary, it's ...
1answer
135 views

### Is every φ above the second level of the arithmetical hierarchy independent of PA?

If I am not wrong, every $\Sigma_n$ (or $\Pi_n$ ) statement $\phi$ is equivalent to a statement that says that a given Turing machine halts (or doesn't halt) on input $C$ using a ...
3answers
124 views

### What is the most mathematically sound way to define the “damage per second” for a weapon?

Consider a weapon firing shots every $f^{-1}$ seconds (i.e. $f$ is the weapon's fire rate). Each shot deals $n$ damage to is target. Consider another weapon firing every $3f^{-1}$ second, but dealing ...
2answers
310 views

### How many decimal places are needed for incremental average calculation?

If using the following formula to incrementally calculate average value: $$m_n = m_{n-1} + \frac{a_{n}-m_{n-1}}{n}$$ And the number of decimal places to where you can save $m_n$ is limited to $d$ ...
1answer
59 views

### An identity involving the powers of a nilpotent element in a unital commutative ring

Suppose $R$ is a commutative unital ring with identity $1$ such that the equation $nx = 1$ has a unique solution for each integer $n \ge 1$, and let $\xi$ be a nilpotent element of $R$ with nilpotency ...
1answer
77 views

### Direct proof of the non-zeroness of an Eisenstein series

Question: Can you show directly from its formula that $G_4(i)\neq0$? Recall that the holomorphic Eisenstein series of weight $2k$ is defined by: G_{2k}(\tau)= \sum_{(m,n)\in\mathbb{Z}^2\setminus ...
2answers
180 views

### How can I write an algorithm to perform the following calculation exactly? (references accepted)

Given natural numbers $N, K, m, C$, with $3^{m/3}K>C$, I want to be able to write an algorithm to exactly compute the number $$\left\lceil \log_3 \left(\frac{N}{3^{m/3}K-C}\right) \right\rceil$$ ...
4answers
486 views

### Long division notation (census of nations)

The Wikipedia article on long division explains the different notations. I still use the European notation I learned in elementary school in Colombia. I had difficulty adapting to the US/UK notation ...
3answers
375 views

### Can two sets have same AM, GM, HM?

Can two set of numbers(same size) have same arithmetic , geometric, and harmonic mean ? When I say different set they must differ by at-least $1$ element and also what if set is not be of distinct ...
5answers
9k views

### How to convert a hexadecimal number to an octal number?

How can I convert a hexadecimal number, for example 0x1A03 to its octal value? I know that one way is to convert it to decimal and then convert it to octal ...
8answers
2k views

### Need faster division technique for 4 digit numbers

I have to divide 2860 by 3186. The question gives only 2 minutes and that division is only half part of question. Now I can't possibly make that division in or less than 2 minutes by applying ...
4answers
333 views

### A prize of \$27,000 is to be divided among three people in the ratio 3:5:7. What is the largest share?

This is not homework; I was just reviewing some old math flash cards and I came across this one I couldn't solve. I'm not interested in the solution so much as the reasoning. Thanks
3answers
162 views

### sum of ten squares

You are given an unlimited supply of $1\times 1,2\times 2,3\times 3,4\times 4,5\times 5,6\times 6$ squares.Find a set of ten squares whose areas add up to $48$.If not the whole solution,even a little ...
5answers
236 views

### Proving that $30 \mid ab(a^2+b^2)(a^2-b^2)$

How can I prove that $30 \mid ab(a^2+b^2)(a^2-b^2)$ without using $a,b$ congruent modulo $5$ and then $a,b$ congruent modulo $6$ (for example) to show respectively that $5 \mid ab(a^2+b^2)(a^2-b^2)$ ...
7answers
150 views

### Rational numbers $\mathbb Q$

$$\Bbb{Q} = \left\{\frac ab \mid \text{$a$ and $b$ are integers and $b \ne 0$} \right\}$$ In other words, a rational number is a number that can be written as one integer over another. ...
5answers
250 views

### What is the remainder of $(14^{2010}+1) \div 6$?

What is the remainder of $(14^{2010}+1) \div 6$? Someone showed me a way to do this by finding a pattern, i.e.: $14^1\div6$ has remainder 2 $14^2\div6$ has remainder 4 $14^3\div6$ has remainder 2 ...
3answers
247 views

### Fractions with radicals in the denominator

I'm working my way through the videos on the Khan Academy, and have a hit a road block. I can't understand why the following is true: $$\frac{6}{\quad\frac{6\sqrt{85}}{85}\quad} = \sqrt{85}$$
5answers
298 views

### Do addition and multiplication have arity?

Many books classify the standard four arithmetical functions of addition, subtraction, multiplication, and division as binary (in terms of arity). But, "sigma" and "product" notation often writes ...
5answers
1k views

### cubic root of negative numbers

excuse my lack of knowledge and expertise in math;) but to me it would came naturally that the cubic root of -8 would be -2 since -2 ^ 3 = -8. but when i check wolfram alpha for cbrt(-8), real it ...
3answers
219 views

### $1 +1$ is $0$ ?​ [duplicate]

Possible Duplicate: -1 is not 1, so where is the mistake? $i^2$ why is it $-1$ when you can show it is $1$? So:  \begin{align} 1+1 &= 1 + \sqrt{1} \\ &= 1 + \sqrt{1 ...
4answers
163 views

### How to think about multiplication with a number $0 <x < 1$?

To clarify my confusion: $88 \cdot 0.732$ is the same as $0.732 \cdot 88$ The latter being: 0.732 eighty-eight times: $0.732 + 0.732 + 0.732 + \cdots + 0.732$ But how to think about \$88 \cdot ...
2answers
189 views

### How to generate two numbers such that the smaller divides the larger

I am creating a children's math game and need an algorithm (that I can write in JavaScript) to generate two numbers such that the smaller always divides the larger. How can I do that?
3answers
74 views

### Repeated nested roots

Quite some years ago, I remember being asked the following question: Suppose $\alpha = \sqrt{2+\sqrt{2+\sqrt{2+\ldots}}}$, what is $\alpha$. The solution was given by squaring $\alpha$ and solving ...
2answers
136 views

### Why $x^{(1/2)2} \neq x^{2(1/2)}$?

I know, probably is a newb. question, but i can't get this $x^{(1/2)2} \neq x^{2(1/2)}$ $x\in\mathbb R^+$. I know $x^{(1/2)2}=(\pm \sqrt{x})^2=+x$ and $x^{2(1/2)}=\pm x$ because ...
5answers
146 views

### Find the possible value from the following.

Find the possible value from the following. I'm not able to end up on a concrete note, as I'm unable to get the essence of question, still not clear to me. $x$, $y$, $z$ are distinct reals such ...
3answers
178 views

### Cancel before multiplying!!

$$\binom{12}6 = \frac{12\cdot11\cdot10\cdot9\cdot8\cdot7}{6\cdot5\cdot4\cdot3\cdot2\cdot1} = 924.$$ Sometimes it's hard to talk students out of computing both the numerator and the denominator in ...
4answers
89 views

### Is it true that $\sum_{k=1}^n(p_k\prod_{i=1}^k(1-p_i)) \stackrel{\mbox{?}}{=} 1 - \prod_{i=1}^n(1-p_i)$

Prove that $$p_1 + \sum_{k=2}^n \left(p_k\prod_{i=1}^{k-1}(1-p_i)\right) = 1 - \prod_{k=1}^n(1-p_k)\ .$$ I'm working in a code where I have to do those computations. I want to see if this ...
2answers
110 views

### If a sum is one, the sum of all products are also one.

Let $p_1,\ldots,p_s$ be $s$ number in the unit interval such that $$p_1+\ldots+p_s=1.$$ Is it then true, that for every $n\geq 1$ we have  \sum_{(k_1,\ldots,k_n)\in \{1,\ldots,s \}^n} p_{k_1}\cdot ...
2answers
131 views

### Solving simple congruences by hand

When I am faced with a simple linear congruence such as $$9x \equiv 7 \pmod{13}$$ and I am working without any calculating aid handy, I tend to do something like the following: "Notice" that adding ...
1answer
105 views

### Least value for addition

We know that $$0\leq a \leq b \leq c\leq d\leq e\,\,\text{ and}\,\, a + b + c + d + e = 100$$. What would be the least possible value of $\,\,a + c + e\,\,$ ? I apologize for poor syntax.
2answers
627 views

### Converting decimal(base 10) numbers to binary by repeatedly dividing by 2

A friend of mine had a homework assignment where he needed to convert decimal(base 10) numbers to binary. I helped him out and explained one of the ways I was taught to do this. The way I showed him ...
2answers
83 views

### integer to float

How can represent any number from 0 to 127 as something between 0 and 1 ? for example 64's equivalent would be 0.5
2answers
113 views

### How to compute the following formulas?

$\sqrt{2+\sqrt{2+\sqrt{2+\dots}}}$ $\dots\sqrt{2+\sqrt{2+\sqrt{2}}}$ Why they are different?
1answer
270 views

### Non Higher-Order Formulation of Gödels Incompletness Theorem

I was just having a look at Gödels incompletness theorem as found in: http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf I noticed that Gödel used a higher order logic. At least ...
2answers
357 views

### Finding four numbers

Find four weights such that given four weights and weighing pan (balance scale) you can measure all weights between $1$ to $80$. I found this one here.Any idea how to solve?
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