Questions on basic arithmetic involving numerical quantities only. Questions involving variable values (other than the result of the operation) should be placed under the (algebra-precalculus) tag.

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0answers
13 views

How to visualize the product of two segments lengths? [duplicate]

So there seems to be easy ways to visualize addition. If three points a, b, and c are on the same straight line respectively, we can say that the sum of the lengths of ...
0
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2answers
36 views

Q: Quadratic Division - How to divide two quadratics?

My studies into graphs and models following examples from Khan Academy has helped me on my goal to learn how to chart and model via the quadratic formula However while I have been successful in ...
0
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2answers
41 views

What is one possible distance (in km) at which I live from Arun’s place?

Michael lives $10$ km away from where I live. Ahmed lives $5$ km away and Susan lives $7$ km away from where I live. Arun is farther away than Ahmed but closer than Susan from where I live. From the ...
2
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2answers
46 views

What is the property of addition called when you break 97 into 100 - 3?

Sometimes it's easier to add numbers when you recognise that they're close to some round number, and then add the differences separately. $$97+198$$ $$=(100-3)+(200-2)$$ $$=(100+200)+(-3-2)$$ ...
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1answer
59 views

Computing Bessel functions in complex interval arithmetic

Does anyone know any resource of algorithms for evaluating the Bessel function of the first kind and second kind under interval-valued domains? In particular, I am interested in complex interval ...
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3answers
34 views

For how many days will the food last in garrison?

A garrison has sufficient food for $75$ soldiers for a period of $90$ days. After $10$ days, one third of the soldiers leave. After another $10$ days, $5$ soldiers return, From this day on, ...
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5answers
67 views

Can a decimal multiplied by another decimal ever equal an integer? [closed]

Any decimal number between 0 and infinity that multiplied by itself or another decimal number. Like 0.4 times 0.25, 2.425 times 2.425 and so on.
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6answers
859 views

Squaring both sides when units are different?

Given $((9) \text{inches})^{1/2} = ((0.25) \text{yards})^{1/2}$, then which of the following statements is true? $((3) \text{inches}) = ((0.5) \text{yards})$ $((9) \text{inches}) = ((1.5) ...
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2answers
36 views

Binary arithmetic - overflow and carryout at same time?

In binary arithmetic, When you subtract 2 signed numbers you must discard the carry out. My question is, is it possible for overflow to occur and a carry out? So, on paper there would be two extra ...
29
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16answers
4k views

How did the rule of addition come to be and why does it give the correct answer when compared empirically?

I'm still a high school student and very interested in maths but none of my school books describe these kind of things, and only say how to do it not the whys. My question is very simple, for example: ...
3
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1answer
19 views

Relationship between operations of a ring

Is there any requirement that the two operations of a ring have to be related to each other, excluding the requirement of distributivity? We all know from grade school that multiplication of integers ...
2
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2answers
38 views

Irreducible vs. reducible fractions

Let $a,b,c,d$ be positive integers. Suppose that $$\frac cd=\frac ab.$$ I want to prove that if $a$ and $b$ are relative primes, then $c/a=d/b$ is an integer. That is, the only way a fraction can be ...
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3answers
134 views

Book on foundational reasoning of standard arithmetics “curriculum”

I am interested in a book that is about arithmetics but the presentation is not just the known to all formulas but the foundational logic behind it. The closest example I can think about is the way ...
3
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1answer
39 views

Simple property of power of -1

Is $(-1)^{a+b} = (-1)^{a-b}$ true $\forall a,b \in \mathbb{Z}$ ? My argument: $$ (-1)^{a-b} = \frac{(-1)^a}{(-1)^b} = \frac{(-1)^a}{(-1)^b}.\frac{(-1)^b}{(-1)^b} = \frac{(-1)^{a+b}}{(-1)^{2b}} = ...
0
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1answer
21 views

Possible Typo in Manipulation of Expressions

Someone claimed that the following two expressions are equal. But after much manipulation, I cannot establish their equality. Is there in fact a typo somewhere? $C(x_1^2+x_2^2)^{-(a+p)/2} x_1^p$ ...
0
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1answer
22 views

Formula for monetary amounts with equal taxes

My colleague ran into the following problem while writing billing software. Suppose you have a monetary amount $M$ and a tax rate $t$. For example, $M = 20.1253$ and $t = 0.07$. The question is ...
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5answers
77 views

Why is the solution to $\sqrt{6-5x}=x$ only $x=1$ and not $x=-6$? [duplicate]

I solved the equation $\sqrt{6-5x}=x$ as follows: $$(\sqrt{6-5x})^2=x^2$$ $$6-5x=x^2$$ $$0=x^2+5x-6=(x+6)(x-1)$$ $$x=-6 \quad \text{or} \quad x=1$$ If I plug in $x=-6$ into the original equation, I ...
0
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1answer
41 views

Is $\displaystyle\lim_{h \to 0} H_n(f(h), g(h)) = H_n(\displaystyle\lim_{h \to 0} f(h), \displaystyle\lim_{h \to 0} g(h))$ true for all $n$?

Consider the limit $\displaystyle\lim_{h \to 0} H_n(f(h), g(h)), $ where $H_n(a, b)$ denotes the $n$th hyperoperation $H_n(a,b) = a \uparrow^{n-2}b$ with both $f(x)$ and $g(x)$ being continuous and ...
1
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3answers
70 views

Problematic square root [duplicate]

Ok, here is what I think. Please correct me if I am wrong. $$\sqrt{9} \neq 3$$ and also $$\sqrt{9} \neq -3$$ Now let's assume, that above statements are false, then we have $-3 = \sqrt{9} = 3$ and ...
1
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1answer
22 views

Inverting the equality which contains the operation of taking integer part

I was recently presented with the following equality $$ n = \left[\frac{w}{2d+a}\right]\cdot \left[\frac{h}{2d+b}\right] $$ where all participating variables are non-negative integers, and $[\ldots]$ ...
1
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1answer
29 views

Integer Y with N Repeating Digits of X?

I have a single Base 10 digit X. I want to return number Y where Y is digit ...
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0answers
25 views

Are there clear, formal definitions for “terms” in subtraction operation?

I tutor children of all ages in Mathematics and I've noticed so many different words thrown around regarding binary operations, particularly with subtraction. For example, when working with a 2nd ...
1
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1answer
45 views

How many non-negative integers less than $10000$ are there such that…

I am stuck with the following two problems : How many non-negative integers less than $10000$ are there such that the sum of digits of the number is divisible by $3$ ? The options are : ...
7
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3answers
169 views

Variety of proofs of a simple proposition in arithmetic

I have a couple of simple proofs of a simple proposition and I'm curious to see the variety of different approaches others would take to prove the same thing. Definition: The dilation of a sequence ...
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3answers
56 views

Is $p-t(p+q)$ the same as $p+t(q-p)$

This might be a silly question, but is $p-t(p+q)$ the same as $p+t(q-p)$? Not that it matters, but this is a formula in linear algebra called two-point form. The original formula is $(1-q)p+tq$ and I ...
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2answers
28 views

Sum based on sub-sums

Let's say, we have three numbers: $a, b, c$ but we know only their sums: $x = a + b, y = b + c.$ Is it possible to find sum $z = a + b + c$? edit - $a, b, c$ are natural numbers
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1answer
70 views

How can I average two integers digit by digit?

I would like to take two integers of similar size (about the same number of digits) and average them, without having to first add them, and then divide the result by two. Let's say I have the ...
-4
votes
3answers
52 views

What will be value of 0/0? [duplicate]

We know $\frac{\text{equal things}}{\text{equal things}}=1$,$\frac{\text{anything}}{0}=\text{undefined}$, and $\frac{0}{\text{anything}}=0$. Then why does $\frac{0}{0}=0$. Could it also be $1$ or ...
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4answers
217 views

35 + 6 = 30 + 11 [closed]

I have a disagreement with regrouping and I was hoping someone with an open mind might help me understand or work with me on this. Essentially I was taught that 35+6 = 30+11 because they both equal ...
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5answers
117 views

How can I compare the numbers $2^{39}$, $5^{19}$ and $52^7$?

I have to compare the numbers $2^{39}$, $5^{19}$ and $52^7$. I don't know how to do that because their exponents don't have anything in common.
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2answers
20 views

Mensuration and similarity

Cone P has a volume of 108cm^3 Calculate the volume of 2nd come , Q , whose radius is double that of cone P and its height is one-third that of cone p Here's my working .... $$V_Q=\frac13 \pi (2r)^2 ...
0
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0answers
7 views

Find interval algorithm problem

I have the following arithmetic problem: What is known condition: m,n [a,b) : a mod(m) = 0 , b mod(m) = 0 [x,y) : x mod(n) = 0 , y mod(n) = 0 b < x What must ...
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1answer
41 views

How to prove that there is no infinite arithmetic progression of perfect squares

How to prove that there is no infinite arithmetic progression of perfect squares This question from a school Olympiad paper ! How can I prove this directly or using contradiction ? For example : 1 ...
113
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13answers
15k views

Why does an argument similiar to 0.999…=1 show 999…=-1?

I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc. Can anyone point me to resources that would explain what the ...
1
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2answers
54 views

What is the value of $xyz$?

Given an expression as : $$xyz+xyz+xyz=zzz$$ where $x,y,z$ are integers and $xyz$ represents a number for example $236$ (not to be confused with $x\times y\times z$), what is the number $xyz$?
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2answers
44 views

Sum of arithmetic sequence inequality in answer

I have been given the question : Find the sum of the arithmetic progression: $$8.5 + 12 + 15.5 + 19 +\dotsb + 103$$ --for clarity there are $27$ terms, as $$\frac{103-8.5}{3.5} = 27$$ ...
5
votes
2answers
101 views

Is $‎‎‎\sqrt[3]{y^3}‎‎‎$ or $\frac{x^2}{x}$ a polynomial?

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Now are $$‎‎‎\sqrt[3]{y^3}‎‎‎,\quad ...
0
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2answers
56 views

Sum of Reciprocal of a sum of square roots

I am trying to find a closed form expression of the following sum: $\sum_{k=1}^{N}\frac{1}{\sqrt{k}+\sqrt{k+3}},\; N>1.$ I tried to determine whether methods used for evaluating more conventional ...
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0answers
24 views

partially ordered group, does x=-x imply x=0?

I have just a simple question: Let (G,+) be a partially ordered Abelian group. Does x = -x imply x = 0 ? If the answer is yes, then how could i prove it? If the answer is no, then a ...
1
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1answer
12 views

Fraction weights and weighted average smaller than sum op parts

I am considering a weighted average $\bar{x} = \frac{w_1x_1+w_2x_2}{w_1+w_2}$, where $x_1<x_2$. I have seen with a lot of numerical experiments that the following relation must hold: ...
2
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2answers
47 views

Proof of $a^{\frac{p-1}{2}} \equiv \left(\frac{a}{p}\right)$ mod $p$

$p$ is a prime, odd integer. $a$ is an integer. we assume that $p$ does not divide $a$. $\left( \frac{a}{p} \right)$ denotes the Legendre symbol. In order to prove $a^{\frac{p-1}{2}} \equiv ...
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0answers
37 views

Recursive division by Burnikel and Ziegler, explaining the breaking down of large numbers

I am looking at Fast Recursive Division by Burnikel and Ziegler. I understand $DivTwoDigitsByOne( ... )$ and $DivThreeHalvesByTwo( ... )$ as they break the numbers down. So, for example, ...
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10answers
6k views

How to prove that all odd powers of two add one are multiples of three

For example \begin{align} 2^5 + 1 &= 33\\ 2^{11} + 1 &= 2049\ \text{(dividing by $3$ gives $683$)} \end{align} I know that $2^{61}- 1$ is a prime number, but how do I prove that ...
1
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1answer
40 views

Prime and Repunits

Prove that: For any integer $n > 5$, if $n$ divides $\dfrac{10^{n-1}-1}{9}$, then $n$ is a prime number. This can also be generalized further as If $n$ is an integer > 5 and divides a ...
3
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1answer
158 views

Polynomials and Arithmetic

Consider the polynomial $$p(x) = a_0 + a_1x + a_2x^2 + · · · + a_nx^n$$ where $a_0, a_1, . . . , a_n ∈ \Bbb Z$. Show that if $p(x_i) = 7$ for 4 distinct integers $x_0, x_1, x_2, x_3$, then $p(z) \neq ...
0
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1answer
49 views

Finding the least number of dots to add into a 10x10 grid

I have a 10x10 grid where are some dots. What is the least number of dots that I need to add in order to have 3 dots in every row and column have odd number of dots in every row and column have ...
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1answer
32 views

LCM confusion question

A section of soldiers are rehearsing for the march past for the National Day parade . If they march in pairs , one soldier will be without a partner . If they match in threes , fives or sevens , they ...
0
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1answer
12 views

Numbers and percentage

When x is decreased by 15% and then increased by 20%, it becomes y. How to find this value of y? Express x:y in its simplest form The first step I done was to take away 15% from X which becomes 85% ...
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2answers
33 views

HCF LCM Question

$540= 2^2 \times 3^3 \times 5$ Find the smallest positive integer $K$ such that $\frac{540}{k}$ is a cube number .
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1answer
36 views

Unable to process Large numbers [closed]

A small spherical cell of diameter $1.616E^{-35}$ is exponentially multiplying as $2^n$ where n is the generation number. The duration of 1 generation is $5.39E^{-44}$ second. And the cells cluster ...