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

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-1
votes
5answers
56 views

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

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.
5
votes
6answers
517 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) ...
0
votes
1answer
12 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 ...
23
votes
15answers
3k 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
votes
1answer
15 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
votes
2answers
27 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 ...
1
vote
3answers
110 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
votes
1answer
37 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
votes
1answer
20 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? $C(x_1^2+x_2^2)^{-(a+p)/2} x_1^p$ ...
-1
votes
0answers
24 views

ratio of two horse speed [on hold]

The ratio of speedof 2 horses is 3:2 in first lap and ratio differ by 4:7 in second lap and ration differ by 8:9 in third lap then find the difference in the ratio all together speed of two horse ...
0
votes
1answer
21 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 ...
3
votes
5answers
63 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
votes
1answer
34 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
vote
3answers
62 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
vote
1answer
20 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
vote
1answer
28 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 ...
0
votes
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
vote
1answer
35 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
votes
3answers
163 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 ...
1
vote
3answers
54 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 ...
1
vote
2answers
23 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
0
votes
1answer
64 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
50 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 ...
-12
votes
4answers
150 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 ...
7
votes
5answers
110 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.
0
votes
2answers
15 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
votes
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 ...
-7
votes
1answer
71 views

Elementary fractions help 4th grade [closed]

Liam puts $10$ marbles in each bag and $4/10$ are stripped. Part a. Liam makes $3$ bags how many are stripped? Part b. Liam used $20$ striped marbles to fill bags how many non striped marbles did ...
-1
votes
1answer
34 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 ...
101
votes
13answers
14k 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
vote
2answers
51 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 $xyz$?
1
vote
2answers
38 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
90 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
votes
2answers
36 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 ...
0
votes
0answers
23 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
vote
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
votes
2answers
42 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 ...
1
vote
0answers
23 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, ...
27
votes
10answers
5k 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
vote
1answer
33 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 ...
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votes
0answers
19 views

Arithmetic problems with understanding

I'm having problems with understanding qn 22. Part (III) don't really know what are they asking :|
3
votes
1answer
156 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
votes
1answer
41 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 ...
0
votes
1answer
22 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
votes
1answer
11 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% ...
1
vote
2answers
32 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 .
2
votes
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 ...
11
votes
6answers
151 views

Simplifying nested square roots ($\sqrt{6-4\sqrt{2}} + \sqrt{2}$)

I guess I learned it many years ago at school, but I must have forgotten it. From a geometry puzzle I got to the solution $\sqrt{6-4\sqrt{2}} + \sqrt{2}$ My calculator tells me that (within its ...
2
votes
2answers
30 views

Greatest Common Divisor of a+b,a-b

Prove that $gcd(a + b, a − b) ≥ gcd(a, b)$ Let $d=gcd(a+b,a-b)$ So $d=m(a+b)+n(a-b) = a(m+n)+b(m-n)$ Which implies $d|a$ and $d|b$ Therefore, $d|gcd(a,b)$ $gcd(a,b)=dx ≥ d= gcd(a + b, a − b)$ Why am ...
0
votes
0answers
35 views

Divide and Conquer division algorithm explained

I am trying to understand the divide and conquer algorithm that is used in the GMP bignum arithmetic library. The code is very optimised and that makes it somewhat hard to understand. the doc does ...