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

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Some geometric equations

Correct me if I'm wrong by saying that the circumference of the circle equals to $180\times 1 $ rad where 180 and 1 are constants and the volume of the sphere equals to $30\times 1$ rad$\times 1^3$ ...
3
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1answer
67 views

Why does $(n+1)^n\lt n^{n+1} \implies \left(1+\frac{1}{n}\right)^n\lt n$?

During an example done in lecture, I encountered an inequality by the form of $$(n+1)^n\lt n^{n+1}$$ My professor immediately simplified it to $$\left(1+\frac{1}{n}\right)^n\lt n$$ I have attempted to ...
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1answer
26 views

Elementary operations

There are four elementary arithmetic operators. Are all operations in mathematics derived from the four elementary arithmetic operators? I'm studying linear algebra and noticed that some exercises ...
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2answers
198 views

Number of sudokus with no consecutive arithmetic progression of length 3 in any row or column.

How many such Sudokus are there? Any reference to papers, books, articles or any insight into the problem will be greatly appreciated. I've tried several search engines, scholarly and not, with no ...
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1answer
27 views

Chopping arithmetic

I would like to use chopping arithmetic for 3 digit chopping for the following: $a)\pi$ and $b) 456788.1234567$ My guess is that it is $3.141$ and $456788.123$, but my book says the pi after ...
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2answers
21 views

if $Re(z\overline w)=|z||w|$ then $w=tz$ where $t\in \mathbb R$

I was given the following exercise in complex numbers: Assume that for some $z,w \in \mathbb C$: $Re(z\overline w)=|z||w|$ Show that this implies $w=tz$ where $t$ is some positive real. What I ...
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1answer
18 views

Difficulty understanding the solution to a problem.

I am studying a book right now, and I'm having a difficulty understanding a (solved) problem regarding congruent modulo. Below I will list the problem and what I have understood of the problem, along ...
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3answers
91 views

Why is $2x-1=7$ not $x=-4 \text{ or } x=4$

How would you explain why $3(2x-1)^2=147$, is $2x-1=7 \text{ or } 2x-1=-7$. But not $2x=8 \text{ so } x=4 \text{ or } x=-4$?
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Calculate relative rate of percentage change

This question could probably just as easily go on quant.SE, but the problem I am having is mathematical in nature, so the domain hopefully shouldn't matter. The problem: Say you have two moving ...
3
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2answers
126 views

Why is $x(x+2)(x-3)$ not $x^2+2x(x-3)$?

How would you explain the principle why $x(x+2)(x-3)$ is not $x^2+2x(x-3)$ but $(x^2+2x)(x-3)$? This may involve the fundamentals of eliminating parenthesis.
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2answers
459 views

How are the elementary arithmetics defined?

In the book Principles of Mathematical Analysis by Rudin, I read that "a < b" is defined this way: if b - a is positive, then a < b or b > a. Then some questions arose to me: we know that ...
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0answers
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Is there a numeral system that makes both addition and multiplication easy?

Decimal positional notation, the system for writing numbers we all use every single day, makes addition very easy by transforming it from a computation to a repeated operation on individual digits ...
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1answer
29 views

Solving a non-linear congruence

How would I go about solving the following: $$x^2+1\equiv2\pmod8$$ I know I can subtract, and I get: $$x^2\equiv1\pmod8$$ I'm not sure if I am allowed to square root both sides, or if I should ...
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2answers
37 views

$\sqrt{1\pm10\varepsilon+\varepsilon^2}=1\pm P(\varepsilon)$. Is there a better way than mine to find $P(\varepsilon)$?

Some days ago we did a classwork, and there was this exercise: Using the limit definition, verify $$\displaystyle \lim_{x\to0} \frac{3x^2-1}{x+1}=-1.$$ From $\displaystyle ...
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1answer
12 views

Quick question about functions distributing over addition, multiplication, minimum and injection.

So consider for example I have the function $$F(x) = -3\cdot x$$ I have to prove that either true or false and then prove my answer if false with a counter example and justify fully a true answer. ...
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1answer
66 views

Expressing a number as a finite sum of terms $a_k/k!$

The original problem is as - Suppose $a_2,a_3,a_4, \cdots a_7$ are integers such that $$\frac{5}{7}=\sum_{k=2}^7 \frac{a_k}{k!}$$ where $0 \le a_j < j$ for $j=2,3,4,5,6,7$ Then find $a_2+a_3+ ...
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2answers
27 views

linear equation for vertical line

If we want to graph a horizontal line, we will do the following: y = 0x + 3 No matter the domain for x, the range for y will always be 3. Therefore, we have a ...
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1answer
55 views

How am I supposed to convert between a teaspoon and a tablespoon?

I've sometimes been having trouble figuring out the conversion between a teaspoon (tsp.) and a tablespoon (tbsp.). How will I know when to multiply or divide by three to do these conversions? I ...
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2answers
55 views

Why are there only parts of the digits of the decimal base (10) in bases lower than the decimal base? [closed]

I've noticed that some bases such as binary (2) and octal (8) have digits that are only part of the decimal base, or 10. In the binary base, only the digits 0 and 1 are in the base and in the octal ...
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1answer
78 views

Proofs involving Fermat's Little Theorem

Let $p$ be a prime and $a$ be an integer such that $p$ does not divide $a$. Suppose $d$ is the smallest positive integer such that $a^d ≡ 1 \ ($mod$ \ p)$. Prove that $d|(p−1)$. So far I've ...
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2answers
42 views

Get values of multiple variables from simple additions

What is the process to get the values of each variable when I only have these calculations to go on? These are the sums: ...
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1answer
33 views

Two questions concerning divisibility

I was looking at some proof questions and had difficulty answering a few of them How do I prove these statements below: 1) $3 \mid (10^{n+1} + 10^n + 1)$ 2) $(a-b) \mid (a^n - b^n)$
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2answers
70 views

Is there a way to simplify this equation

Is there a way to simplify this equation? $$ CE = 2 FD \sin \left( \arctan \left( \frac{AF}{FD} \right) - \arccos\left( \frac{AB}{\sqrt{AF^2+FD^2}}\right) \right) + \sqrt{AF^2+FD^2-AB^2} $$ Edit: ...
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36 views

Modular arithmetic Proofs

For all $a$, $m \in\mathbb{Z}$, prove that For all $x \in [a]$, where $[a]$ is the congruence class of $a \pmod m$, $\quad\gcd(x,m)=\gcd(a,m)$. I have no idea where to start for this.
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45 views

Modular Arithmetic Involving Remainders

Using modular arithmetic, solve the following: Find the remainder of $(2014^{2015} \cdot 2016^{2017}) + 2018^{2019}$ when it is divided by 13. I have no idea where to start. I've tried putting this ...
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3answers
47 views

Mowing the lawn alone vs. together

The question is: Nils mows the lawn in 3 hrs alone. Nils and Jonas mow the lawn together in 2 hrs. How quickly would Jonas mow the lawn alone? My thinking: If Nils mows the lawn alone in 3 hrs, he ...
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2answers
711 views

What's wrong with this proof that commutativity is implied by the other field axioms?

I seem to have found a proof that the commutativity of $+$ follows from the other field axioms. It is as follows: Let $(k,+,\cdot)$ be a structure satisfying all field axioms except commutativity of ...
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1answer
45 views

How do expressions like “more than” and “is more than” have different meanings?

I've looked up in my present math book that expressions like (1) "less than" and "is less than" and (2) "more than" and "is more than" have different meanings. I saw that "less than" indicates ...
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1answer
38 views

A simple arithmetic problem

In a field grass grows at uniform rate ; If the field can feed 36 cows for 4 days , 21 cows for 9 days , how many cows can be feed for 18 days ? My solution : Let $x$ be the initial amount of ...
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1answer
45 views

binary addition

Can any direct me to any resources online that teach how to approach binary addition such as this/ working with more complex binary arithmetic? I know the basics of binary addition and carrying the ...
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2answers
43 views

Square root and principal square root confusion

A few months ago I asked a question about the $\pm$ symbol because I was confused about it... I still carry the same confusion (which really bugs me) but I think the real confusion has to do with the ...
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3answers
59 views

Numerical problems [closed]

Arrange the following in ascending order 3 to the power 34, 2 to the power 51, 7 to the power 17. How? Also, please explain.
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1answer
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is this set closed under addition?

I have some revision questions in my maths books and I'm a bit stuck on this one. Is $S=\{n^2:n \in \mathbb{Z}\}$ closed under the usual addition. I know that for it to be closed the sum of any 2 ...
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2answers
98 views

Origin or author of 'Japanese Multiplication Method'

What is the origin or author of the method 1 shown in the following image? Notes Also known as Japanese Multiplication Method for Kids.
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Appreciating the Distributive Law

I'm going to introduce my middle school students to the distributive law in arithmetic, in a meaningful way so that they understand its importance and value. I need some interesting examples and ...
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What are different ways to do the partial-quotients method?

I've seen this method in a math book and I was probably obsessed with it. To cut to the chase, I need to know the different ways to do this. Now, what are some specific ways you have?
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1answer
80 views

Integral solutions of $x^5-27y^3=2x$

Find all integers $x$ and $y$ such that $x^5-27y^3=2x$.
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2answers
126 views

Two math professors problem

My friend asks me a question from internet. The question is as follows Two math professors, professor Uno and professor Dos, play chess at the park while reminiscing about their past. Prof. ...
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6answers
82 views

How to know a number is divisible by a given number without using a calculator?

My question is simple and comes from my curiousity during studying math. How to know a number is divisible by $7$ or $13$ without using a calculator? For example, how do we decide intuitively that ...
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2answers
41 views

solve this congruence please.

$7^{n+1} -(n+1)\times 7^{n} -1 ≡ 0 $ (mod 4) with the variable n as an exponent you can't use a modulo 4 table, which is why it bothers me a bit.i tried messing around with it and i got that this ...
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1answer
25 views

reducing the modulus of a Dirichlet character

Let $\chi$ be a Dirichlet character modulo $N$. Let $M$ be a positive divisor of $N$ such that $$\text{radical}(N)=\text{radical}(M).$$ Is $\chi$ be a character modulo $M$? Best regards.
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1answer
23 views

if $p=(a+ib)(c+id)$ and $p^2 = a^2 + b^2$ then $p\mid a$ & $p\mid b$

We're working on Gauss integers... p is an odd prime such that $p \not\equiv 1 \pmod 4$. We want to prove that if there is $(a,b,c,d) \in \mathbb{Z}^4$ such that $$p = (a+ib)(c+id) \text{ ...
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1answer
38 views

how to calculate gross salary when net salary and percentage of deduction is known

When net salary is 10350 and 13.75% pf deducted how gross amount is calculated.. For eg: If gross salary is 12000, pf is 13.75% then net salary= 12000* 13.75% = 10350 But how to ...
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$\lim_{n\to\infty}\sqrt{6}^{\ n}\underbrace{\sqrt{3-\sqrt{6+\sqrt{6+\dotsb+\sqrt{6}}}}}_{n\text{ square root signs}}$

We have the following representation of pi: $$\pi=\lim_{n\to\infty}2^n \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\dotsb+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}}}}}}_{n\text{ square root ...
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2answers
40 views

Proving arithmetical properties for non-natural numbers

Sorry if my question is dumb but here it is: I know how to prove all of the arithmetical properties such as $(a^{m})^{n}=a^{mn}$ and $a^{m}a^{n}=a^{m+n}$ and $a(b+c)=ab+ac$ etc. For numbers that ...
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How do you track each investor's equity stake when the total amount changes?

Let's say there's a casino with a bankroll of $0. User1 invests $100 and now owns $100/100. User2 invests ...
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1answer
352 views

Need help with GRE question

I encountered a question while preparing for GRE and am stuck. In an examination paper of 5 Questions, 5 percent of the candidates answered all of them and 5 percent none. Of the rest, 25% ...
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1answer
82 views

Converting double precision IEEE 754 hex to base 10 with repeating decimals

The number is 0x4001 8CCC CCCC CCCC. So far I have the stored exponent as 1000000000 which equals ...
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5answers
231 views

How do I solve this fraction addition problem?

$4\frac{2}{9} + -9\frac{1}{2}$ yeilds result of $-5\frac{13}{18}$ but WolframAlpha says the answer is $-5\frac{5}{18}$ fixed.
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Calculate Hourly vacation accrual

I'm try to calculate vacation accrual for 40 hour week for an employee who is allowed 15 days vacation per yer.if the number of business hours per year is 2080,that means his accrual rate is 40/2080= ...