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

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Extensions by recursive definitions

In the Wikipedia entry on Extension by definitions I learn that an explicit definition in the language of a theory $T$ yields a conservative extension $T'$ of $T$. I wonder if this eventually does ...
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2answers
69 views

Squared binomial paradox?

When you square this $$(5-2)^2$$ you will get 49 $$ 5^2 - 2 * 5 * (-2) + (-2)^2$$ $$25 + 20 + 4 = 49$$ but if you do it like this (5-2) * (5-2) you will get 9 $$ 5(5-2) - 2(5-2)$$ $$25-10-10+4$$ ...
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2answers
34 views

Looking for a Formula to apply to a set of numbers (input) that will output a certain result.

Sorry for the crude title: I'm looking for a formula to apply to each element of an "input set" of numbers that will output elements in another "output set" with the following characteristics: The ...
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2answers
50 views

Automorphisms of $\langle \mathbb{N}, \cdot \rangle$

It is an elementary fact that multiplication in $\mathbb{N}$ is commutative: $$(\forall n,m)\ n \cdot m = m \cdot n$$ This - among other things - implies that the representation of an $n \in ...
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4answers
455 views

Expression with last digits different

Given the expression: $$1234567893 \times 1234567894 - 1234567895 \times1234567892$$ Is it correct to say that the answer is $ (3 \times 4) - (5 \times 2) $? If so, why?
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5answers
143 views

How $\sqrt{2}=1+\frac{1}{\sqrt{2}+1}$?

I have found it in the chapter about chain fractionals. I am unable to transform it to such state. $$\sqrt{2}=1+\sqrt{2}-1=?=1+\frac{1}{\sqrt{2}+1}$$
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1answer
50 views

Recursive definitions of $n<m$, $n\mid m$, and $n \bmod m$

Without referring to the apparatus of (primitive) recursive functions one can introduce addition into the language of successor arithmetic by two additional axioms which naturally reflect the essence ...
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1answer
42 views

Why is every number which ends in 5 divisible by 5?

Is there more of an answer to this which is more than just 'it does'?
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3answers
253 views

Derivation of the “Combined Work Formula”

Before I get to my question, some background: Person $A$ can paint a fence at the rate $9 \frac{hour}{fence}$ (or equivalently $\frac{1}{9} \frac{fence}{hour}$) Person $B$ can paint a fence at the ...
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2answers
185 views

Why we can't define $\frac{1}{0}$ to be $1$ (or anything else), but we can define $1^0$ to be $1$?

We know that we can't define division by zero "in any mathematical system that obeys the axioms of a field", because it would be inconsistent with such axioms. (1) Why can we define $a^0$ ($a\neq 0$) ...
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0answers
26 views

Adding a fixed quantity of something to two different sized containers yields a different result

For example: An object has 29,880 health. It starts out at 10% of that, and in order to get it to full health you need to add 30 health packs to it. So we can calculate that each health will add 897, ...
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1answer
83 views

Finding distance using rates of change — best approach?

The question: A man drives from state $A$ to state $B$ going $60 \frac{miles}{hour}$. Then he returns from state $B$ to state $A$, driving $45 \frac{miles}{hour}$. His total driving time is $2.5 ...
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2answers
38 views

Overall difference in percent

I want to calculate the total difference in % between two investments {A,B} in the following scenario: In year t=0 revenue A is 70 % smaller than revenue B. Every year the revenue from A further ...
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1answer
84 views

Is this real number an integer?

Is this real number : $$\Big(2+\frac{10}{9}\sqrt{3}\Big)^{1/3}+\Big(2-\frac{10}{9}\sqrt{3}\Big)^{1/3}$$ an integer ? I've tried different factorization, but nothing seems to work.
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2answers
74 views

Proving that a number is non-negative?

The numbers $a$,$b$ and $c$ are real. Prove that at least one of the three numbers $$(a+b+c)^2 -9bc \hspace{1cm} (a+b+c)^2 -9ca \hspace{1cm} (a+b+c)^2-9ab$$ is non-negative. Any hints would be ...
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1answer
49 views

Find out the no of digits in product between some prime.

How many digits are there in? $2^{17}*3^{2}*5^{14}*7$. help me.
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1answer
21 views

Arithmetic regarding area under curve

I am looking at a textbook example but I am not able to go from where I have set the question mark and to the where the arrow is pointing. Could someone please explain where the x's are coming from ...
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6answers
2k views

Is there a way to prove that $2+2$ really equals $4$?

In elementary school, one learns that $2+2=4$ by experiment (putting two apples next to two other apples), and maybe also from some addition table to be memorized. But is there any approach that ...
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2answers
47 views

Different arithmetics

The original Peano axioms were based on a single unary operator $\operatorname{succ}$ and one second-order induction axiom: $\lbrace \operatorname{succ} \rbrace + \operatorname{IND}_2$ Peano ...
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2answers
35 views

Arithmetic simplification

I am asked to find $\frac{d^2y}{dx^2}$, and prove that $\frac{d^2x^2+y^2=a^2}{dx^2}$=$-\frac{a^2}{y^3}$, This is how I have proceeded: $2 y \frac{dy}{dx}+2 x=2 a \frac{da}{dx}$ => ...
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1answer
40 views

Simplifying an equation step by step

I am not able to go from the left hand side to the right hand side. Could someone take me through the steps, I tried using common denominator (y^2-x)^2, but was not able to get the result on the right ...
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5answers
81 views

Determine variables that fit this criterion…

There is a unique triplet of positive integers $(a, b, c)$ such that $a ≤ b ≤ c$. $$ \frac{25}{84} = \frac{1}{a} + \frac{1}{ab} + \frac{1}{abc} $$ Just having trouble with this Canadian Math ...
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0answers
53 views

Inverse element of “-”

What is meant by the inverse element of "-"? There is a statement in my book that says there exists an inverse element of "-" in $\mathbb{R}$ and I have to mark it true or false. I know that the ...
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1answer
41 views

Comparing Fractional Numbers

Does a formula exist for comparing two fractional numbers, without resolving to using anything other than integers and fractions? (Thus not real numbers). In other words: given $\dfrac{a}{b}$ and ...
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5answers
77 views

Arithmetic Progression.

Q. The ratio between the sum of $n$ terms of two A.P's is $3n+8:7n+15$. Find the ratio between their $12$th term. My method: Given: $\frac{S_n}{s_n}=\frac{3n+8}{7n+15}$ ...
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14answers
6k views

Logic behind dividing negative numbers

I've learnt in school that a positive number, when divided by a negative number, and vice-versa, will give us a negative number as a result.On the other hand, a negative number divided by a negative ...
3
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2answers
157 views

Simple subtraction that I can't figure out. [duplicate]

A bat and a ball cost £1.10 in total. The bat costs £1 more than the ball. How much does the ball cost? The answer to this question is somehow 5p. How?!! Should it not be 10p?
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1answer
118 views

Having problem with 10's complement subtraction

From what I've found, to find A - B using 10's complement; where A and B are decimals Let A = 215 , B = 155 Find 10's complement of B = (1000 - 155) = 845 Add 10’s complement of B to A If it ...
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3answers
276 views

How to prove: $\left(\frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt[4]{25}-\sqrt[4]{125}}}-1\right)^{4}=5$?

Question: show that: the beautiful ${\tt sqrt}$-identity: $$ \left({2 \over \sqrt{\vphantom{\Large A}\, 4\ -\ 3\,\sqrt[4]{\,5\,}\ +\ 2\,\sqrt[4]{\,25\,}\ - \,\sqrt[4]{\,125\,}\,}\,}\ -\ ...
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2answers
55 views

Common divisor of the form d = ax+by

There is a theorem that says that every pair of integers $a$ and $b$ has a common divisor $d$ of the form $d = ax+by$ where $x$ and $y$ are integers. Is it true that $d$ is also definitely the ...
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4answers
126 views

Explaining multiplication of fractions

The best way I've been able to describe multiplication is as $$ a\times b = \sum^a_{i=1} b$$ But my definition does not account for things such as $2.99792458\times8.987551787$ and ...
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1answer
42 views

Vector Image Representation on 640X480 Screen

Im trying to learn Computer Graphics. I have the following statement For the representation of vector images, we assume that a typical image consists of 500 lines [BHS91]. Each line is defined by its ...
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3answers
241 views

Algorithm for multiplying numbers

Background Today I had to explain to some kid how to multiply numbers with multiple digits in them. Then I recalled, that some other day I answered this question describing one of the numerous ...
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4answers
77 views

Conjecture for product of binomial coefficient

Is it true that for any $n, k\in\mathbb N$ $$\frac{(kn)!}{k!(n!)^k} = \prod_{l=1}^k {{ln-1}\choose{n-1}} \quad?$$ I tested it for some small $k$ and $n$, but I don't know how to prove that it is true ...
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0answers
50 views

Given a set of nonnegative numbers, put $\pm$ between them to minimize the magnitude of the result

Let's say I have a finite set of non-negative numbers. I have to put $+$ or $-$ between the numbers, in order to minimize the absolute sum.(i.e the sum has to be closest to 0) For example: the set: ...
2
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1answer
142 views

Should BODMAS not be BODMSA?

Let me and blindly follow for a second BODMAS. $$ \begin{align*} 1-3+2 &=1-(3+2)\\ &=1-5\\ &=-4 \end{align*} $$ (The brackets are just to make the error clear - I wouldn't write them in ...
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2answers
98 views

Prove that $a^3 + 2b^3 + 4c^3 − 6abc \neq 0$ if at least one of $a$, $b$, and $c$ are non-zero [closed]

Prove for $a, b, c \in \mathbb Q$ that $a^3 + 2b^3 + 4c^3 − 6abc \neq 0$ if at least one of $a$, $b$, and $c$ are non-zero without resorting to field theory or linear algebra.
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2answers
62 views

What kind of mathematical operation is used to repeatedly increase a number by a certain percentage?

I am sure that this is an easy question to answer for most of you. I need to take a number, let's say $10$, and then increase it by a percentage, let's do $25\%$. $10 \times 1.25 = 12.5$ Easy ...
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2answers
105 views

Find a number by the decimal part of its square root [duplicate]

I have a math problem consisting of two questions: can we find a number N knowing only the decimal part of its square root up to a precision (only an approximation of the decimal part because the ...
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2answers
186 views

What is the name for a number plus itself?

I'm asking out of curiosity more than anything else. A number multiplied by itself is squared, is there a specific mathematical term for a number plus itself? $(n*n = n^2, n+n=?)$
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2answers
64 views

Evaluate $\frac{\frac{1}{1}}{\frac{1}{5^{-2}}}$

I solved a question in the Manhattan GRE 5 pound book (specifically the 11th question in the Exponents and Roots section). I evaluated $\frac{\frac{1}{1}}{\frac{1}{5^{-2}}}$ as $5^{-2}$ and then ...
2
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2answers
78 views

square root solutions

Is there a specific rule to get square root of any non-negative number?. The main reason why I'm asking this is that my maths teacher told me there is only one solution can be contained for any ...
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2answers
41 views

Greatest possible length to measure given lengths

I was given a question to find a greatest possible length to measure $495$, $900$,$1665$ (in centimetres). The solution is finding GCF or GCD or HCF (highest common factor) of these numbers which is ...
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10answers
305 views

Difference between $\sqrt{x^2}$ and $(\sqrt{x})^2$

According to my logic, $$\large\sqrt{x^2} = x^{2\times \frac{1}{2}} = x = x^{\frac{1}{2}\times 2}={(\sqrt{x})}^2$$ But when I look at the graphs of these guys, they're totally different. Edit: ...
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3answers
35 views

Percentages of $\$100,000$

$\$6,200$ is $6.2\%$ of $\$100,000$. That leaves $\$93,800$ as $93.8\%$ of that $\$100,000$. But when I take $\$93,800$. and multiply it by $6.2\%$, I get $\$99,615.60$ instead of $\$100,000$. Why is ...
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0answers
55 views

Number theory/proof help appreciated! Includes working.

my question concerns number theory and proofs. I have shown my working for the following two questions but I'm not too sure if I am correct - I feel as if this might be a trick question. Any feedback ...
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3answers
106 views

What is the smallest number x such that $180\times x$ is a perfect cube?

What is the smallest number x such that $180\times x$ is a perfect cube?
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5answers
325 views

Mathematical proof for order of operations

I was watching this YouTube video and at around 40:40 the speaker himself states that he does not know why we have the order of operations we have today. This got me thinking and I realize that I ...
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1answer
48 views

Is this correct - Rs.1 is equal to 1 paisa

yesterday got a SMS, and can any one explain it.... ...
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2answers
59 views

A property of proportions: if $a/b=c/d$, then $(ma+nb)/(pa+qb)$ is equal to $ (mc+nd)/(pc+qd)$

If $\large\frac{a}{b}=\frac{c}{d}$ how we can obtain $\displaystyle{\frac{ma+nb}{pa+qb}=\frac{mc+nd}{pc+qd}}$? I can get $\large\frac{ma}{qb}=\frac{mc}{qd}$ and $\large\frac{nb}{pa}=\frac{nd}{pc}$ , ...