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

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5
votes
4answers
398 views

General formula of repeated roots.

Prove that $$\underbrace{\sqrt{k\sqrt{k\sqrt{k\sqrt{\cdots\sqrt{k}}}}}}_{n\text { times}}=k^{1-1/2^n}$$ How do I derive this formula?
3
votes
2answers
54 views

Square grid , sum of elements

I am trying to solve the following problem : Find all the positive integers $n$ and $k$ such that it is possible to write integers in an $n \times n$ grid so that the sum of all elements in the grid ...
0
votes
2answers
49 views

How do break down this addition?

I've been given the following expression: $2(a + b) + (n + 1)(2a + c) + 2n(2a + d + b) + (a + r)$ And I've been told that it can be simplified to: $n(6a + 2b + c + 2d) + (5a + 2b + c + r)$ I've ...
4
votes
3answers
338 views

The proof of $\sqrt{2}$ is not rational number via fundamental theorem of arithmetic.

I assume that $\sqrt{2}$ is positive number satisfies $(\sqrt{2})^2=2$. proof) Let $m$, $n$ as natural number,$\ $ $M$ is the number of prime factor of $m$,$\ $ $N$ is also the number of prime ...
0
votes
1answer
107 views

Divisibility rule for 22

Divisibility rule for 22: Under what conditions a natural number $N$ is divisible by $22$ ? My thought is The divisibility rule for $22$ is that the number is divisible by $2$ and by $11$. ...
1
vote
1answer
22 views

Calculation of a value for number divisible by 11

Calculates the value $x$ for the number $M=5278x$ is divisible by 11 my attempt, $11\mid M=5278x \Longleftrightarrow (5-2)+(7-8)+x=2+x$ is multiple of $11$ $ 2+x$ is multiple of $11 ...
2
votes
2answers
80 views

Divisibility rule of 11

Let $M$ be a natural number with $n+1$ digits; represented by $M=a_{n}a_{n-1}\cdots a_{2}a_{1}a_{0}$ Show $M$ is divisible by $11$ if and only if ...
20
votes
10answers
3k views

Get $5$ by doing any operations with four $7$s

How can one combine four sevens with elementary operations to get $5$? For example $$\dfrac{(7+7)\times7}{7}$$ (though that does not equal $5$). I am not able to do this. Can you solve it or prove ...
3
votes
0answers
78 views

What was babylonians estimation for square root 3?

We see a lot of papers and talk about ancient Babylonians exactness of calculating the value of square root of 2. For example: ...
2
votes
1answer
35 views

Represented in basis X

Let ABCD represent the digits of the starting number. The four digit number would be represented in basis $X\in \mathbb{N}$ by : $$\textrm{ABCD}=X^{3}.A+ X^{2}.B+ X^{1}.C+ X^{0}.D$$ Am I ...
1
vote
2answers
101 views

How to simplify $\sqrt{\sqrt{5}+1} \cdot \sqrt{\sqrt{5}-1}$?

This is the original problem: $\sqrt{\sqrt{5}+1} \cdot \sqrt{\sqrt{5}-1} = x$. I'm really confused about how to solve this problem, I come as far as saying this: $\sqrt[4]{5} + \sqrt{1}\cdot ...
3
votes
4answers
53 views

What's the intuition for why repeated div and mod converts a number to another base?

This guide here tells how to base 10 number to binary. For example, for the first bit, you take $356_{10} \div 2 = 178 \, R \, 0$. Because the remainder was 0, the first bit is $0$ and we recurse ...
2
votes
1answer
207 views

Turing machines that compute $\pi$

For each $K > 0$ there is a brut force Turing machine $\pi_K$ that "computes" the first $K$ digits of $\pi$ starting on the blank tape (all $b$s) with $K+1$ states $S \in \mathsf{S} = ...
0
votes
3answers
121 views

Particular number is divisible by 11

Let $\mathcal{N} \ $ be a natural number of the form $\mathcal{N}=\textrm{dcba}$ ($a$ being the number of units $b$ the tens digit $c$ the hundreds digit and $d$ the thousands digit). On what ...
0
votes
0answers
16 views

Achieving a polynomial that maps from $GF(p^q)$ to {0,1} with the same probability

I am using an arithmetic circuit, which can compute polynomials over the field $GF(p^q)$. I need a polynomial that maps any element from the field to an element from $\{0,1\}$, I need that the range ...
0
votes
0answers
55 views

Arithmetic Turing machines

Consider the family $T_{1}$ of Turing machines with two tape symbols $b,1$ ($b$ the blank symbol). The family $T_{1}$ is Turing complete. Identify the tape with $\mathbb{Z}$ and let $0\in \mathbb{Z}$ ...
2
votes
2answers
110 views

congruence issue

I need to understand why this : $$(1+4+\ldots+4^{n−1})\equiv n \pmod3$$ Is that because \begin{align} 1&\equiv -2 \pmod3\\ 4&\equiv 1 \pmod3\\ 4^{2}&\equiv1 \pmod3\\ ...
0
votes
0answers
46 views

Redundant Binary Representation

Is it possible to have a technique using Redundant Binary Representation so that repeated addition can be obtained with no carry propagation time?
3
votes
2answers
127 views

Why isn't the identity $\sqrt{ab}$ = $\sqrt{a} \cdot \sqrt{b}$ always true?

If we take $a=b=-1$ then the L.H.S. is $1$ but the R.H.S. is $-1$. Is this identity not applicable for complex numbers? How to prove this and prove that this is not applicable for some complex ...
0
votes
2answers
90 views

Finding a formula for a pattern

I have this pattern which is an infinite sequence (I have placed commas so it's easy to see the pattern)... $1 ,1 2, 1 2 3, 1 2 3 4, 1 2 3 4 5 ...$ Is there any formula governing this sequence, ie, ...
0
votes
4answers
61 views

An example of how to solve equation over $\mathbb{Z}$

I found an example of how to solve equation over $\mathbb{Z}$ Example Solve the equation over $\mathbb{Z}$ : $$xy + 1 = 3x + y. $$ $$ xy = 1 + 3x + y \Longleftrightarrow (x-1) (y-3) = ...
0
votes
2answers
54 views

How many boys, girls, men and women are there?

In a village, there are exactly $10$% more boys than girls; $15$% more women than men; $20$% more children than adults. The population is less than $6000$. Solution: $b = g + 0.1g$--------(i), ...
16
votes
3answers
282 views

Is$\frac{\sqrt{a}}{\sqrt{b}}$ the same as $\sqrt{\frac{a}{b}}$?

My idea is that the two functions are not the same since for the first function, the domain of the function is only non negative reals for the numerator and positive reals for the denominator. ...
0
votes
2answers
35 views

multiply fraction with what number to get a whole number?

I'm solving some programming puzzle and it has come down to this: I've a fraction, say 12/13, and I need to multiply it with a smallest possible natural number (say x) to get a whole number. How do I ...
3
votes
2answers
44 views

Number of $1$s in the binary representation of $n$

Trying to define the function $b(n)$ which counts the number of $1$s in the binary representation of $n$ arithmetically I came up with the following definition: $$b(n)=m :\equiv (\exists k_1\dots ...
2
votes
3answers
473 views

A hard square root question

This is my first question on StackExchange. So my question is: If $$x = \frac{\sqrt{\sqrt5 +2}+\sqrt{\sqrt5-2}}{\sqrt{\sqrt5 + 1}} + \frac{\sqrt{\sqrt5 +2}+\sqrt{\sqrt5-2}}{2\sqrt{\sqrt5 + 1}} - ...
7
votes
3answers
154 views

Confused about the $\pm$ sign?

I have multiple questions about the $\pm$ sign, since it seems to confuse me in general... Question 1: Say I have $15=\pm(a+x)$, Can I use the distributive property so it becomes $15=\pm a \pm x$? ...
1
vote
2answers
78 views

Compensation Question

I want to create a compensation system which takes into account two variables. Lets say I have $1M to distribute among ten employees who produce widgets. I want to compensate each employee by two ...
1
vote
1answer
69 views

Prove an addition property of Natural numbers

Prove: For any $x,y \in \mathbb{N}, y \neq x+y$. I'm only suppose to use the Peano axioms as defined here http://aleph0.clarku.edu/~djoyce/numbers/peano.pdf and the properties of addition in ...
2
votes
6answers
84 views

Why is the result of $-2^2 = -4$ but $(-2)^2 =4$?

I am really new into math, why is $-2^2 = -4 $ and $(-2)^2 = 4 $?
0
votes
1answer
26 views

How can we generate a $2$-digit number $XY$ on base $B$, such that $BX+Y=Y^X$?

For example, $25$ on base $10$ is equal to $5^2$. This should be pretty easy to solve using fairly simple arithmetic. But I'm finding it hard to generate any other solutions besides the one ...
5
votes
5answers
338 views

The number $(3+\sqrt{5})^n+(3-\sqrt{5})^n$ is an integer

Prove by induction that this number is an integer: $$u_n=(3+\sqrt{5})^n+(3-\sqrt{5})^n$$ Progress I assumed that it holds for $n$ and I tried to do it for $n+1$ but the algebra gets quite messy and ...
0
votes
3answers
32 views

How to solve for x in this eq

I'm doing a physics E&M problem, but I'm stuck on a math part. I can't remember how to solve for x in this instance. ${x\over 2} = {0.04-x\over 5}$
0
votes
1answer
22 views

How to break changes in ratios into two changes?

I am running into a real world problem. But I think this is more like a math problem. So here it is. Suppose I have $$A = B + C.$$ The $A, B, C$ in this period are called $A_{1}, B_{1}, C_{1}$. ...
2
votes
3answers
61 views

Ambigous question regarding how to view surds with numbers infront

Say I want to multiply 2 by 5$\sqrt3$ . Do I firstly do 2 * 5, then 2 * 3? I'm not sure about the order of operations here. Such a dumb question, I know. Edit - can someone show me the systematic ...
0
votes
0answers
38 views

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 ...
0
votes
2answers
70 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 $$ 5(5-2) - 2(5-2)$$ $$25-10-10+4$$ ...
0
votes
2answers
36 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 ...
3
votes
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 ...
9
votes
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?
5
votes
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. $$\sqrt{2}=1+\sqrt{2}-1=?=1+\frac{1}{\sqrt{2}+1}$$
1
vote
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 ...
0
votes
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'?
0
votes
3answers
283 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 ...
3
votes
2answers
186 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 $a^0$ ($a\neq 0$) ...
0
votes
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, ...
0
votes
1answer
84 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 ...
1
vote
2answers
39 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 ...
0
votes
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.
2
votes
2answers
75 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 ...