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

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2
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4answers
1k views

Limit of $\sqrt{x^2-6x+7}-x$ as x approaches negative infinity

What is $\lim\limits_{x\to-\infty}(\sqrt{x^2-6x+7}-x)$ ? Don't understand how to approach this question
3
votes
1answer
157 views

Understanding countable ordinals (as trees, step by step)

Even though ordinal numbers – considered as transitive sets – are perfect non-trees, it is worth (and natural) to visualize them as trees, starting from the finite ones which are given as ...
6
votes
1answer
58 views

Defining natural numbers without $0$ or $1$.

Let's define Peano's axioms having $2$ as the first number: $\newcommand\Nt{\mathbb N''}2\in\Nt$. $\newcommand\next{\mathop{\mathrm{next}}}\forall n\in\Nt:\next n\in\Nt$ (or $\next:\Nt\to\Nt$). ...
3
votes
2answers
68 views

Why is the coefficient in front of $\sqrt n$ always 1 in the intermediate terms for finding the continued fraction expansion of $\sqrt n$?

After playing around on paper for a bit, I came up with a short python generator to find the continued fraction expansion of $\sqrt n$. I understand why it gets the right answer when it gets an ...
1
vote
2answers
196 views

why does double rounding 9.46 give 10 but “regular” rounding gives 9?

What's the correct way to round, or estimate, a number to a specified precision? Starting with wikipedia: Rounding a number twice in succession to different precisions, with the latter ...
3
votes
0answers
138 views

Visualizations of ordinal numbers

I find this picture of the ordinal numbers up to $\omega^\omega$ rather hard to grasp: I wonder if the following might be a more compelling way to visualize ordinal numbers up to $\omega^\omega$: ...
0
votes
4answers
101 views

Why does $ \frac {\frac {1}{\sqrt{x}}}{x} = \frac {\sqrt{x}}{x^2} $?

A homework question recently asked for me to simplify: $\frac{1}{\sqrt{7}} \div {7}$ It's easy to see that this becomes $\frac{1}{7\sqrt{7}}$ But according to wolfram alpha this is also equal to ...
0
votes
1answer
128 views

The sum of two numbers equals 318, express the product of the numbers according to the lowest

I didn't understand the question on the title, what am I supposed to do?
2
votes
1answer
91 views

Show that $\frac{\sqrt[n]a}{\sqrt[n]{ab}+\sqrt[n]a+1}+\frac{\sqrt[n]b}{\sqrt[n]{bc}+\sqrt[n]b+1}+\frac{\sqrt[n]c}{\sqrt[n]{ac}+\sqrt[n]c+1}=1$ [closed]

If $$\sqrt[n]{{abc}} = 1,$$ Prove that $$\frac{\sqrt[n]a}{\sqrt[n]{ab}+\sqrt[n]a+1}+\frac{\sqrt[n]b}{\sqrt[n]{bc}+\sqrt[n]b+1}+\frac{\sqrt[n]c}{\sqrt[n]{ac}+\sqrt[n]c+1}=1.$$
1
vote
0answers
328 views

Property of arithmetic means?

$a,b,c,d \geq 0.$ It seems to me that this inequality is true and equality holds when $a=b=c=d$? $$\dfrac{a+b}{2}\dfrac{b+c}{2}\dfrac{c+d}{2}\dfrac{d+a}{2}\leq ...
0
votes
0answers
43 views

LCM of these polynomials

I'm having a hard time wrapping my head around how to get the lcm of these polynomials $$h(x+h+1), x+h+1, h(x+1)$$
0
votes
2answers
39 views

How do you simplify this expression?

$$\lim_{h\to0}(\frac{x}{h(x+h+1)} + \frac{1}{x+h+1} - \frac{x}{h(x+1)})$$ I know the answer is $$\frac{1}{(1+x)^2}$$ But I can't get there
0
votes
2answers
111 views

Combining products of like terms in a division

I was seeing an example on a book which says that: $$\frac{t}{t(x+t+1)} = \frac{1}{x+t+1}$$ The instructions read: "combine products of like terms" What exactly is that? Why does it work and why ...
1
vote
3answers
52 views

I need some basic introduction to limits

So, I know you can obviously cut out a value if it is multiplying and dividing something at the same time, right? Like: $$\frac{4h-2xh-h^2}{h} = \frac{h(4-2x-h)}{h} = 4-2x-h$$ But then I saw this ...
0
votes
0answers
53 views

Basic question about simplifying a square root

I just wanted to know how to get from $\sqrt{12}$ to $2\sqrt{3}$ Because my buddy was teaching me math the other day and gave me a list with some basic exercises to do, one of which is to solve ...
3
votes
2answers
43 views

Question about arc length.

I am having some trouble finishing an arc length problem. Specifically, what is $\int_{0}^{1}|x'(t)| dt=?$ Is it just $\int_{0}^{1} |x(t)| dt=|x(1)-x(0)|$? If so why?
0
votes
0answers
43 views

Proof that $ k^2<2^k$ [duplicate]

I'm struggling with this problem of proof by induction: For any natural number $k\geq 5$, prove that $k^2<2^k$. I assumed that $k^2<2^k$ I want to show that $(k+1)^2<2^{k+1}$ The ...
1
vote
1answer
91 views

Solving $[x]+[x]=[2x]$

Solving the equation $[x]+[x]=[2x]$ Since $[x]$ is the greatest integer function. I tried, $\forall x\in\mathbb{N}$, we have $[x]=x$ and $[2x]=2x$ this implies that $[x]+[x]=[2x]$, but if ...
2
votes
2answers
147 views

Sum of the reciprocals of divisors of a perfect number is $2$?

How do I show that the sum of the reciprocals of divisors of a perfect number is $2$? I tried $d_i\mid n$ with $i\in\mathbb{N},\;d_i\leq n$ then ...
2
votes
6answers
141 views

Proof that $n^3-n$ is a multiple of $3$. [duplicate]

I'm struggling with this problem of proof by induction: For any natural number $n$, prove that $n^3-n$ is a multiple of $3$. I assumed that $k^3-k=3r$ I want to ...
-1
votes
3answers
79 views

How does these square roots work? [closed]

$$\tag{1}2\sqrt{90} - 5\sqrt{160} + 3\sqrt{250} - 2\sqrt{40} = ?$$ $$\tag{2}\sqrt[3]{a\cdot b^2} \cdot \sqrt[4]{a^3 \cdot b \cdot c^2} = ?$$ $$\tag{3}\frac 6{2 \cdot \sqrt{3} - 3} = ?$$ picture
7
votes
4answers
6k views

Factorial, but with addition [duplicate]

Is there a notation for addition form of factorial? $$5! = 5\times4\times3\times2\times1$$ That's pretty obvious. But I'm wondering what I'd need to use to describe $$5+4+3+2+1$$ like the ...
0
votes
0answers
24 views

vector addition, would like to check my answers

If U = ( 1, 2 ) V = ( 3, -4 ) Is the answer to 2u + 1/2v ( 3.5, 2 ) I did the following: ...
5
votes
4answers
3k views

What's the algebraic property where you can flip the fractions in an equation?

Earlier in algebra, we spent over 20 minutes trying to figure out $$ \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{R_e} \,\,\,\, \text{solve for }R_2 $$ when the teacher said "What you start out with is ...
6
votes
2answers
221 views

struggle simplifying $\sqrt{9+\sqrt{5}}$

I need to simplify $\sqrt{9+\sqrt{5}}$ I already do this (proven it) $\sqrt{9-4\sqrt{5}}=2- \sqrt{5}$ But I couldn't when apply to ...
0
votes
0answers
667 views

Splitting an array into two subarrays with minimal sum

My question is if given an array,we have to split that into two sub-arrays such that the absolute difference between the sum of the two arrays is minimum with a condition that the difference between ...
0
votes
1answer
46 views

How to calculate sum?

As you can see in my screenshot, I take 25% from score and sum the result which come up to 64. My question is that how can I obtain the same 64 from sum of full(175) and score (144)? I have tried ...
0
votes
1answer
880 views

Time and Work in Unitary Method

Let us suppose, A and B can do a given work in 12 and 18 days respectively. They work alternately for equal period of time. And A started the work. Now, what is the time taken by A and B to complete ...
5
votes
1answer
67 views

$x;y;z\in \mathbb{Z}$ such that $(x-y)(y-z)(z-x)=x+y+z$. Prove that : $27\mid x+y+z$

Let $x;y;z\in \mathbb{Z}$ such that $(x-y)(y-z)(z-x)=x+y+z$. Prove that : $27\mid x+y+z$ Thanks :) P/s : I don't have any ideas about this problem..!!
0
votes
3answers
38 views

Adding and subtracting roots

I have to find (a/b), b=(1-(√3/3)), a= (1+ (1/√3)) I know I can just punch this into my calculator. However, my teacher says we need to know how to add these and subtract these but just as said that, ...
2
votes
2answers
60 views

show that $ \forall n \in \mathbb N$, $9\mid\left(10^n + 3\cdot4^{n+2} +5\right)$

Using congruence theory, show that $ \forall n \in \mathbb N$, $9\mid\left(10^n + 3 \cdot 4^{n+2} +5\right)$. The proof is quite simple with induction, but how can it be proved with congruences?
0
votes
3answers
148 views

how to solve factorial involving multiplication

I am trying to solve this question but not able to find any helpful material. It involves factorial with multiplications, $$\frac{8!}{5!}\cdot \frac{7!}{7!10!}$$ I tried crossing 8 and 5 and 7 with ...
1
vote
2answers
207 views

Inequality with cube roots

$$(\sqrt{n}+1)^{1/3}-(\sqrt{n}-2)^{1/3} \geq (\sqrt{n+1}+1)^{1/3}-(\sqrt{n+1}-2)^{1/3}$$ $$n\in \mathbb{N}$$ I come upon this inequality when trying to use prove series declension for the Leibnit'z ...
3
votes
3answers
114 views

Find $a,b \in \mathbb{Z}^+$ such that : $\frac{a^{2}-2}{ab+2}\in \mathbb{Z}$

$1$. Find $a;b\in \mathbb{Z}^+$ such that : $\frac{a^{2}-2}{ab+2}\in \mathbb{Z}$ $2$. Find $m;n>1$ such that : $2^m+3^n=k^2$ $(k\in \mathbb{Z})$ Problem 1. I thought : ...
1
vote
1answer
75 views

The arithmetic mean of $X$ when arithmetic mean of $X^2 = 29$.

Sorry if my question is a beginner because of my mathematical knowledge is low. arithmetic mean is : $$ \overline x=\dfrac{x_1+x_2+\cdots+x_n}n $$ What method can solve it? $$ ...
0
votes
2answers
75 views

Addition and subtraction with exponents

I'm doing an Advanced Functions course right now, and I'm wondering about something. Look at this here evaluation/simplification that I did: http://puu.sh/5w3XQ.png What I'm wondering is about the ...
7
votes
4answers
350 views

How to solve the equations $\sqrt{x-3}+\sqrt{y-3}=\sqrt{y-12}+\sqrt{z-12}=\sqrt{z-27}+\sqrt{x-27}=12$

Let $x,y,z\in R$, and $$\begin{cases} \sqrt{x-3}+\sqrt{y-3}=12\\ \sqrt{y-12}+\sqrt{z-12}=12\\ \sqrt{z-27}+\sqrt{x-27}=12 \end{cases}$$ Find the $x,y,z$. My try: I want use The geometry to ...
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votes
2answers
87 views

we need to find $m+n$.

I just dont understand this question, could any one tell me how to solve this one? A pen costs $13$ dollar and a notebook costs $35$ dollar, let $m$ be the maximum number of items that can be ...
-2
votes
2answers
63 views

A mathematical puzzle.

A man is driving from A to B. He travels half the distance at a constant speed of 30km/h. If he wishes to have an average speed, over the whole journey, of 60km/h, at what speed must he travel for the ...
0
votes
1answer
29 views

An inequality involving quadratic integers

Let $d$ be a positive integer which is not the square of any integer, $x,y \in \mathbb Z$ and $u:=x+y \sqrt d$ $u$ is such that $$ u \geq 1 \;\text{and} \; |u \overline u|=|(x+y \sqrt d)(x-y \sqrt ...
6
votes
0answers
219 views

The structure of countable ordinals

Consider the recursively defined hyperoperation sequence $\circ_i$ $$\begin{array}{rcrclclcl} x& \small{+}&(y\ {\small+}1)&:=&x& &&{\small+}&1\\ x& ...
12
votes
2answers
163 views

A problem in fractions from a very old arithmetic textbook

Similar in vein to a problem I posted before here, I would be interested if anyone can give me any pointers as to how one might solve this question from the same arithmetic textbook: "Simplify ...
1
vote
2answers
62 views

Recursive definition of recursively defined operations

The recursive definitions of addition, multiplication, and exponentiation usually stop after exponentiation ("${\small+}1$" to be read as "the successor of"): $x \boldsymbol{+} (y\ {\small+}1) := (x ...
2
votes
4answers
45 views

how do we prove $p|q\cdot r \rightarrow p=q$ or $p=r$ (all primes)?

I know this is one of the most fundamental basis of arithmetic but I can't find the result by myself. how do we prove $p|q\cdot r\rightarrow p=q$ or $p=r$? ($p, q, r$ being prime numbers)
1
vote
1answer
107 views

Evaluating $\int\,\cos(x)\cos(\omega x)\,dx$ using trigonometric addition formulas

I'm looking to solve the integral $$\frac{1}{\pi}\int_{-\frac{\pi}{2}}^\frac{\pi}{2}\, \cos(x)\cos(\omega x)\,dx$$ by rewriting the terms using the trigonometric addition formulaes. It should end ...
2
votes
3answers
97 views

What's the difference between arithmetic mean and average?

I'm trying to intuitively understand an average / arithmetic mean: Here's my attempt: In front of me, I see 1 thermos, two computer mice, two pens, and an iPhone. If I sum those, I get ...
1
vote
2answers
130 views

How can you prove that a value raised to a fraction($\frac{1}{2}$ for example), is $\sqrt{x}$?

I know that if you raise a value to $\frac{1}{2}$ for example, you take the square root, but that is not what I am asking, what I am asking is; what are you actually doing when raising a value to ...
2
votes
1answer
449 views

Prove by induction that $a-b|a^n-b^n$ [duplicate]

Given $a,b,n \in \mathbb N$, prove that $a-b|a^n-b^n$. I think about induction. The assertion is obviously true for $n=1$. If I assume that assertive is true for a given $k \in \mathbb N$, i.e.: ...
5
votes
1answer
108 views

Determine if the number $ \sqrt{8+2\sqrt{10+2\sqrt{5}}} - \sqrt{8-2\sqrt{10+2\sqrt{5}}} $ is rational

$ \sqrt{8+2\sqrt{10+2\sqrt{5}}} - \sqrt{8-2\sqrt{10+2\sqrt{5}}} $ I have tried to raising it to the square, but I can't obtain the result. $ \sqrt{8+2\sqrt{10+2\sqrt{5}}} - ...
0
votes
1answer
53 views

What are the smallest possible theories?

Im wondering how we could define a general form for the smallest possible theory in some formal language. In other words, if we have the formal language of first order logic, what is the smallest set ...