Questions about numbers expressible as the quotient of two integers. For questions on determining whether a number is rational, use the (rationality-testing) tag instead.

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How do you calculate certain variables of two or more events that occur simultaneously compared to the same events happening subsequently.

Say you have two hoses, A and B, that fill up a pool of equal size at different rates. Hose A fills up a pool in 10 mins, hose B in 20 mins. Thus A = 1p/10m, B = 1p/20m. Lets say that Hose A filling ...
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1answer
44 views

$a, b, x \in \mathbb{Q}$ with $a \neq 0$. Is the $\frac{b}{a}$ the only possible value for x in $a \cdot x = b$

I have an exercise in my last assignment for calculus which is the following: Let $a, b, x \in \mathbb{Q}$ with $a \neq 0$. Use only the field axioms and the properties which we showed in class ...
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1answer
48 views

Show using ordering axioms that $x^2 < y^2$ for $x, y \in \mathbb{Q}$, with $0 < x < y$

I have an exercise in my last assignment of calculus: Show using ordering axioms that $$x^2 < y^2$$ for $x, y \in > \mathbb{Q}$, with $0 < x < y$ This is my solution: We have that ...
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0answers
63 views

Set theory proof question on rational numbers

I was assigned a problem by my Discrete Mathematics professor that goes as follows: Prove that on $\mathbb{Q}$ (the set of all rational numbers), the relation "$<$" satisfies " $< \circ <~ = ...
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1answer
54 views

How can I integrate $\int{1\over 2x+2}$

$$\int{1\over 2x+2}$$ Method 1 $$\int{1\over 2x+2} = \frac 12\int{1\over x+1} = \frac 12 ln(x+1) + c $$ Method 2 $$\int{1\over 2x+2} = \frac 12\int{2\over 2x+2} = \frac 12 ln(2x+2) + c $$ Wolfram ...
9
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3answers
1k views

If $ f(x \cdot f(y) + f(x)) = y \cdot f(x) + x $, then $f(x)=x$

Let $ f : \mathbb{Q} \rightarrow \mathbb{Q} $ be a function which has the following property: $$ f(x \cdot f(y) + f(x)) = y \cdot f(x) + x \;,\; \forall \; x, y \in \mathbb{Q} $$ Prove that $ f(x) = ...
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4answers
1k views

Are there any bases which represent all rationals in a finite number of digits?

In base 10, 1/3 cannot be represented in a finite number of digits. Examples exist in many other bases (notably base 2, as it's relevant to computing). I'm wondering: does there exist any base in ...
4
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6answers
121 views

Find $\lim_{x \to \infty} \left(\frac{x^2+1}{x^2-1}\right)^{x^2}$

How to calculate the following limit? $$\lim\limits_{x \to \infty} \left(\frac{x^2+1}{x^2-1}\right)^{x^2}$$
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1answer
33 views

for $p$ given, $\zeta_p$ a primitive root of unity, fow which $d\in \mathbb{Z}$ does $\zeta_p \in \mathbb{Q}(\sqrt{d})$?

Here is a question that I am trying to answer: Let $p$ be a prime greater than $2$. For which $d \in \mathbb{Z}$ contains $\mathbb{Q}(\sqrt{d})$ a primitive root of power $p$? What I did If ...
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3answers
113 views

Show that $\frac{1}{a}+\frac{1}{b}\not=\frac{1}{a+b}$

Problem Assume that $a,b\in\mathbb{R}-\{0\}$ and that $a+b\not=0$. Prove that $\frac{1}{a}+\frac{1}{b}\not=\frac{1}{a+b}$. My Proof Let's assume that $\frac{1}{a}+\frac{1}{b}=\frac{1}{a+b}$, then ...
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3answers
54 views

Finding a sequence of sets whose intersection is a null set

Find a sequence of sets $I_n=\{r:r \in \mathbb{Q}, a_n\le r \le b_n\} $ in $\mathbb{Q}$, where $a_n, b_n \in\mathbb{Q}$ such that $$I_{n+1} \subset I_n\forall n\in\mathbb{N}$$ $\lim_{n \to ...
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3answers
55 views

Find the fractional representation $p/q$…

Been trying to get some sort of solution for this for hours now, with no avail. Find the fractional representation $p/q$, with $p \in \mathbb{N}$ and $q \in \mathbb{N}$, of the rational number whose ...
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1answer
27 views

A Elementary fact but proof needed

Let $n,q\in\mathbb{N}$, $r\in\mathbb{R}$ and $m,p\in\mathbb{Z}$ such that $\frac{m}{n}<r<\frac{m+1}{n}$ and $|\frac{p}{q}-r|<\min(r-\frac{m}{n};\frac{m+1}{n}-r)$. It does seem obvious that we ...
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3answers
66 views

If an object halves its speed every second (but never gets to 0), will it eventually get from point A to point B?

There is a ball that starts at point A on a line and moves toward point B. Every second, it moves half of the distance left, but never stops moving: Etc. Would the ball ever reach point B? In one ...
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4answers
40 views

infinity sum of the fractional

Can anyone explain how to simplify $ \frac{2}{3} + \frac{6}{9} + \frac{12}{27} + \frac{20}{81} + \frac{30}{243} + . . . $ I have no any idea since i dont have pattern i can't do it with integral or ...
5
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0answers
48 views

Is there any elegant formalization of fractional numbers?

The question is just what is on the title, but I'll describe the context for completion: Natural numbers can be encoded quite elegantly on the Lambda Calculus as church numbers, that is, a function ...
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1answer
44 views

Fraction in other bases

How to convert a base 10 fraction into fraction in other bases?. For example base 10 fraction 17/94, How we convert this 17/94 into base 2 fraction ?
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1answer
78 views

Find the functions

Find all the functions $ f : \mathbb{Q} \rightarrow \mathbb{Q} $ with the following property: $$ f(x + 3f(y)) = f(x) + f(y) + 2y, \: \forall x, y \in \mathbb{Q} $$
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4answers
113 views

How come $\left(\frac{n+1}{n-1}\right)^n = \left(1+\frac{2}{n-1}\right)^n$

I'm looking at one of my professor's calculus slides and in one of his proofs he uses the identity: $\left(\frac{n+1}{n-1}\right)^n = \left(1+\frac{2}{n-1}\right)^n$ Except I don't see why that's ...
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2answers
24 views

Compare colon notation with fraction

I'm working on a job interview test and there is one answer which I just don't get. The test states that statement below is true. To me it just seems wrong. No box is provided to check. Then how do I ...
2
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2answers
53 views

simplifying and factoring a fraction

how i get $\frac{(a+b)^2+(a+c)^2+(b+c)^2}{2}$ from $\frac{a^4}{(a-b)(a-c)}+\frac{b^4}{(b-a)(b-c)}+\frac{c^4}{(c-a)(c-b)}$ assuming that $a\ne b\ne c\ne a$ i tried to make $$\begin{align} ...
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vote
5answers
511 views

Un-Simplifying a fraction, i.e. computing partial fraction decomposition

$\frac{3x^2+17x}{x^3+3x^2+-6x-8}$ I need to find the value of C in the form of $\frac{A}{x+1} + \frac{B}{x-2} + \frac{C}{x+4}$ which is based on the fraction give at the top. I can get so far to do ...
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vote
1answer
10 views

Explaining the non-application of the multiplication law of logarithms, when logs are in the denominators.

I have an A' Levels student who had to solve the following problem: $ log_2 x + log_4 x = 2$ This was to be solved using the Change of base rule, and then substitution, as follows: $ \frac{1}{log_x ...
3
votes
1answer
43 views

Algebraic number with bounded coefficients

How many algebraic numbers $z$ are there satisfying $P(z)=0$ where $P(z)$ is some polynomial with integer coefficients of degree less than or equal to $n$ such that the absolute value of every ...
0
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1answer
25 views

Is there a value for $a$ other than a factor or a multiple of $c$ in $\frac{a}{b}=\frac{c}{d}$

Suppose $a,b,c,d$ to be whatever quantities whatsoever that satisfy the proportion $\frac{a}{b}=\frac{c}{d}$. Is there a value for $a$ other than a factor or a multiple of $c$. Or, is there a value ...
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0answers
18 views

GCD and fraction problem

If x/y = 1/a + 1/b + 1/c and GCD of a , b and c is 9 then find a) minimum of x and y which do not cause x/y repeating decimal b) the best of x and y that cause x/y nearly to 3/10 many ...
0
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2answers
76 views

Let $S=\{x\in\mathbb Q\mid x>2\}$. Prove $\inf S = 2$.

Okay, so I think I kind of get this one already. Since 2 is the lowest rational number in the set that's less than $x$, then $\inf S = 2$. But is there is any other way to explain this? I feel like ...
2
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1answer
38 views

Proof for number of rational ordered pairs on a line

It is given that the function $y=ax+b,\; a \neq 0$ has an ordered pair $(x,y)=( \sqrt{2}, 0)$. Prove that $y=ax+b$ does not have two or more rational ordered pairs. From the above I know that ...
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2answers
44 views

Does there exist $a,b,c,d$ such that $\frac{a+b+c+d}{4}$ is an integer?

Let $a,b,c,d$ be defined as such: $$\{a,b,c,d\} \geq 1,\\ a\neq b\neq c\neq d,\\ a \not\in \{bx,cx,dx\},\\ b \not\in \{ax,cx,dx\},\\ c \not\in \{ax,bx,dx\},\\ d \not\in \{ax,bx,cx\},\\ \{a,b,c,d\} ...
3
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1answer
61 views

When is $(12x+5)/(12y+2)$ not in lowest terms?

I am struggling to solve this problem and would appreciate any help: When is $\frac{12x+5}{12y+2}$ NOT in lowest terms? (x,y are nonnegative integers) I have found that it is not in lowest terms for ...
4
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3answers
66 views

Proving $\lim _{x\to \infty }\left(\frac{\sqrt{x+1}-\sqrt{x-2}}{\sqrt{x+2}-\sqrt{x-3}}\right) = \frac35$

$$\lim _{x\to \infty }\left(\frac{\sqrt{x+1}-\sqrt{x-2}}{\sqrt{x+2}-\sqrt{x-3}}\right)$$ Can someone help me to solve it? result of online calculator: 3/5
4
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2answers
370 views

Equivalent of adding to a denominator?

Given the inequality $\frac{n}{m} \ge \frac{1}{2}$, I want to add $1$ to both $n$ and $m$: $$\frac{n+1}{m+1}.$$ What would be the equivalent operation on the RHS of the equation? Adding $1$ to $n$ ...
5
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1answer
76 views

Is $\Bbb Q$ homeomorphic to $\Bbb Q^2$? [duplicate]

It's an easy excercise in set theory to exhibit a bijection $\Bbb Q \cong \Bbb Q\times \Bbb Q$. However, none of the bijections I'm aware of respect the topologies on $\Bbb Q$ and $\Bbb Q^2$, ...
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2answers
101 views

How is $\frac{1-x}{x^2-1}=\frac{1}{x+1}$?

When integrating $\int \frac{1-x}{x^2-1} dx$ Maple rewrote it as $-\int\frac{1}{x+1}dx$ How is $\frac{1-x}{x^2-1}=\frac{1}{x+1}$?
0
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1answer
43 views

Image of ring homomorphism $\phi : \mathbb{Z}[t] \to \mathbb{Q}$?

Here is a problem I face practicing the theory of rings: Define $\phi : \mathbb{Z}[t] \to \mathbb{Q}$, a ring homomorphism (it does map $1$ to $1$). I'm trying to show that if $\phi(t)=\frac{u}{v}$ ...
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6answers
118 views

Why is $\mathbb Q $ (rational numbers) countable? [duplicate]

By definition, a set $S$ is called countable if there exists an bijective function $f$ from $S$ to the natural numbers $N$. If we take a function $g\colon\mathbb{Z\times N\to Q}$ given by $g(m, n) = ...
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1answer
53 views

Shorten $\frac{n}{n^\frac{1}{2}}$?

I have a short question to solve my problem. Can I simplify $\frac{n}{n^\frac{1}{2}}$ ? Thanks already for answers.
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1answer
24 views

Rational number in $\mathbb{Z}[\omega]$ should be integer.

Let $\omega = \cos \frac{2\pi}{p} + i \sin \frac{2\pi}{p}$ for some prime number $p > 2$. Then how to prove that if $q \in \mathbb{Q} \cap \mathbb{Z}[\omega]$, $q$ must be integer.
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1answer
29 views

Rearranging equation

I'm reading a textbook in which an equation is rearranged and I'm failing to see how they've done it. I've tried writing it down step by step in my notebook but can't come up with the right answer. ...
0
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1answer
92 views

Fractions vs Decimal numbers

I want to know if there is any difference between Fractions and Decimal numbers, are Decimal numbers just Fractions that are written in a different way according to a predefined rule: using "a group ...
2
votes
2answers
164 views

Why must the decimal representation of a rational number in any base always either terminate or repeat?

Wikipedia makes the following statement about rational numbers. The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same ...
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1answer
42 views

Fractional Exponents and Fractions

When dealing with fractional exponents like in the question below, how do you combine them so the two "n's" in the first fraction become one? ((how do i combine $4/3$ with $1/3$)) The aim is to end ...
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5answers
47 views

need help on this fraction equation $2/5 = 2/3 - r/5$

$$\frac{2}{5} = \frac{2}{3} - \frac{r}{5}$$ I'm trying to find $r$. Can anyone give me a step by step?
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1answer
44 views

What is the smallest fraction produced by a sum of fractions with bounded denominator?

For $x$ a sum of fractions: $$ x = \sum_{i=1}^{N}\frac{a_i}{b_i} $$ for all $a_i, b_i \in \mathbb{Z}$ with $ 0 < b_i \leq D$ and $N$ are non-zero positive integers, I know that the denominator of ...
0
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1answer
33 views

Why do the denominators of two fractions with numerators $1$ add up to a third fraction that have the special things below?

I found out that the denominators of two fractions with numerators $1$ add up to a third fraction that has the sum in the numerator and the product in the denominator. For example, ${1\over ...
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2answers
26 views

How do I solve this equation?

I have an equation, where I need to find n, that I need help solving. I already cheated a little bit by using a CAS (Maple) to solve the equation, so i know what the result should be, but I need to ...
24
votes
8answers
2k views

How to make sense of fractions?

Can anybody explain what a fraction is in a way that makes sense. I will tell you what I find so confusing: A fraction is just a number, but this number is written as a division problem between two ...
0
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2answers
51 views

variables to the power of a fraction

I have this question for advanced math, I can't seem to get my head around. $$\frac{x^{5/2}}{(x^{1/3})^4}$$
6
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1answer
316 views

Will 0.99999999 eventually become equal to 1?

I am currently learning about fractions, and there is something that I am finding it hard to make sense of. When a fraction it added to the right of the decimal point, the number becomes slightly ...
0
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2answers
36 views

what is the default order and direction of operation?

I have a division like this 16/8/4/2 what is the default way to do calculations when the bracket is not specified . Method 1 : Is it correct to go from right to left like [16/ (8 / { 4 / ...