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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Why is the $\mathbb Z$-module $\mathbb Q$ projective in category $\sigma[\mathbb Q]$?

We know that $\mathbb Q$ is not a projective $\mathbb Z$-module, but, I've read this paper and it says in Example 2.11 that it's easy to check that "$\mathbb Q$ as $\mathbb Z$-module is projective ...
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4answers
58 views

Multiplying whole number with fractions.

I'm looking at a solution to a math problem and there are obviously some rules regarding multiplication of fractions that I don't know. Can someone make any sense of this? $$s_n = 625 \cdot ...
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1answer
46 views

Finding all possible pairs of positive integer values

The ratio of the sum of two positive integers to their difference is $7:5$. If the the sum of the two numbers is at most $25$, find all possible values for the pair of numbers. Let $m$ be the first ...
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1answer
60 views

About subspaces of $\mathbb{R}$ as vector space over $\mathbb{Q}$.

In many texts is noted the analogy between the transcendence degree of a field extension and the dimension of a vector space, so I'm tempting to use such analogy to better understand the structure of ...
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3answers
127 views

All sets of rational numbers are bigger than the set containing infinite integers - or are they?

Intro This started with me learning the different types of infinity. I like to call them types instead of sizes due to the fact, that infinite is defined by being endless or "not-finite" - meaning ...
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1answer
26 views

Fraction Transforms

Here's a number theory problem I'm having some difficulty with: Say we transform a fraction by the following rule: we start with some fraction $\frac{m}{n}$ with $m > n$ and then convert it to ...
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5answers
76 views

How to compute $\frac{t^2}{t+1}$ to the form $\frac{1}{t+1} +t -1$

One of my attempts would look like below. $\frac{t^2}{t+1}$ = $\frac{t \times t+1-1}{t+1}$ = 1+ $\frac{t-1}{t+1}$ = $\frac{t-1+1-1}{t+1} + 1$ = $\frac{-2}{t+1} +2$ Also, I put into t an arbitrary ...
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3answers
31 views

Adding fractions with exponents

$$3^5 + {1\over3^5}=?$$ My first instinct was to rewrite the second term as $3^{-5}$. Since the base is $3$, rewrite as $3^{5+-5}$. It simplifies to $3^0= 1$. Apparently this is incorrect. Can anyone ...
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2answers
43 views

Reciprocal over a summation [closed]

Is this statement true? Can we take reciprocal over a summation? $$\frac 1{\sum_{n=1}^\infty\frac 1{(n+1)^3}}=\sum_{n=1}^\infty (n+1)^3$$
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0answers
55 views

Natural bijection between $\mathbb{N}$ and algebraic numbers?

Q. Is there a canonical, explicit bijection between the natural numbers $\mathbb{N}$ and the algebraic numbers? The earlier MSE question, "Bijection for algebraic numbers," does not quite ...
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4answers
61 views

Characterization of the $x$ such that $\sin(x)$ is rational?

For $x \in [0,\pi/2]$, $\sin(x)$ ranges over $[0,1]$. So every rational number in $[0,1]$ is the sine of some $x \in [0,\pi/2]$. Q. Is there any characterization of the $x$ for which $\sin(x)$ is ...
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2answers
37 views

Adding drawnfractions (Very simple question)

I have been surprised not being able to solve this: ...
0
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1answer
35 views

Recursive formula for partial fraction decomposition of a specific kind of fractions

I need to make a partial fraction decomposition of the following fraction : $$ \frac{1}{(x-a)^2(x-b)^2(x-c)^2(x-d)^2(x-e)} $$ The problem is that Wolfram doesn't give any answer : ...
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3answers
83 views

If $\frac{a}{b}=\frac{x}{y}$, is $\frac{x-a}{y-b}=\frac{x}{y}$? [closed]

Does this hold? $b,y \neq 0$, $b \neq y$.
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1answer
130 views

For each irrational number $b$, does there exist an irrational number $a$ such that $a^b$ is rational?

It is well known that there exist two irrational numbers $a$ and $b$ such that $a^b$ is rational. By the way, I've been interested in the following two propositions. Proposition 1 : For each ...
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4answers
29 views

Simplifying $\frac{3(a^{1/4}+4)}{2a-32a^{1/2}}$

I have a fraction $\frac{3a^{1/4}+12}{2a-32a^{1/2}}$ which I have factored out into $\frac{3\left(a^{\frac{1}{4}}+4\right)}{2a-32a^{\frac{1}{2}}}$, but checking out W|A I also get that there ought to ...
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2answers
82 views

Dedekind Cut Roots

I was studying Dedekind Cuts in my history of math class and was looking at the following formulas for the multiplication of Dedekind Cuts: $L_\sqrt2 = \{ r \in \mathbb{Q} : r^2 < 2 $ and $ r>0 ...
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1answer
31 views

Rational Zeros Theorem, get the possible solutions

In my book there is a section about rational numbers. The first example show that:6^(1/3) cannot represent a rational number. In the proof it says that the ...
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4answers
83 views

What is the reciprocal of $(-1/2)^k$?

What is the reciprocal of $(-1/2)^k$? The answer is meant to be $2^{-k}$ as if you flip something upside down the power becomes negative. However, I am not sure what happens to the negative in front ...
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4answers
39 views

Algebraic fractions: addition

I have very elementary question about adding algebraic fractions. Now, I know the following: $$\frac{a}{c} + \frac{b}{d} = \frac{da + cb}{dc}$$ My question is however, how given this expression: ...
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0answers
42 views

How many elements are in the following set?

The set is $$\{ x \in Q:x^2 =64/25 \} $$ I thought the answer was $\{ \frac{8}{5}, -\frac{8}{5} \}$ but I am told there are in fact 4 distinct elements: $$\{ \frac{8}{5}, \frac{8}{-5}, \frac{-8}{5}, ...
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1answer
58 views

Set in $\mathbb{R}^2$ that contains no (non-null) measurable rectangles [closed]

From Torchinsky's Real Variables text, presented in a chapter on abstract Fubini's Theorem. Define the following set: $$E := \mathbb{R}^2 \setminus \{(x,y) \in \mathbb{R}^2: x-y \in \mathbb{Q}\}.$$ ...
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1answer
48 views

algebra exponents and fractions

I could be over thinking or tired... But I am to embarrassed to ask my prof. this probably very simple algebra rule I am ignorant of... Also this is just a snip-it from a inductive proof example. ...
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1answer
56 views

Dedekind cuts in Rudin's PMA

I'm working on Appendix to chapter I of Rudin's Principles of mathematical analysis and I have the following problem: Given a positive cut $\alpha$ and a rational $x>1,$ how can I prove that there ...
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1answer
45 views

Annuity present value formula explanation

Could somene please explain me how the formula evolves, ie. how does the fraction flip, etc? Thank you in advance!
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1answer
22 views

Pre-Algebra Fractional Exponent Question

Why does $t^{\frac{3}{2}} \cdot t^{\frac{1}{2}} = t^2$? What I tried to do was multiply the exponents together $\frac{3}{2} \cdot \frac{1}{2} = \frac{3}{4}$ so my final answer was $t^{\frac{3}{4}}$ ...
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1answer
38 views

Limit of sequences and integers

If $a$ is a non zero real number , $x \ge 1$ is a rational number and $(r_n)$ is a sequence of positive integers such that $\lim _{n \to \infty}ax^n-r_n=0$ , then is it true that $x$ is an integer ?
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3answers
36 views

How to calculate value of expressions when $a = 22$

$a = 22$ Round the answer to three significant figures: $\dfrac{77}{3a}$ for this one I am not sure if I do $\dfrac{77}{3(22)} = 1.17$ or $\dfrac{77}{3(22)} = 56$. Sorry if this is written in a ...
2
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1answer
27 views

How exactly is this happening?

I was studying Derivative and my book says if: Then its derivative is: I can't understand how the writer has changed the first derivative fraction into the second one. In other words, how did he ...
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3answers
97 views

Find rational points on $x^2 + y^2 = 3$ and on $x^2 + y^2 = 17$

$(a)$ Find all rational points on the circle $x^2 + y^2 = 3$, if there are any. If there is none, prove so. $(b)$ Find all rational points on the circle $x^2 + y^2 = 17$, if there are any. If there ...
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0answers
30 views

How to calculate the Integer portion of a fraction using only +, -, $\div$ and *?

I made something in excel that calculates the days left until a given date, and from that how many weeks were left. I had it so that 9 days displayed as 1.2 using this formula: ...
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0answers
54 views

Irrational numbers to irrational powers being rational?

So some of you may be familiar with the proof that some irrational numbers to irrational powers are rational, that is: if $A = \sqrt2^\sqrt{2}$ then it follows that $A^\sqrt{2} = 2$. So, I've found a ...
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0answers
34 views

Rational number in form $(a^x+b^y)/(c^z+d^w)$

Find all positive integers $(x,y,z,w)$ such that for any positive rational number $r=p/q$, there exist positive integers $(a,b,c,d)$ for which $$r=\frac{a^x+b^y}{c^z+d^w}.$$ For instance, for ...
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4answers
145 views

Find All Dimensions such that Volume of Box = Surface Area

A rectangular prism has integer edge lengths. Find all dimensions such that its surface area equals its volume. My Attempt at a Solution: Let the edge lengths be represented by the variables $l, w, ...
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1answer
23 views

Second Order Approximation for a Polynomial

if I have an expression: $L=\frac{12a^3d^3-4wa^3d^2+16a^2d^2-4wa^2d+6ad+1}{12a^3d^3-4wa^3d^2-4a^2wd+16a^2d^2+7ad-aw+1}$ what is the second order approximation in $\frac{d}{w}$? I know that ...
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1answer
59 views

Is a subset of $\mathbb{Q}\times\mathbb{Q}$ that all variants of an exponentiation equation have answers in it, infinite?

Note that we have: $$A=\{(a,b)\in\mathbb{Q}\times\mathbb{Q}~|~\text{Both equations}~a+x=b, b+y=a~\text{have answers in }~\mathbb{Q}\}=\mathbb{Q}\times\mathbb{Q}$$ ...
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0answers
17 views

Extracting a function of a variable from an expression

I have this expression: $\frac{d+2wd}{2w+3wd-3d-w^2-1}$ Is there anyway I can write it just as a function of f(d)? [To me this looks like it is already a function of d, but I want to confirm if ...
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3answers
116 views

Why$ 1/12$ is NOT an irreducible basic fraction?

I'm trying to solve this problem. A fraction $m/n$ is basic if $0 \le m < n$, It is irreducible if $\gcd( m,n ) = 1$ (greatest common divisor) In the example, when $n=12$, irreducible basic ...
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2answers
46 views

Find the fraction that creates a repeating decimal that repeats certain digits

Is there any way to find the fraction $x/y$ that, when converted to a decimal, repeats a series of digits $z$? For example: ${x}/{y} = z.zzzzzzzz...$ or with actual numbers, $x/y = 234.234234234...$ ...
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3answers
61 views

Expressing $\frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{5}}$ with rational Denominator

could you please help me express this with a rational denominator $\frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{5}}$ Thank you
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3answers
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How to simply this fraction with irrational denominators? [closed]

How to simplify? $\frac{1}{1+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{5}} + \frac{1}{\sqrt{5}+\sqrt{7}} \frac{1}{\sqrt{7}+3}$
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1answer
58 views

How to solve for $x$ in ${\sqrt{9+2x}} - {\sqrt{2x}} = \frac{5}{\sqrt{9+2x}}$

How can I solve for $x$ in the following equation? ${\sqrt{9+2x}} - {\sqrt{2x}} = \frac{5}{\sqrt{9+2x}}$
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2answers
47 views

Seeing the plane as a four (or more) dimensional vector space on $\mathbb Q$

As I was trying to answer a question about the enumeration of circuits one can build with a set of miniature train track elements, I realized that all plane positions that could be reached had ...
2
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3answers
70 views

How to simplify $(a^2+ab+b^2)/(a+\sqrt{ab}+b)$

How can I simplify as much as possible: $$\frac{a^2+ab+b^2}{a+\sqrt{ab}+b}$$ Also, first post here, looking forward to sticking around!
2
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2answers
88 views

What fraction is $\frac{2}{5}$ of $\frac{3}{4}$?

$\frac{2}{5}$ of blood donors at a centre have group O blood. $\frac{3}{4}$ of these donors are under 30. What fraction of the group O blood donors at the centre are under 30? What I did was divide ...
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1answer
873 views

Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$

About a month ago, I got the following : For every positive rational number $r$, there exists a set of four positive integers $(a,b,c,d)$ such that ...
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3answers
90 views

“Canceling out” in division doesn't always work the same way does it?

I've been working on Nested Fractions at the Khan Academy. Recently I was doing a routine problem and came to the correct conclusion but I realized I didn't understand why I wouldn't keep dividing. ...
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2answers
354 views

Sum of series with binary parity in the numerator

I'm now stuck with this question, and I don't even know where to start: Find sum of series$$\sum_1^\infty \frac{f(n)}{n(n+1)}$$, where f(n) - number of ones in binary representation of n. I wish I ...
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1answer
33 views

$5.30$ converted to a fraction or mixed number in lowest terms

$5.30$ converted to a fraction or mixed number in lowest terms The correct answer they got on my worksheet is $5 \frac3{50}$, but I get $5 \frac{15}{50}$.
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3answers
73 views

How to put a fraction in simplest form, such as $140/255$?

Given the fraction $$\dfrac{140}{255}$$ How do I find a common factor so it can be easily simplified? I have already tried $2$, $3$ and $4$.