Questions on fractions, numbers of the form $p/q$ where $p$ and $q$ are integers, and $q$ is not zero.

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20
votes
2answers
1k views

Why is the decimal representation of $\frac17$ “cyclical”?

$\frac17 = 0.(142857)$... with the digits in the parentheses repeating. I understand that the reason it's a repeating fraction is because $7$ and $10$ are coprime. But this...cyclical nature is ...
8
votes
1answer
724 views

Prove that any rational can be expressed in the form $\sum\limits_{k=1}^n{\frac{1}{a_k}}$, $a_k\in\mathbb N^*$

Let $x\in\mathbb{Q}$ with $x>0$. Prove that we can find $n\in\mathbb{N}^*$ and distinct $a_1,...,a_n \in \mathbb{N}^*$ such that $$x=\sum_{k=1}^n{\frac{1}{a_k}}$$
28
votes
3answers
6k views

Can you raise a number to an irrational exponent?

The way that I was taught it in 8th grade algebra, a number raised to a fractional exponent, i.e. $a^\frac x y$ is equivalent to the denominatorth root of the number raised to the numerator, i.e. ...
33
votes
7answers
1k views

Bad Fraction Reduction That Actually Works

$$\frac{16}{64}=\frac{1\rlap{/}6}{\rlap{/}64}=\frac{1}{4}$$ This is certainly not a correct technique for reducing fractions to lowest terms, but it happens to work in this case, and I believe there ...
56
votes
14answers
3k views

Why rationalize the denominator?

In grade school we learn to rationalize denominators of fractions when possible. We are taught that $\frac{\sqrt{2}}{2}$ is simpler than $\frac{1}{\sqrt{2}}$. An answer on this site says that "there ...
21
votes
1answer
617 views

Simplify $\left({\sum_{k=1}^{2499}\sqrt{10+{\sqrt{50+\sqrt{k}}}}}\right)\left({\sum_{k=1}^{2499}\sqrt{10-{\sqrt{50+\sqrt{k}}}}}\right)^{-1}$

Simplify $$\frac{\displaystyle\sum_{k=1}^{2499}\sqrt{10+{\sqrt{50+\sqrt{k}}}}}{\displaystyle\sum_{k=1}^{2499}\sqrt{10-{\sqrt{50+\sqrt{k}}}}}$$ I don't have any good idea. I need your help.
4
votes
2answers
362 views

Approximation of irrationals by fractions

If $\alpha$ is an irrational, and I'm trying to judge the suitability of of a rational $p/q$ as its approximation by the error $\Delta = |\alpha - p/q|$. For a given denominator $q$, I am finding a ...
8
votes
2answers
269 views

Writing $1$ in form of $\frac{1}{t_1}+\cdots+\frac{1}{t_n}$ [duplicate]

Possible Duplicate: Prove that any rational can be expressed in the form $\sum\limits_{k=1}^n{\frac{1}{a_k}}$, $a_k\in\mathbb N^*$ Can anyone help me with this problem? It's a little ...
7
votes
1answer
207 views

Number to the exponent divided by exponent value

Can someone explain including working out how to solve this? $$\dfrac{5^x}{x} = 79.85957$$ I know that the answer is $x = 3.5$, but how does one normalise the equation so that the x is on one side?
4
votes
3answers
342 views

Constructing Farey sequences inductively

Objective: I'd like to prove that $F_{n+1}$ (the Farey sequence of order $n+1$) is obtained form the Farey sequence $F_n$ of order $n$ by adding all fractions of the form $\frac{a+c}{b+d}$ when ...
2
votes
8answers
420 views

Why Not Define $0/0$ To Be $0$?

For every number $x$, $x\times 0=0$, hence $\dfrac{0}{0}$ can be any number! So $\dfrac{0}{0}$ "is knows as indeterminate" [1]. But what if we define it to be $0$? I already have an answer, but ...
0
votes
5answers
121 views

largest fraction less than 1 [duplicate]

What is the mathematically rigorous way to answer the question: "what is the largest fraction less than 1"? (or to explain why it cannot be answered in the manner worded).
37
votes
1answer
799 views

Why does this ratio of sums of square roots equal $1+\sqrt2+\sqrt{4+2\sqrt2}=\cot\frac\pi{16}$ for any natural number $n$?

Why is the following function $f(n)$ constant for any natural number $n$? $$f(n)=\frac{\sum_{k=1}^{n^2+2n}\sqrt{\sqrt{2n+2}+{\sqrt{n+1+\sqrt ...
20
votes
4answers
622 views

Is $\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}$ true for $m\in\mathbb N$?

Question : Is the following true for any $m\in\mathbb N$? $$\begin{align}\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}\qquad(\star)\end{align}$$ Motivation : I reached ...
17
votes
11answers
3k views

Why is $\frac{1}{\frac{1}{X}}=X$?

Can someone help me understand in basic terms why $$\frac{1}{\frac{1}{X}} = X$$ And my book says that "to simplify the reciprocal of a fraction, invert the fraction"...I don't get this because isn't ...
5
votes
8answers
365 views

If $\,\,x+\dfrac{1}{x}=5,\,\,$ find $\,\,x^5+\dfrac{1}{x^5}$.

If $x>0$ and $\,x+\dfrac{1}{x}=5,\,$ find $\,x^5+\dfrac{1}{x^5}$. Is there any other way find it? $$ \left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=23\cdot 110. $$ Thanks
5
votes
4answers
549 views

What is the non-trivial, general solution of these equal ratios? [closed]

Provide non-trivial solution of the following: $$\frac{a}{b+c}=\frac{b}{c+a}=\frac{c}{a+b}$$ $a=?, b=?, c=?$ The solution should be general.
1
vote
1answer
40 views

What is the chances of a duplicate in this equation

I'm not very good at math; However I have a scenario where I'm trying to find the chance of duplicate for randomly generated data. In a nuttshell I have a "bag" with 62 different items, lets say a ...
20
votes
2answers
1k views

Why is $\pi$ irrational if it is represented as $c/d$?

$\pi$ can be represented as $C/D$, and $C/D$ is a fraction, and the definition of an irrational number is that it cannot be represented as a fraction. Then why is $\pi$ an irrational number?
7
votes
4answers
1k views

How to simplify $\frac{\sqrt{4+h}-2}{h}$

The following expression: $$\frac{\sqrt{4+h}-2}{h}$$ should be simplified to: $$\frac{1}{\sqrt{4+h}+2}$$ (even if I don't agree that this second is more simple than the first). The problem is ...
5
votes
1answer
213 views

When is the $lcm$ of a fraction sum the actual denominator.

Consider a sum $$\frac{a}{b}+\frac{c}{d} = \frac{x}{y}$$ where each fraction is reduced. Alternatively using the familiar process of lowest common denominators, we have $$\frac{a}{b}+\frac{c}{d} = ...
3
votes
2answers
171 views

How do you calculate how many decimal places there are before the repeating digits, given a fraction that expands to a repeating decimal?

If you have a fraction such as $$\frac{7}{26}=0.269230\overline{769230}$$ where there are a number of digits prior to the repeating section, how can you tell how many digits there will be given just ...
2
votes
3answers
3k views

How do I go about simplifying this complex radical?

I'm having a difficult time trying to simplify the radical below. When I type the radical to the left into my calculator, I arrive at the correct answer to the right. But I cannot seem to figure out ...
2
votes
1answer
387 views

Upper bound/exact length of decimal expansion of simple fraction

E.g. 1/8=0.125 has three decimals when written out in base 10, but what is a good example of a simple fraction where the decimal sequence terminates but is very large? Is there some sort of rule ...
1
vote
3answers
2k views

Solving a literal equation containing fractions.

I know this might seem very simple, but I can't seem to isolate x. $$\frac{1}{x} = \frac{1}{a} + \frac{1}{b} $$ Please show me the steps to solving it.
0
votes
5answers
452 views

Is this a weighted average/percentage problem?

Let's say a Marketing company has a total turnover of 10000 \$ There are 3 salesmen A,B,C with the following turnovers A = 2000 $ B = 3000 $ C = 5000 $ Now, ...
44
votes
3answers
4k views

Why do we miss 8 in 0.012345679…, 98 in 0.0001020304050607080910111213…, and so on in fractions like 1/81, 1/9801, and so on?

I've seen this happen that when you divide by a fraction using the square of any set of nines in the denominator depending on how many there are like ${1\over 99^2}={1\over 9,801}$, you get ...
12
votes
3answers
217 views

Any 'odd fraction' can be represented as the finite sum of different 'odd unit fractions'?

Let us call a fraction whose denominator is odd 'odd fraction'. Also, let us call an odd fraction whose numerator is $1$ 'odd unit fraction'. Then, here is my question. Question : Is the ...
15
votes
2answers
592 views

What is the max of $n$ such that $\sum_{i=1}^n\frac{1}{a_i}=1$ where $2\le a_1\lt a_2\lt\cdots\lt a_n\le 99$?

What is the max of $n$ such that $$\sum_{i=1}^n\frac{1}{a_i}=1$$ where $a_{i}\ (i=1,2,\cdots,n)$ are integers which satisfy $2\le a_1\lt a_2\lt\cdots\lt a_n\le 99$ ? Also, I need how to prove that ...
13
votes
11answers
5k views

Dividing by 2 numbers at once, what is the answer?

Let's say i have 4/1/5. or 4 divided by 1 divided by 5. Are there any rules that i am allowed to use to stop any mistakes?, for example this has 2 solutions, 4/5 , and 20. Edit: Thanks for your ...
4
votes
2answers
267 views

Bound on lcm of denominators of rational numbers that sum to 1.

This is related to the question If a finite set of rational numbers sums to one, does one of the rationals have a denominator equal to the LCM of all the denominators? Suppose $1 = ...
2
votes
1answer
141 views

Existence of a simultaneous rational approximation of real numbers in (0,1)

I have a simple question the rational approximation of real vectors. Dirichlet's simultaneous approximation theorem states: Given any $d$ real numbers $\alpha_1,\ldots,\alpha_d$ and for every ...
11
votes
8answers
3k views

Is 1 divided by 3 equal to 0.333…?

I have been taught that $\frac{1}{3}$ is 0.333.... However, I believe that this is not true, as 1/3 cannot actually be represented in base ten; even if you had ...
11
votes
1answer
745 views

Interesting pattern in the decimal expansion of $\frac1{243}$

There appears to be an interesting pattern in the decimal expansion of $\dfrac1{243}$: $$\frac1{243}=0.\overline{004115226337448559670781893}$$ I was wondering if anyone could clarify how this ...
8
votes
4answers
177 views

Primary/Elementary Pedagogy: What is the rationale for the absent '+' in mixed fractions?

Why are elementary students taught to represent one and a half as 1 1/2 rather than 1 + 1/2? This mode of expression seems standard throughout at least North America. I think it is bad pedagogy for a ...
8
votes
4answers
198 views

What is the 'physical' explanation of a division by a fraction?

For example, dividing by 2, means we cut something in two. But dividing by 0.5, can only be explained with multiplying something by 2. So, is there a "physical" explanation of dividing by 0.5? Is it ...
3
votes
3answers
609 views

How to find a “simple” fraction between two other fractions?

If we have two fractions $a = { a_1 \over a_2} $ and $c = {c_1 \over c_2}$ with $a<c$, how to find the fraction $b = { b_1 \over b_2 }$ , $a < b < c$ for which some measure of ...
3
votes
2answers
357 views

Adding a different constant to numerator and denominator

Suppose that $a$ is less than $b$ , $c$ is less than $d$. What is the relation between $\dfrac{a}{b}$ and $\dfrac{a+c}{b+d}$? Is $\dfrac{a}{b}$ less than, greater than or equal to ...
2
votes
7answers
156 views

Why is $1 / x^y = x ^ {-y}$

I've known this rule for a long time, but I never got to understand why is that and how it works, could anyone explain to me how it is done? Any help is appreciated.
2
votes
2answers
95 views

Assuming there exist infinite prime twins does $\prod_i (1+\frac{1}{p_i})$ diverge?

Assume there are an infinite amount of prime twins. Let $p_i$ be the smallest of the $i$ th prime twin. Does that imply that $\prod_i (1+\frac{1}{p_i})$ diverges ?
1
vote
3answers
11k views

Most efficient method for converting flat rate interest to APR.

A while ago, a rather sneaky car salesman tried to sell me a car financing deal, advertising an 'incredibly low' annual interest rate of 1.5%. What he later revealed that this was the 'flat rate' ...
10
votes
2answers
550 views

Is there any toy for learning algebraic manipulation of fractions?

Is there any toy for learning algebraic manipulation of fractions? If you don't know of any, how would you design one? What I'm imagining is something similar to a Rubik's cube whose manipulation ...
6
votes
9answers
2k views

Are all integers fractions?

In a college class I was asked this question on a quiz in regards to sets: All integers are fractions. T/F. I answered False because if an integer is written in fraction notation it is then ...
6
votes
6answers
348 views

How to show that $\frac{x^2}{x-1}$ simplifies to $x + \frac{1}{x-1} +1$

How does $\frac{x^2}{(x-1)}$ simplify to $x + \frac{1}{x-1} +1$? The second expression would be much easier to work with, but I cant figure out how to get there. Thanks
5
votes
2answers
174 views

Fraction raised to integer power

if I have $(p/q)^n$ where $p,q,n$ are integers and $p/q$ is a... I don't know what you call it. Not a whole number, but something like 15/7 where you can't reduce it any more and it's non-integer. Can ...
4
votes
3answers
395 views

Fractions with radicals in the denominator

I'm working my way through the videos on the Khan Academy, and have a hit a road block. I can't understand why the following is true: $$\frac{6}{\quad\frac{6\sqrt{85}}{85}\quad} = \sqrt{85}$$
3
votes
3answers
126 views

What is $\lim_{n \to \infty} \sum_{x=0}^{n-1} \frac{n-x}{n+x}$?

These are two little questions that came to mind while I was looking at this problem. What is $\displaystyle \lim_{n \to \infty} \sum_{x=0}^{n-1} \frac{n-x}{n+x}$? I am fairly certain that the ...
3
votes
1answer
298 views

length of period

Each rational number (fraction) can be written as decimal periodic number. Does exists a method or hint that show how long will be the period of arbitrary fraction. For example $1/3=0.3333...=0.(3)$ ...
3
votes
1answer
103 views

$S^{-1}B$ and $T^{-1}B$ isomorphic for $T=f(S)$

Let $f:A\to B$ be a homomorphism of rings, $S$ be a multiplicatively closed subset of $A$ and $T=f(S)$. Then $S^{-1}B$ and $T^{-1}B$ are isomorphic as $S^{-1}A$-modules. First we define the ...
3
votes
1answer
628 views

Nested Division in the Ceiling Function

During class, we were introduced to a proof that used the ceiling function. We assumed (without proof) that: $$ \left\lceil{\frac{n}{2^i}}\right \rceil= ...