Questions tagged [fractions]

Questions on fractions, i.e. expressions (not values) of the form $\frac ab$, including arithmetic with fractions. Not to be confused with the tag (rational-numbers): fractions denote rational numbers, but the same rational number may be written in different ways as a fraction.

Filter by
Sorted by
Tagged with
95
votes
4answers
33k 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. $\...
38
votes
6answers
2k views

Finite Sum $\sum\limits_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}$

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 $(\star)$ by ...
88
votes
15answers
21k 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 ...
24
votes
7answers
3k views

Why are we allowed to cancel fractions in limits?

For example: $$\lim_{x\to 1} \frac{x^4-1}{x-1}$$ We could expand and simplify like so: $$\lim_{x\to 1} \frac{(x-1)(x^3 + x^2 + x + 1)}{x-1} = \lim_{x\to 1} (x^3 + x^2 + x + 1) = (1^3 + 1^2 + 1^1 + ...
38
votes
2answers
4k 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?
35
votes
12answers
4k 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 ...
30
votes
2answers
2k 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 ...
44
votes
7answers
2k 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 ...
8
votes
1answer
2k 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 $$x=\sum_{k=1}^n{\frac{1}{a_k}}$$
4
votes
2answers
1k views

When can't $dy/dx$ be used as a ratio/fraction?

By searching this question, I found this: Can I ever go wrong if I keep thinking of derivatives as ratios? However, the answers don't have what I'm looking for! (Edit: Meaning, a counterexample. ...
9
votes
10answers
8k 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 ...
4
votes
6answers
6k views

Proof of dividing fractional expressions

For dividing two fractional expressions, how does the division sign turns into multiplication? Is there a step by step proof which proves $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}...
47
votes
1answer
1k 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 k}}}}{\sum_{k=1}^{n^2+2n}\sqrt{\sqrt{2n+2}-{\sqrt{n+1+\...
4
votes
2answers
843 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 $p$...
41
votes
2answers
4k views

Weird large K symbol

Take a look at this symbol: $$ \pi=3 + \underset{k=1}{\overset{\infty}{\large{\mathrm K}}} \frac{(2k-1)^2} 6 $$ Does it look familiar to you? If so please help me!
10
votes
5answers
801 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 ...
5
votes
4answers
1k views

How do we find a fraction with whose decimal expansion has a given repeating pattern?

We know $\frac{1}{81}$ gives us $0.\overline{0123456790}$ How do we create a recurrent decimal with the property of repeating: $0.\overline{0123456789}$ a) Is there a method to construct such a ...
5
votes
2answers
19k views

Can we ever get an irrational number by dividing two rational numbers?

If we try to divide any two random arbitrarily long rational numbers like 103850.2387209375029375092730958297836958623986868349693868398659825528365... and <...
5
votes
4answers
293 views

Proof of one inequality $a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$

How to prove this for positive real numbers? $$a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$$ I tried AM-GM, CS inequality but all failed.
7
votes
1answer
256 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?
26
votes
4answers
1k views

Inequality with five variables

Let $a$, $b$, $c$, $d$ and $e$ be positive numbers. Prove that: $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a}\geq\frac{a+b+c+d+e}{a+b+c+d+e-3\sqrt[5]{abcde}}$$ Easy to show ...
42
votes
6answers
3k views

What is $\cdots ((((1/2)/(3/4))/((5/6)/(7/8)))/(((9/10)/(11/12))/((13/14)/(15/16))))/\cdots$? [duplicate]

What does this number equal if it goes on forever? $$\frac{\frac{\frac{\frac{\frac{1}{2}}{\frac{3}{4}}}{\frac{\frac56}{\frac78}}}{\frac{\frac{\frac{9}{10}}{\frac{11}{12}}}{\frac{\frac{13}{14}}{\frac{...
24
votes
1answer
860 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.
21
votes
5answers
2k views

If $\dfrac{a}{b} = \dfrac{c}{d}$ why does $\dfrac{a+c}{b + d} = \dfrac{a}{b} = \dfrac{c}{d}$?

Can anyone prove why adding the numerator and denominator of the same ratios result in the same ratio? For example, since $\dfrac{1}{2}=\dfrac{2}{4}$ then $\dfrac{1+2}{2+4}=0.5$.
10
votes
4answers
15k views

Finding modular of a fraction

Im really into cryptography and to find the private key of a message I need to use modular arithmetic. I understand how modular arithmetic using a clock with whole numbers. But I get really stuck when ...
11
votes
1answer
349 views

Does the sum of the reciprocals of composites that are $ \le $ 1

The sum itself: $$ \frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{12}+\frac{1}{14}+ \frac{1}{15}+ \frac{1}{39}... \le 1 $$ These are all sums of reciprocals of composites that ...
7
votes
4answers
381 views

Prove $\sum{\frac{1}{(x+2y)^2}} \geq\frac{1}{xy+yz+zx}$

Let $x,y,z>0$: Prove that: $\frac{1}{(x+2y)^2}+\frac{1}{(y+2z)^2}+\frac{1}{(z+2x)^2} \geq\frac{1}{xy+yz+zx}$ I tried to apply Cauchy - Schwarz's inequality but I couldn't prove this inequality!
4
votes
5answers
352 views

What is the value of $\frac{\sin x}x$ at $x=0$?

On plotting graph for $\frac{\sin x}{x}$ using Wolfram|Alpha and Google, got that : also, I can get the value of $\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1$ using squeeze theorem and as illustrated ...
9
votes
3answers
16k views

Detecting that a fraction is a repeating decimal

Given any fraction where both the numerator (N) and denominator (D) are both positive and are both whole numbers. Without manually dividing N by D, is it possible to pre-determine if the resulting ...
4
votes
7answers
576 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 don'...
1
vote
5answers
916 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).
336
votes
18answers
37k views

Find five positive integers whose reciprocals sum to $1$

Find a positive integer solution $(x,y,z,a,b)$ for which $$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$ Is your answer the only solution? If so, show why. I was ...
58
votes
3answers
9k views

Why do we miss 8 in the decimal expansion of 1/81, and 98 in the decimal expansion of 1/9801?

Why do we miss $8$ in the decimal expansion of $1/81$, and $98$ in the decimal expansion of $1/9801$? I've seen this happen that when you divide in a fraction using the square of any number with only ...
23
votes
12answers
4k 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 ...
13
votes
4answers
10k views

Length of period of decimal expansion of a fraction

Each rational number (fraction) can be written as a decimal periodic number. Is there a method or hint to derive the length of the period of an arbitrary fraction? For example $1/3=0.3333...=0.(3)$ ...
11
votes
2answers
279 views

If $\frac {1}{2+a} + \frac {1}{2+b} + \frac {1}{2+c} = 1$, prove $\sqrt{ab} + \sqrt{ac} + \sqrt{bc} \leq 3$

Let $a,b,c$ be non-negative numbers such that $$\frac {1}{2+a} + \frac {1}{2+b} + \frac {1}{2+c} = 1.$$ Prove that $ \sqrt{ab} + \sqrt{ac} + \sqrt{bc} \leq 3 $.
4
votes
0answers
214 views

Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $?

Let $f_- (n) = \Pi_{i=0}^n ( \sin(i) - \frac{5}{4}) $ And let $ f_+(m) = \Pi_{i=0}^m ( \sin(i) + \frac{5}{4} ) $ It appears that $$\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $$ Why is that so ? ...
4
votes
4answers
173 views

Let $(x, y, z)\in \Bbb R$, prove that $\big(\frac{2x-y}{x-y}\big)^2+\big(\frac{2y-z}{y-z}\big)^2 +\big(\frac{2z-x}{z-x}\big)^2 \geqslant 5$ [closed]

Olympiad Inequation Let $x$, $y$ and $c$ be distinct real numbers. Prove that: $$\left(\frac{2x-y}{x-y}\right)^2+\left(\frac{2y-z}{y-z}\right)^2+\left(\frac{2z-x}{z-x}\right)^2 \geqslant 5.$$ ...
7
votes
8answers
8k views

Effect of adding a constant to both Numerator and Denominator

I was reading a text book and came across the following: If a ratio $a/b$ is given such that $a \gt b$, and given $x$ is a positive integer, then $$\frac{a+x}{b+x} \lt\frac{a}{b}\quad\text{and}\...
5
votes
3answers
1k 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
0answers
646 views

Why is $\inf g \sup g = \frac{9}{16} $?

Consider this question here : Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $? Call that conjecture about $\frac{5}{4} $ conjecture $1$. Let $g(n) = \prod_{i=0}^n (\sin^2(n) + \frac{9}{16}) ) $ ...
2
votes
3answers
51 views

Is there a general relation between $a/b$ and $(a+c)/(b+c)$ where $a,b,c > 0 $?

Is there a general relation between $a/b$ and $(a+c)/(b+c)$ where $a,b> 0$ and $c\geq 0$ ? Is there a general proof for that relation ?
6
votes
2answers
170 views

Prove that $\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}}+\frac{1}{\sqrt{1+c^2}}\le\frac{3}{2}$

For positive real numbers with $a+b+c=abc$ prove that $$\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}}+\frac{1}{\sqrt{1+c^2}}\le\frac{3}{2}$$ I made the substitution $a=\tan(\alpha), b = \tan(\beta),...
4
votes
3answers
1k views

If the sum of two irreducible fractions is an integer, then the denominators are equal

I have to show the following:"If the sum of two irreducible fractions with positive denominators is an integer, then the denominators are equal." $$\frac{a}{b}+\frac{c}{d}=k, \text{ where k an integer ...
2
votes
1answer
2k 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= \left\lceil{\frac{\lceil{\frac{n}{2}}\rceil}{...
1
vote
3answers
282 views

sum $\sum _{n=2}^{\infty}\frac{(-1)^n}{n^2+n-2}$

I have the following series: $\sum _{n=2}^{\infty}\frac{(-1)^n}{n^2+n-2}$ I am not able to do the telescoping process in the above series. I converted that into following partial fraction: $\sum _{n=...
-2
votes
2answers
81 views

How to simplify the series of $x + x^{-1} + (x + x^{-1})^{-1} + (x + x^{-1} + (x + x^{-1})^{-1})^{-1}$? [closed]

$$x + \frac1x + \cfrac1{x + \frac1x} + \cfrac1{x + \frac1x + \cfrac1{x + \frac1x}} +\cfrac1{x + \frac1x + \cfrac1{x + \frac1x} + \cfrac1{x + \frac1x + \cfrac1{x + \frac1x}} } + \cdots $$ How can this ...
9
votes
2answers
3k 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 ...
5
votes
6answers
256 views

How $\frac{1}{\sqrt{2}}$ can be equal to $\frac{\sqrt{2}}{2}$?

How $\frac{1}{\sqrt{2}}$ can be equal to $\frac{\sqrt{2}}{2}$? I got answer $\frac{1}{\sqrt{2}}$, but the real answer is $\frac{\sqrt{2}}{2}$. Anyway, calculator for both answers return same numbers.
5
votes
4answers
713 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.