Questions on fractions, which are expressions (not values) of the form $\frac pq$.

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88
votes
12answers
10k views

Can you be 1/12th Cherokee?

I was watching an old Daily Show clip and someone self-identified as "one twelfth Cherokee". It sounded peculiar, as people usually state they're "1/16th", or generally $1/2^n, n \in \mathbb{N}$. ...
61
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 ...
47
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 ...
39
votes
1answer
866 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 ...
37
votes
6answers
1k views
36
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 ...
30
votes
3answers
7k 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. ...
28
votes
14answers
6k views

Logic behind dividing negative numbers

I've learnt in school that a positive number, when divided by a negative number, and vice-versa, will give us a negative number as a result.On the other hand, a negative number divided by a negative ...
24
votes
8answers
1k 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 ...
23
votes
4answers
731 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 ...
23
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?
22
votes
1answer
644 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.
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 ...
18
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 ...
18
votes
7answers
1k views

Simple proof that $8\left(\frac{9}{10}\right)^8 > 1$

This question is motivated by a step in the proof given here. $\begin{align*} 8^{n+1}-1&\gt 8(8^n-1)\gt 8n^8\\ &=(n+1)^8\left(8\left(\frac{n}{n+1}\right)^8\right)\\ &\geq ...
18
votes
5answers
535 views

Can you prove why consecutive diagonal intersection points show decreasing fractions inside a rectangle?

When I was in third grade, I was playing with rectangles and diagonal lines, and discovered something very interesting with fractions. I've shown several math teachers and professors over the years, ...
15
votes
2answers
602 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 ...
15
votes
1answer
247 views

Closed-form of infinite continued fraction involving factorials

Is there a closed form of this: $$ 1!+\dfrac{1}{2!+\dfrac{1}{3!+\dfrac{1}{4!+\ldots}}} $$
14
votes
4answers
335 views

Show the identity $\frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a}=-\frac{a-b}{a+b}\cdot\frac{b-c}{b+c}\cdot\frac{c-a}{c+a}$

I was solving an exercise, so I realized that the one easiest way to do it is using a "weird", but nice identity below. I've tried to found out it on internet but I've founded nothingness, and I ...
13
votes
11answers
4k 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 ...
13
votes
2answers
189 views

Why do I get $0.098765432098765432…$ when I divide $8$ by $81$?

I got this remarkable thing when I divided $16$ by $162$, or, in a simplified version, $8$ by $81$. It's $0.098765432098765432\cdots$, or more commonly known as $0.\overline{098765432}$, with all the ...
12
votes
4answers
1k views

Is Cantor's diagonal argument dependent on the base used?

Applying Cantor's diagonal argument to irrational numbers represented in binary, one and only one irrational number can be generated that is not on the list. Wikipedia image: But if you change ...
12
votes
2answers
474 views

proof of $\sum\nolimits_{i = 1}^{n } {\prod\nolimits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } = 1$

i found a equation that holds for any natural number of n and any $x_i \ne x_j$ as follows: $$\sum\limits_{i = 1}^{n } {\prod\limits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } ...
12
votes
3answers
224 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 ...
11
votes
8answers
4k 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
758 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 ...
10
votes
2answers
348 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 ...
10
votes
4answers
903 views

How do I rewrite -100+1/2 as the mixed number -99 1/2?

This has been bugging me for some time now, so I ask you to try to help me realize what is going on here. I just can't get my brain around this. I have a proper fraction and a negative integer. The ...
10
votes
2answers
561 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 ...
9
votes
0answers
253 views

To how many decimals is $\sum_ {k=1}^\infty \frac{k}{\sqrt{k!}} = \frac{49850839\,\pi}{29567947}$ correct?

Consider: $$\sum_ {k=1}^\infty \frac{k}{\sqrt{k!}} = \frac{49850839\,\pi}{29567947}$$ This is, as far as I'm able to check with my software, correct to at least 167 decimals. If anyone has the ...
8
votes
3answers
650 views

Numbers whose self and reciprocal are finitely decimally expressable that are close to one?

How would I go about finding numbers x such that x and 1/x are finitely decimally reciprocal and are also close to 1? I'm not entirely certain how to phrase this question, but take for example 2. 2 ...
8
votes
4answers
203 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 ...
8
votes
2answers
270 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 ...
8
votes
3answers
281 views

Why do these fractions give $99…9$?

Today, as usual, we were doing all those boring numerical computations in our calculators. It all started when my professor replaced $0.2$ with $1/5$. I got into calculating the unit fractions one by ...
8
votes
1answer
525 views

Does it make sense to multiply slopes?

Multiplying fractions is a regular occurance. If those fractions are considered slopes, does it make any sense? For example, if these fractions are slopes,$\frac{9}{8} \times \frac{49}{48},$ does the ...
8
votes
1answer
158 views

$\frac{x}{10!} = \frac{1}{8!} + \frac{1}{9!}$

I have a pretty simple straightforward question. Q) Find the value of $x$ in the following: $$\frac{x}{10!} = \frac{1}{8!} + \frac{1}{9!}$$ Instinctively, I do the quickest thing I know how to ...
8
votes
1answer
798 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 $$x=\sum_{k=1}^n{\frac{1}{a_k}}$$
8
votes
4answers
188 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
1k views

Remainder when $20^{15} + 16^{18}$ is divided by 17

What is the reminder, when $20^{15} + 16^{18}$ is divided by 17. I'm asking the similar question because I have little confusions in MOD. If you use mod then please elaborate that for beginner. ...
8
votes
2answers
97 views

Integrating $\int{\frac{\sqrt{1-x^2}}{(x+\sqrt{1-x^2})^2} dx}$

I am a little bit lost with integral: $$\int{\frac{\sqrt{1-x^2}}{(x+\sqrt{1-x^2})^2} dx}$$ I have already worked on in and done substitution $x = \sin(t)$: This brings me to: ...
8
votes
1answer
157 views

cyclic permutations of periods of recurring fractions

In base 10, the recurring bits of the fractions $\frac{1}7,\ldots,\frac{6}7$ are cyclic permutations of each other. e.g. $$\frac{1}{7}=0.(142857)$$ $$\frac{2}{7}=0.(285714)$$ ...
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 ...
7
votes
5answers
404 views

Solving inequality $\frac{2x}{x-2}>1$

I'm trying to solve $$\frac{2x}{x-2}>1$$ but I can't seem to get the correct answer. I'm doing something wrong but I don't know what; that is why I'm asking. This is what I've got: ...
7
votes
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 ...
7
votes
1answer
141 views

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer. I've tried to bring all fractions under commmon denominator and it didn't helped me much. With guessing I find out ...
7
votes
2answers
270 views

Rationalizing the denominator 3

It is a very difficult question. How can we Rationalizing the denominator? $$\frac{2^{1/2}}{5+3*(4^{1/3})-7*(2^{1/3})}$$
7
votes
1answer
209 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?
7
votes
3answers
163 views

Why does partial fraction decomposition always work?

Say you have a function $p(x)/q(x)$ for some polynomials $p(x)$ and $q(x)$ and $p$ has a lower degree than $q$. Say $q$ has degree three and $p$ has degree two. If you partially decompose it, you'll ...
7
votes
2answers
216 views

How to best understand Euclid's definition of equal ratios? How does it relate to Dedekind cuts?

This is something I've been wondering about. When I think of "ratios" $x/y$ and $z/w$ as being "equal", with $x$, $y$, $z$, and $w$ being real numbers, this means the results of dividing the real ...
6
votes
9answers
1k 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 ...