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.

learn more… | top users | synonyms (1)

2
votes
2answers
33 views

decimal to fractions

When being asked how to solve the Arithmetic Means of 8, 7, 7, 5, 3, 2, and 2, I understand that adding these numbers then dividing by 7 (the amount of numbers) gives me the decimal 4.85714... But ...
2
votes
2answers
79 views

Here are two fractions, $\frac{2}{3}$,$\frac{7}{8}$, which of these fractions are closer to $\frac{3}{4}$?

I've been throwing this question around my family. No one has a clue, therefore can someone help? I'm pretty sure this will be easy to do
0
votes
2answers
57 views

Fractions of an amount [closed]

I need help with the following problem: Catalin works in an office. One week he divides his time between these tasks: $\frac{1}{4}$ of his time in meetings $\frac{5}{8}$ of his time writing ...
2
votes
1answer
74 views

Are fractions with zero divisors in the denominator never well defined?

Are fractions with zero divisors in the denominator never well defined? I know that for a fraction in modular arithmetic to be well defined, the denominator must not be a zero divisor, e.g: $$ x ...
0
votes
2answers
35 views

Is there such a thing as complex rational numbers and does it have the same properties as the usual complex numbers as extension of the real numbers?

I've been wondering if there is any use to defining a set that is isomorphic to $\mathbb{Q}^2$ (in the same way that $\mathbb{C}$ is isomorphic to $\mathbb{R}^2$). I immediately see a problem with ...
0
votes
4answers
75 views

Dividing fractions in real life scenario / application

First of all sorry if this question sounds too stupid or offends anyone. One apple divide by two you get half an apple. $\large{\frac{1}{2} = 0.5}$ I couldn't get my head around with dividing ...
4
votes
6answers
449 views

Is $15/52$ equal to $17/59$?

Is $\frac{15}{52} = \frac{17}{59}$? I typed it into the calculator and found: $$\frac{15}{52} = 0.2884615 $$ $$\frac{17}{59} = 0.2881356 $$ So I thought they were different. But then my friend said ...
3
votes
4answers
385 views

SAT Maths Question About Fractions

Whilst revising, a problem caught my eye and I cannot seem to find an answer. I am usually bad at these types of questions. On a certain Russian-American committee, $\frac23$ of members are men, ...
0
votes
3answers
68 views

$(5x +1) ÷ (3x)$ is not a polynomial?

On the Mathwarehouse page on polynomial equations, it gives this expression as an antiexample, something that is not a polynomial: $(5x +1) ÷ (3x)$ However, it also says on the same page that if it ...
3
votes
2answers
95 views

Showing $r\in\mathbb Q\setminus \{0\}\implies e^r\notin \mathbb Q$

For a given $n>0$, let $\displaystyle J_n:x\to \frac{1}{n!}\int_{-x}^x(x^2-t^2)^ne^tdt$ a. Prove that there exists $A_n,B_n\in \mathbb R_n[X]$ such that $\forall x\in \mathbb R^+, ...
2
votes
1answer
19 views

Fractions and Largest Common Multiple, Algebra, Numerator and Denominator Identical Numbers?

This is the question find $x$ of equation: $$\frac{5x-2}{5} - \frac{2x+3}{2} = 3$$ I tried multiplying this all by 10, the LCM. It ended with: $x -x=49.$ How do you solve this without cancelling ...
3
votes
1answer
85 views

Is the set of real numbers really uncountably infinite?

The proof that the set of real numbers is uncountably infinite is often concluded with a contradiction. In the following argument I use a similar proof by contradiction to show that the set of ...
0
votes
5answers
47 views

How does this seemingly-trivial simplification work?

In a section on inductive proofs in the book Modelling Computing Systems: Mathematics for Computer Science (Muller, Struth) there is a simplification that is assumed to be trivial, but that I can't ...
7
votes
2answers
8k views

Is a non-repeating and non-terminating decimal always an irrational?

We can build $\frac{1}{33}$ like this, $.030303$ $\cdots$ ($03$ repeats). $.0303$ $\cdots$ tends to $\frac{1}{33}$. So,I was wondering this: In the decimal representation, if we start writing the ...
1
vote
1answer
44 views

Can we write “fractional root” symbol in math?

Fractional exponents are legit but I have never seen fractional roots, so I just wonder if we can write fractional roots such as this: It sometimes can be convenient to think about too.
5
votes
1answer
128 views

Roots of a Cubic Polynomial with Elementary Symmetric Polynomial Coefficients

Let $R_n$ be a set of $n$ distinct nonzero rational numbers. Let $e_k$ be elementary symmetric polynomials over $R_n$---i.e. $e_0=1$, $e_1 = \sum_{1\le i\le n} r_i$, $e_2 = \sum_{1\le i<j\le n} r_i ...
0
votes
3answers
29 views

Ordering an even and odd fraction that are close

We know that $1/4 < 5/11 < 1/2$. I did it this way from small to large: $$\frac{1 \cdot 3}{4 \cdot 3} = \frac{3}{12}$$ $$\frac{5}{11}$$ $$\frac{1 \cdot 6}{2 \cdot 6} = \frac{6}{12}$$ It is ...
0
votes
1answer
25 views

Simplification imaginary fractions

In an exercise, a partial fraction expansion has to be done. I have no problem with that, but one of the last steps includes a simplification as follows: \begin{equation*} \left( -\frac 12 - \frac 16 ...
2
votes
3answers
85 views

Is a prime to the power of a fraction always irrational?

Let $p$ be a prime number and let $x$ be a faction, i.e. $x \in \mathbb{Q} - \mathbb{N}$. It seems to be the case that $p^x$ is always irrational. How do I prove this?
0
votes
1answer
34 views

Question on rounding off rule

There is a specific rule(Banker's Rule I think) for rounding of numbers that end in 5. The rule is that we add 1 to the preceding digit of it's odd but keep it as it is if it's even. It's always ...
43
votes
1answer
864 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 ...
0
votes
1answer
49 views

Why does $ \frac {a}{b}$ of $c$ mean $ \frac {a}{b} \cdot c$ [closed]

When someone writes "$ \frac {a}{b}$ of $c$", why is the preposition "of" interpreted as multiplication of $c$ by $a/b$?
-1
votes
3answers
60 views

How to show simple inequality of fractions

If $$\frac {a}{a+b}<\frac{a'}{a'+b'}$$ then how can I show that $$\frac {a}{a+2b}<\frac{a'}{a'+2b'}\ \forall\ a,b,c>0$$ I tried puitting in a constant k so $$\frac ...
0
votes
1answer
19 views

Farey Sequence implemenatation

I'm trying to use the Farey sequence to get the next lowest reduced fraction in a list. For example, for $n = 8$, we have $\dots, \frac13, \frac38, \frac25, \frac37, \frac12, \dots$ So let's take ...
2
votes
2answers
30 views

Sign of fractional exponent [duplicate]

What is the sign of $-1^{\frac{2}{3}}$? I thought it was positive 1 because it involves squaring, but that doesn't seem to be the case. Why?
8
votes
3answers
106 views

Minimal $ab$ for Rational Number $a/b$ in an Interval

Given rational numbers $L$ and $U$, $0<L<U<1$, find rational number $M=a/b$ such that $L \le M<U$ and $(a\times b)$ is as small as possible---$a$ and $b$ are integers. For example, If ...
2
votes
1answer
168 views

Tetration and its inverse to various exponents

I've recently seen in my studies tetration, or the next operation in the addition, multiplication, exponentation... series. I've also heard much discussion about how to extend this operation to ...
1
vote
3answers
84 views

Closed form of a sum of ratios of integers

I am computing in a program this sum (does it have a "name"): $$\sum_{\alpha=2}^{K} \frac{\alpha-1}{\alpha}$$ is there a way to avoid the sum, term by term, and use a more compact closed form ?
3
votes
1answer
36 views

Real numbers and rationals - Decimal Expansion

How would one endeavor to show that A real number is rational if and only if its decimal expression ends in recurring digits?
5
votes
2answers
51 views

What automorphisms exist on the abelian group of positive rationals under multiplication?

Consider the abelian group $(\mathbb{Q}_{>0}, \times)$. What automorphisms exist for this group? I can only think of the trivial one and of $\phi(q) = \frac{1}{q}$. If we relax the problem to ...
1
vote
2answers
48 views

Solving equations including floor function.

I got a little trouble solving equations that involve floor function in an efficient way. For example : $$ \left\lfloor\frac{x+3}{2}\right\rfloor = \frac{4x+5}{3} $$ In the one above, I get that ...
4
votes
4answers
93 views

How does $-\frac{1}{x-2} + \frac{1}{x-3}$ become $\frac{1}{2-x} - \frac{1}{3-x}$

I'm following a solution that is using a partial fraction decomposition, and I get stuck at the point where $-\frac{1}{x-2} + \frac{1}{x-3}$ becomes $\frac{1}{2-x} - \frac{1}{3-x}$ The equations are ...
2
votes
1answer
36 views

Approximating non-rational roots by a rational roots for a quadratic equation

Let $a,b,c$ be integers and suppose the equation $f(x)=ax^2+bx+c=0$ has an irrational root $r$. Let $u=\frac p q$ be any rational number such that $|u-r|<1$. Prove that $\frac 1 {q^2} \leq |f(u)| ...
1
vote
1answer
45 views

Is every homeomorphism of $\mathbb{Q}$ monotone?

It is well known that every continuous injective map $\mathbb{R}\rightarrow\mathbb{R}$ is monotone. This statement is false for maps $\mathbb{Q}\rightarrow\mathbb{Q}$. (That is becaus $\mathbb{Q}$ is ...
5
votes
0answers
77 views

Prove that the square root of any irrational number is irrational.

The problem I'm having with this proof is that I'm not sure if my proof actually proves the theorem correct or if I'm using circular reasoning. Theorem: Prove that the square root of any irrational ...
1
vote
1answer
28 views

rational numbers as upper limit of a summation?

a quick question: Is it a legit way to use a fraction as the upper limit of a summation? Given is a frequency $f$ and a sample rate $f_s$. I want to use a sum like this: $\sum_{k=1}^{\frac{f_s}{2f}} ...
0
votes
5answers
50 views

canceling double fractions how?

I had this example: $$ \frac{\frac{11}{5}}{2} = \frac{11}{10} $$ then: $$ \frac{2\frac{1}{5}}{2} = \frac{11}{10} $$ $$ \frac{1}{5} \not= \frac{11}{10} $$ is this right canceling of double ...
1
vote
2answers
31 views

If $\frac{a}{b}=\frac{b}{c}=\frac{c}{d}$, prove that $\frac{a}{d}=\sqrt{\frac{a^5+b^2c^2+a^3c^2}{b^4c+d^4+b^2cd^2}}$

What I've done so far; $$\frac{a}{b}=\frac{b}{c}=\frac{c}{d}=k\\ a=bk, b=ck, c=dk\\ a=ck^2, b=dk^2\\ a=dk^3$$ I tried substituting above values in the right hand side of the equation to get ...
0
votes
0answers
28 views

Long division for multipolynomial expression, little o notation

I have this expression: $$\mathrm{Exp}=\frac{d^3(-12a^4)+d^2(4a^4-16a^3)+d(4a^3-6a^2-a)}{d^3(-12a^4+12a^3)+d^2(4a^4-20a^3+16a^2)+d(4a^3-11a+7a)+(1-2a+a^2)}$$ Is there any way I can take the second ...
2
votes
4answers
15k 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' ...
-2
votes
4answers
80 views

How to prove that $\frac{\ln 12}{\ln 18}$ is irrational witout using the change of base rule? [closed]

I have to show that $\frac{\ln 12}{\ln 18}$ is irrational by using change of base rule. At the beginning I have proved that $\ln r$ is irrational for any rational $r$, $r\ne 1$. Then using this we ...
1
vote
1answer
37 views

Compare density of rationals to the density of integers

Is is possible to somehow quantitatively compare the density of rational numbers to the density of integer numbers, ascribing to the both a number characterizing the density?
3
votes
2answers
562 views

How can the decimal expansion of this rational number not be periodic?

I just noticed that dividing $1 \div 998$ gives me the apparently non-periodic $$0.001002004008016032064\ldots ,$$ which is $$10^{-3} + 2\times 10^{-6} + 4\times10^{-9} + 8\times 10^{-12} + \cdots = ...
4
votes
2answers
1k views

Hint for: Prove any terminating decimal can be represented as a rational number

I am currently working on a problem from Hardy and have been stuck trying to figure out what to do. I was wondering if someone could provide me with a hint that may help jump-start my thought process. ...
1
vote
1answer
24 views

Order approximation for rational polynomial

I have this fraction: $\frac{(-12a^3)d^3 + (4wa^3 - 16a^2)d^2 + (5wa^2 - 8a)d - a^2w^2 + 2aw - 1}{(- 12wa^4 + 12a^3)d^3 + (4a^4w^2 - 20a^3w + 16a^2)d^2 + (4a^3w^2 - 11a^2w + 7a)d + a^2w^2 - 2aw + 1}$ ...
17
votes
7answers
8k views

What rational numbers have rational square roots?

All rational numbers have the fraction form $$\frac a b,$$ where a and b are integers($b\neq0$). My question is: for what $a$ and $b$ does the fraction have rational square root? The simple answer ...
72
votes
24answers
13k views

Why do we still do symbolic math?

I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical one. Numerical computations, to my understanding, never ...
2
votes
1answer
51 views

High School Probability and Contradiction

So I recently came across this question (2(a)) that my friend who teaches high school math posed to me. I thought the solution could be found by using the identities $P(B\,|\,A) = \dfrac{P(A\cap ...
0
votes
0answers
21 views

Calculating enrichment

My question concerns how enriched something is as im trying to combine several lists of uneven group size and the answer is escaping me. So basically, I have 6 groups and I want to compare them with ...
0
votes
2answers
47 views

$\frac{6}{4 \times 2} + \frac{7}{5 \times 2} + … + \frac{21}{19 \times 2}$

I got this exercise from school and I have no idea what notion to use, it resumes to Harmonic series, I can't find a generic answer. Do you have any idea? $\frac{6}{4 \times 2} + \frac{7}{5 \times 2} ...