Questions on fractions, numbers of the form $p/q$ where $p$ and $q$ are integers, and $q$ is not zero.
2
votes
1answer
33 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
2answers
27 views
Solving Simple Mixed Fraction problem?
How do you wrap your head around mixed fraction, does anyone knows how to figure out, can someone give me an example how it can be solved?
1
vote
1answer
84 views
Am I right or is Wolfram right?
Let ${a_n}$ be a sequence whose corresponding power series $A(x)=\sum_{i\geq 0}a_ix^i$ satisfies
$$A(x)=\frac{6-x+5x^2}{1-3x^2-2x^3}$$
Determine a recurrence relation that ${a_n}$ satisfies.
I ...
0
votes
0answers
29 views
Solving $m$ in $m = \lim_{n\to\infty}\prod_{k=x+1}^n\, 1+\dfrac{(k+x)^2}{2^{k-x}}$ from $n$ and $x$
How should one proceed in order to solve $m$, where $x$ is an integer
$$m = \lim_{n\to\infty}\prod_{k=x+1}^n\, 1+\dfrac{(k+x)^2}{2^{k-x}} $$
from $n$ and $x$ in an unconditional form, such as, for ...
0
votes
0answers
17 views
Computability of division of large numbers
What is the largest computable mathematical division in terms of the number of digits that can be handled by a typical desktop computer using the best available big number libraries, assuming input is ...
1
vote
1answer
32 views
Series of Fractional Sums
What is the sum of $$\frac{1}{2\times 5} + \frac{1}{5\times 8}+ \frac{1}{8\times 11}+...+ \frac{1}{2009\times 2012}?$$
What is the easiest way to solve this kind of problem?
3
votes
5answers
40 views
Simplify with fractional exponents and negative exponents
I am trying to simplify
$$ \left(\frac{3x ^{3/2}y^3}{x^2 y^{-1/2}}\right)^{-2} $$
It seems pretty simple at first. I know that a negative exponent means you flip a fraction. So I flip it.
$$ ...
5
votes
3answers
108 views
Using the hypothesis $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ to prove something else
Assuming that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$$
Is it possible to use this fact to prove something like:
...
1
vote
3answers
52 views
Simplifying fractions - Ending up with wrong sign
I've been trying to simplify this
$$
1-\frac{1}{n+2}+\frac{1}{(n+2) (n+3)}
$$
to get it to that
$$
1-\frac{(n+3)-1}{(n+2)(n+3)}
$$
but I always end up with this
$$
1-\frac{(n+3)+1}{(n+2)(n+3)}
$$
Any ...
6
votes
4answers
64 views
$20^{15} + 16^{18}$ is divided by 17
What is the reminder, when $20^{15} + 16^{18}$ is divided by 17.
I'm asking the similar question because I have little confusions in MOD.
If you use mod then please elaborate that for beginner.
...
4
votes
3answers
35 views
Why does the Denominator of the Denominator go to the Numerator?
Quite blindly I've learnt a basic rule about fractions:
The Denominator of the Denominator goes to the numerator.
I'm confused about it and I'll give an example as to why.
Imagine the following: ...
1
vote
2answers
35 views
Proving that $\frac{\sigma_{n-1}}{\omega_n} = n$ in $\mathbb{R}^n$
If $\sigma_{n-1}$ was the surface area of the unit sphere in $\mathbb{R}^n$ and $w_{n}$ was the area of the unit ball in $\mathbb{R}^n$, my lecture notes prove that $$\frac{\sigma_{n-1}}{\omega_n} = ...
1
vote
2answers
86 views
when the numerator is less than the denominator
when the numerator is less than the denominator the result is always between 0 and 1?
for example if I have a number like
x/y where x<y then the result will be ...
0
votes
1answer
50 views
how to solve this math problem for homework [closed]
the fifth graders vote to determine where they want to go on a field trip. one-third of fifthe graders want to go bowling. two-sixths of the fifth graders want to goto the zoo. one-sixth of the ...
2
votes
2answers
27 views
Fraction question
Alvin and Bob had a total of 60 marbles. Alvin gave 1/4 of his marbles to Bob. Bob then gave 1/3 of the total number of marbles he had to Alvin. In the end, each of them had the same number of ...
2
votes
2answers
32 views
Simple fractions question
Mary and Henry shared a collection of stamps. Mary had 7/10 of the total number of stamps. If Mary gave 38 stamps to Henry, she will have thrice the number of stamps Henry have. How many stamps did ...
2
votes
4answers
40 views
Dividing and multiplying surds - Rule
What rule/process allows me to take this equation:
$$6x^{2} \cdot \sqrt{\frac{y}{x}}$$
And simplify it to become:
$$6x \cdot \sqrt{xy}$$
1
vote
2answers
41 views
Recurring decimals to fraction
$.\overline{36}=\frac{36}{99}$
$2.1\overline{36}=2\frac3{22}$
The part I do not understand however, is "you could used 1) to speed up the working of 2)" which is written in the book.
How would ...
2
votes
2answers
156 views
How to prove the fraction identity without using calculator
How to prove without calculator that
$$ \frac{1}{1001} + \frac{1}{3001} > \frac{1}{1000}$$
7
votes
16answers
1k views
Why is $\;n^2-\frac{n^2}{2} =\frac{n^2}{2}\;$? [closed]
Could someone please expand on how to get from $\;\displaystyle\left( n^2-\frac{n^2}{2}\right)\;$ to $\;\left(\dfrac{n^2}{2}\right)\;?\;$
I can't seem to wrap my head around that.
3
votes
2answers
32 views
To split in to partial fractions, the expression $\frac{1}{x^2(x+a)^2}$
To split in to partial fractions, the expression $\frac{1}{x^2(x+a)^2}$
$\frac{1}{x^2(x+a)^2}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{(x+a)}+\frac{D}{(x+a)^2}$
One method of finding the values of the ...
2
votes
1answer
28 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
43 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 ...
1
vote
2answers
49 views
How can be done by the method of mathematical induction?
We are given that $P(x+1)-P(x)=2x+1$
We also know that $P(0)=1$
We want to prove that $P(2004)=(2004)^2 +1$
Can someone explain how can be solved with mathematical induction?
Thank you in advance!
5
votes
2answers
49 views
Find non-repeating decimal between two fractions
I have two fractions which, when evaluated, may or may not be non-terminating (they go on forever). They're both between 1 and 0.
I need to find the shortest decimal number that lies between these ...
0
votes
2answers
43 views
When is a fraction simplified?
When is a fraction simplified?
"A fraction is simplified if the numerator and denominator do not have any common factors other than 1."
This is what I read on this website: ...
0
votes
2answers
70 views
Check my answers to these problems on rationalizing denominators? [closed]
I just need to check my answers for these problems. I did the work already on paper. I'm not trying to get free answers because I know that would hurt me in the long run. So here's the questions:
...
0
votes
1answer
43 views
Ring of fractions in $\mathbb{Z}/35\mathbb{Z}$
How can I determine $S^{-1}(\mathbb{Z}/35\mathbb{Z})$, where $S$ consists of of all elements of $\mathbb{Z}/35\mathbb{Z}$ except $0,5,10,15,20,25,$ and $30$?
1
vote
2answers
58 views
most efficient way to convert a number into a fraction
supposing I have a decimal like
$$ 0.30000000000000027$$
What would be the best way to know the same number but in a fraction way like we know
$\dfrac{1}{3} > 0.30 > \dfrac{1}{4}$
because ...
1
vote
1answer
32 views
Integer+fraction vs Top-heavy fraction
What is the name of a fraction like this:
$$\frac{22}{7}$$
as opposed to one like this:
$$3\frac{1}{7}$$
I've never actually had to describe this until today. Not only have I no idea how to ...
3
votes
2answers
48 views
Separating $\frac{1}{1-x^2}$ into multiple terms
I'm working through an example that contains the following steps:
$$\int\frac{1}{1-x^2}dx$$
$$=\frac{1}{2}\int\frac{1}{1+x} - \frac{1}{1-x}dx$$
$$\ldots$$
$$=\frac{1}{2}\ln{\frac{1+x}{1-x}}$$
I ...
0
votes
3answers
50 views
system of equations with three equations.
We have to find all real solutions to this system of equations:
$$x=\frac{4z^2}{1+4z^2},y=\frac{4x^2}{1+4x^2},z=\frac{4y^2}{1+4y^2}$$
1
vote
2answers
94 views
Why are fractions with a negative denominator valid?
Whenever in a fraction, there is $0$ in the denominator, the fraction becomes $\infty$ or indeterminate. But why do we consider those fractions valid that have some negative numbers in the denominator ...
0
votes
3answers
98 views
$3 \dfrac12$ divided by $\dfrac45\,$ ; why do I get 4.3?
This has been bothering me a lot, this is my thinking:
$3\dfrac12 \implies \dfrac72 \implies \dfrac{35}{10}$
similarly $\dfrac45 \implies \dfrac{8}{10}$
So
...
3
votes
3answers
102 views
Lottery Odds as Multiples of Fractions
I run a Lottery syndicate for the UK lottery, and we play 30 lines per draw.
The odds of winning £10 (3 matching numbers) is deemed to be 1 in 56.7 (or 1/57 for the purposes of this question).
...
0
votes
1answer
52 views
summation of fractions and inequalities
I am trying to prove that $\sum_{i=1}^{n}\frac{1}{a_i}\leqslant 2$, where all $a_i$ are less than 1000, and all $a_i$ have a lowest common multiple greater than 1000.
This is what I have done so far:
...
0
votes
2answers
59 views
Simplifying $\frac{4}{2x-7}-\frac{3}{(2x-7)^2}$
A homework question asks me to "perform the addition or subtraction and simplify"
$$
\begin{gather}
\frac{4}{2x-7}-\frac{3}{(2x-7)^2}=4(2x-7)-3=8x-28-3=8x-31 \\
8x=31 \\
x=\frac{31}{8}
\end{gather}
...
0
votes
2answers
53 views
Aproximate calculation in decimals
I am trying to refresh on precision of calculations.
If we have the decimal fractions:
$.234673$, $.322135$, $.114342$, $.563217$ each known to be correct to six figures why are each of the decimals ...
6
votes
5answers
201 views
An interesting problem with fractions
These are a few examples of how "forbidden" procedures can lead us to the correct answer:
$$\displaystyle\frac{1\not4^1}{2\not8_2} = \frac{11}{22}=\frac{1}{2}$$
...
9
votes
7answers
1k views
Is 1 divided by 3 equal to 0.333…?
I have been taught that $\frac{1}{3}$ is 0.333.... However, I believe that this is not true, as 1/3 cannot actually be represented in base ten; even if you had ...
1
vote
3answers
49 views
Simplify a sum of fractions
I am stuck trying to get from:
$$\frac{pZ(a)}{pZ(a) - (1-p)Z(b)} - \frac{p(pZ(a) - (1-p)Z(b))}{pZ(a) - (1-p)Z(b)} $$
to
$$\frac{p(1-p)(Z(a) - Z(b))}{pZ(a) - (1-p)Z(b)} $$
Obviously my problem is ...
0
votes
2answers
228 views
How to enter subscript characters in WolframAlpha? [closed]
I'm trying to enter equations like this in WolframAlpha.
How do I format this?
1
vote
2answers
43 views
Monotonicity of a fraction
If I have a fraction $f(x) = \dfrac{n(x)}{d(x)}$, where $n(x)$ increases monotonically and $d(x)$ decreases monotonically; as functions of $x$.
Can I be sure that $f(x)$ increases monotonically as a ...
5
votes
1answer
61 views
finite field to rational fraction
Suppose I have a number $n\in\mathbb F_p$, i.e. an element of the finite field obtained by arithmetic modulo some (odd) prime $p$. I'm looking for a way to find a simple description of $n$ as a ...
3
votes
1answer
69 views
Given two ratios $\frac{p_i}{q_i}$, what is $\frac{p_1+p_2}{q_1+q_2}$ in their terms
I am ashamed to say that I cannot figure this one out:
I am given two ratios $\dfrac{p_i}{q_i}$ where $i=1$, $2$. (We just know the ratios and not the numbers $p_i, q_i$. What I mean by this is ...
0
votes
1answer
50 views
Inventory of Clocks and Frequency of Chimes
How do you determine the hours for which the clocks chime?
6
votes
9answers
318 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 ...
2
votes
1answer
49 views
Which numbers will remain if I keep removing the second third of the remaining interval?
Inspired by this Google Code Jam problem - Vanishing Numbers
There is a pool of numbers which are arbitrary decimal fractions from
the interval (0, 1). In the first round of the game the middle ...
0
votes
2answers
662 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' ...
4
votes
3answers
101 views
How does $({{n/e})^n} / ({({n/{2e}})^n})$ simplify to $2^n$ (MIT OpenCourseware 6.006)
As stated in the title, how is the following simplification performed?
$$\frac{\left(\frac{n}{e}\right)^n}{\left(\frac{n}{2e}\right)^n}=2^n$$
This was shown by a student in this Recitation video ...
