A fraction (from Latin fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples:
1
2
{\displaystyle {\tfrac {1}{2}}}
and
17
3
{\displaystyle {\tfrac {17}{3}}}
) consists of a numerator displayed above a line (or before a slash like 1⁄2), and a non-zero denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.
In positive common fractions, the numerator and denominator are natural numbers. The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. The denominator cannot be zero, because zero parts can never make up a whole. For example, in the fraction 3/4, the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates 3/4 of a cake.
A common fraction is a numeral which represents a rational number. That same number can also be represented as a decimal, a percent, or with a negative exponent. For example, 0.01, 1%, and 10−2 are all equal to the fraction 1/100. An integer can be thought of as having an implicit denominator of one (for example, 7 equals 7/1).
Other uses for fractions are to represent ratios and division. Thus the fraction 3/4 can also be used to represent the ratio 3:4 (the ratio of the part to the whole), and the division 3 ÷ 4 (three divided by four). The non-zero denominator rule, which applies when representing a division as a fraction, is an example of the rule that division by zero is undefined.
We can also write negative fractions, which represent the opposite of a positive fraction. For example, if 1/2 represents a half dollar profit, then −1/2 represents a half dollar loss. Because of the rules of division of signed numbers (which states in part that negative divided by positive is negative), −1/2, −1/2 and 1/−2 all represent the same fraction — negative one-half. And because a negative divided by a negative produces a positive, −1/−2 represents positive one-half.
In mathematics the set of all numbers that can be expressed in the form a/b, where a and b are integers and b is not zero, is called the set of rational numbers and is represented by the symbol Q, which stands for quotient. A number is a rational number precisely when it can be written in that form (i.e., as a common fraction). However, the word fraction can also be used to describe mathematical expressions that are not rational numbers. Examples of these usages include algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as
2
2
{\textstyle {\frac {\sqrt {2}}{2}}}
(see square root of 2) and π/4 (see proof that π is irrational).
Homework Statement
In the section of the book of integration by parts, there is an exercise that I don't even know how to tackle anymore. It's this:
\int \frac{xe^{2x}}{(1+2x)^2} dx
Homework Equations
{uv} - {\int v{du}}
The Attempt at a Solution
u = x ;
du = 1...
Can someone please help me with this problem. Please see the attached image.
The integration when removing the min and max is also provided, how is the answer x2/2 - x3/6? Do we not do anything with the 1 - 1/2 part? and the +2 part? Are we only looking at (x2-2x)/2?
I appreciate the help.
Now this is a pretty straight forward question. And I just want to make sure that I am not doing anything stupid.
But when doing partial fraction expansions of the type
\frac{K}{s^{2}+2\zeta\omega_{n}s+\omega_{n}^{2}} Shouldnt I always be able to factor the denominator into the following...
Can anyone help with the following question - see attachment.
Answer C is correct.
I have been told that it can be solved thus:
As our compound contains carbon and hydrogen we need to calculate these masses and deduct them from 128 to give us the mass of the chlorine and bromine...
Homework Statement
I want the simplify the following equation
Homework Equations
\frac{x}{{p}^{2}}+\frac{n-x}{{(1-p)}^{2}}, where p=\frac{x}{n}The Attempt at a Solution
I got this \frac{{n}^{2}}{x}-\frac{{n}^{2}}{n-x}, I don't think this is the right answer.
As part of a project I have been working on I fin it necessary to manipulate the following expression.
e^(icx)/(x^2 + a^2)^2 for a,c > 0
I would like to decomp it into the form
A/(x^2 + a^2) + B/(x^2 + a^2) = e^(icx)/(x^2 + a^2)^2
but I am having trouble getting a usable outcome.
\displaystyle\frac{1}{998,001} \;=\;0.\overline{000\,001\,002\,003\,004\,005\, \cdots\,996\,997\,999}\, \cdots
The decimal representation contains all the 3-digit numbers except 998
. . and the 2997-digit cycle repeats forever.
This is just one of a family of such fractions.
Can you determine...
Hi guys I have a question here relating integration by partial praction..
the question said what is the antiderivative of
x^2+2x-1/2x^3 +3x^2 - 2x
valid only when x > 1/2.
anyway i had poor background in math and working hard to catch up...
I don't understant why "valid only...
Homework Statement
x^2-x-13/(x^2+7)(x-2)
hello i am having trouble solving this problem.. could anyone please show me how to do this step by step? i know polynomial long division is required before it can be converted to partial fractions.
I also know the answer is 2x+3/x^2+7 - 1/x-2...
Homework Statement
Let ΔgM represent the difference in the gravitational fields produced by the Moon at the points on the Earth's surface nearest to and farthest from the Moon. Find the fraction ΔgM /g, where g is the Earth's gravitational field. (This difference is responsible for the...
Homework Statement
I want to check the answer to this question.
⌠ 2 x dx / (x+1) (x+2)
⌡ 1
Homework Equations
The Attempt at a Solution
For partial fraction I got A= -1 and B = 2
My final answer is -ln 2 + ln4 - ln3 = ln 4/ ln2 * ln 3
One of the major concerns in any Cosmic Microwave Background (CMB) anisotropy analysis is to determine which fraction of the observed signal is due to some foreground contaminant. Two sources of foreground contamination have been firmly identified: the diffuse Galactic emission and unresolved...
Homework Statement
\frac{2e^3}{((s^2)-6s+9)*s^3}
you can factorize the denominator into s,s,s,(s-3),(s-3)
that gives you 5 residuals.
the first 3 should all be the same value but that's apparently not correct, so where
am I going wrong?
Homework Statement
Julie tightened herself fastened a set of bathroom spring scales to a platform and went down a 30 degree slope. No friction was present and the platform supporting the scales was HORIZONTAL. what fraction of her true weight was indicated by the scales?
The Attempt...
Homework Statement
I'm supposed to decompose 1 / x(x2 + 1)2
Also, we haven't learned matrices yet so I can't use that technique to solve it.
Homework Equations
None.
The Attempt at a Solution
1 / x(x2 + 1)2 = A/x + (Bx + C) / (x2 + 1) + (Dx + E) / (x2 + 1)2
I multiplied...
This program has been assigned in my intro to programming course, which assumes no previous knowledge of programming. Up to this point we've only been required to write functions that accomplish specific tasks within a program. This is the first time we've been asked to design and write a full...
Homework Statement
Approximately what fraction of molecules of a gas (assumed ideal) have velocities for
which the angle φ lies between 29.5° and 30.5°, while θ lies between 44.5° and 45.5°?
Homework Equations
The Attempt at a Solution
What does the question even mean...
I was wondering what fraction of 3*3 square matrices are singular? I guess if the elements are real then the answer will be vanishingly small.
If however, the elements are integers, is there a way to work this number out?
Homework Statement \stackrel{lim}{x\rightarrow}∞ \sqrt{x^2+x}-x
I'm taking the limit of a rational function as x approaches ∞. However, within the problem, there is an algebraic arrangement I'm having trouble with. How would I get from the first fraction to the second fraction?
Applying...
I know that you cannot view \frac{dy}{dx} as a fraction, with dy and dx meaning infinitesimal changes in their respective variables, but what I do not understand is why not.
Can anyone explain to me why you can't (correctly) view the derivative as a fraction?
I want to split the fraction:
2a/((a+1)(a2+4))
I have tried using partial fractions, but came to something that was nonsense, and my question is why that is. Why doesn't partial fractions work in this case, from a mathematical point of view, and is it still possible to split up the fraction...
Homework Statement
Carbon 14 has a half-life of 5730 years, what fraction will have decayed in 2300 years?
Homework Equations
Nf/No =e^-kt
The Attempt at a Solution
Nf/No =e^-1.21 x 10^-4 x 2300
=0.068585374
However if I use Nf =No(1/2)^2300/5730
I get 0.757125224
I...
Basically i have this code, obviously the fraction is in display maths, but i want BIG products too, with sub/super scripts above and below rather than next to it. Thanks
$$
pr(X=k)=\mu\frac{\prod^{k-1}_{i=1}\{1-\mu+(i-1)\theta\}}{\prod^{k}_{i=1}\{1+(i-1)\theta\}}, \quad \mbox{for k$\geq$ 1,}
$$
A block slides a distance d down a frictionless plane and then comes to a stop after sliding a distance s across a rough horizontal plane, as shown in the accompanying figure.
What fraction of the distance s does the block slide before its speed is reduced to one-third of the maximum speed it...
This is throwing me through so many loops.
I have the equation 1/(x^3 + xa^2).
I can not for the life of me decompose this equation.
I use 1(x^3 + xa^2) = A/x + (Bx + C)/(v^2+a^2)
I can get A=1/a^2, but from there progress stops.
All examples on the internet and books only have...
Homework Statement
I am told this is wrong, but I am convinced it is right. I just need to show how but don't have a clue.
Homework Equations
(ab)/(2c(d+e))+(ab)/(2c(d-e))=(abd)/(c(d^2-e^2))
The Attempt at a Solution
I do not have a clue where to start. I am clueless as...
Homework Statement
Use the definition of a derivitive to find f'(x).
f(x)=1/(1-4X)
Homework Equations
Lim as h approaches 0 [f(x+h)-f(x)]/h
The Attempt at a Solution
I know that the answer is supposed to be -1/(1-4X)2 but I keep getting 4/(1-4X)2. This is what I have done so far (I...
Hello,
Is there any general formula for the partial fraction of the following function:
\frac{1}{(ax_1+1)(ax_2+1)\cdots (ax_L+1)}
I can work for L=3, but it get involved for larger L!
Thanks in advance
Homework Statement
I came across a problem that I can't solve
and it is ∫dx/x(x^2 + 4)^2
Homework Equations
None
The Attempt at a Solution
So I'm pretty sure this is to be solved by partial fraction since I am on a chapter on
Integration by partial fraction.
so I started with...
So do to a few issues I have definitely missed some thing in my materials class so I hope I'm not going to be told I'm an idiot. Please forgive me if I seem dense I am a single mother with a one year old who is also tackling an engineering degree ^_^'''
So without further ado... I am working...
Hi Everyone,
I am trying to calculate the mass of oxygen and keep running into dead ends and was wondering if anyone could offer any insight.
I am trying to find the resulting mass of oxygen in a 6m x 6m x 6m enclosure at standard temperature and pressure when the mole fraction of oxygen...
Homework Statement
Simplify
\frac{2}{49^{1/3}+7^{1/3}+1}Homework Equations
The Attempt at a Solution
I can simplify
\frac{2}{49^{1/3}+7^{1/3}}
but I don't have idea how to do this one..
EDIT: I have found it. No need to reply this. Thanks :)
how would one evaluate this without using trig substitution? Is it possible to make one integral out of this?
{int[(y^2 + a1^2)^-1]dy +c1}/{int[(x^2 + a2^2)^-1]dx +c2} +c3
the numbers behind the 'a's and 'c's are supposed to be subscripts.
Also, how would one deal with this...
Homework Statement
∫(1/(x^2-x+1)dx
Homework Equations
No idea
The Attempt at a Solution
I tried this by the subsitution method but that attempt was feeble as it only complicated the integral even further.
let t=x^2-x+1
this integral can neither be split into partial...
Homework Statement
The electric field of an electromagnetic wave is given by;
E = Re(\frac{1}{\sqrt{13}}E_{0}(2\widehat{x}+ 3i\widehat{y})e^{i(kz-wt)})
Identify the polarization state.
Homework Equations
I = |E_{x}|^{2} + |E_{y}|^{2}
Q = |E_{x}|^{2} - |E_{y}|^{2}
U = |E_{a}|^{2} -...
This problem comes from an engineering exercise (hence the C/R which you can ignore). I want to see if I got it right.
http://img51.imageshack.us/img51/3989/mama1h.jpg
http://img851.imageshack.us/img851/3401/mam2.jpg
Is it possible to have an solution to this sort of integral? And if not, why not?
\int_0^\infty \frac{e^{-ax}}{e^{-bx}+e^{-cx}}dx
Is a Taylor expansion the only way forward?
Many thanks
David
Hi all, am stuck with the integral with fraction in exponential
The equation
I=∫exp(bz)/(a+iz)*exp[(6*a^2-ik*w^2*z)/(z^2+a^2)]*dz
I already tried to partial fraction the 2nd exponential term, then i tried to perform integration by parts but it doesn't work well.i tried substitution too...
Q: Given a fraction A/B when does there exist a finite group G and an automorphism f s.t. exactly A/B elements of G are mapped to their own inverses (f(a) = a-1? If so how can we find the group? Does anything change if we allow infinite groups?
I have a friend who was preparing for an intro...
Homework Statement
What fraction of the air molecules in a house must be pushed outside while the furnace raises the inside temperature from 17.0° C to 28.0° C? The pressure does not change since the house is not 100% airtight.
Homework Equations
v2/v1=17+273/28+273
=.963
Why is my...
Homework Statement
\frac{\frac{2}{3}x(x^{2}+4)^{1/2}(x^{2}-9)^{-2/3}-x(x^{2}-9)^{1/3}(x^{2}+4)^{-1/2}}{x^{2}+4}
The Attempt at a Solution
\frac{x(x^{2}+4)^{-1/2}(x^{2}-9)^{-2/3}(\frac{2}{3}(x^{2}+4)-(x-9))}{x^{2}+4}
Then:
\frac{x(\frac{2}{3}(x^{2}+4)-(x-9))}{(x^{2}+4)^{3/2}(x^{2}-9)^{2/3}}...
Homework Statement
Show that n/(n+1)!=(1/n)-(1/(n+1)!)
I am totally lost on the algebraic steps taken to come to this conclusion. It is for an
Infinite series.
Thanks
Homework Statement
H(z) = \frac{6-z^{-1}}{1+0.5z^{-1}} + \frac{2}{1-0.4z^{-1}} = k + \frac{A}{1+0.5z^{-1}} + \frac{2}{1-0.4z^{-1}}
where A = (6-z^{-1}) is evaluated at z^{-1}=-2.
Homework Equations
Partial fraction expansion.
The Attempt at a Solution
Why is z^{-1} set equal to -2? I...
I was reading and came across a formula that didn’t look right so I tried to simplify the left side to match the right side; however, my fraction algebra is a little rusty and could use some guidance. I did get the 2 fractions to a common denominator; however, the best I could do was factor out...
I need help with Iterated Integration. I'm trying to integrate the following equation:
∫ylnx/x from e^y to y for dx, y>0
(sorry about the messy equation; first time user)
I know how to integrate ln, but the fraction has me thrown for a loop. I tried U-substitution, but the equation kept...
Homework Statement
i need to integrate:
\frac{1}{r(u-r)}dr
Homework Equations
u is a constant
The Attempt at a Solution
im not sure if i should decompose the fraction. i tried that, but it didnt seem to be of any help.
im studying for my circuits midterm and the proff has handouts with questions and answers but not detailed answers. i can't figure out how he went from a fraction to an answer.
(-j2)(2+j2)/-j2+2+j2 the answer on the paper is 2-j2
i do not know what I am allowed to do with the 2 next to...
\lim_{n\rightarrow ∞}\frac{3^n+2*5^n}{2^n+3*5^n}
I tried using l'hopitals rule and got
\lim_{n\rightarrow ∞}\frac{3^n*ln(3)+2(5^n*ln5)}{2^n*ln(2))+3(5^n*ln5)}
I'm not quite sure if that is the right way to approach this. This problem is an early one in the assignment so I assume that...