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
I(alpha) = ∫1/((x+alpha^2)(x+1)) dx between the limits of 0 and infinity
Evaluate the integral above depending on the parameter alpha using partial fractions.
The Attempt at a Solution
1/((x+alpha^2)(x+1)) = A/(x+alpha^2) + B/(x+1)
1 = A(x+1) + B(x+alpha^2)...
I am trying to separate out
\[
\frac{s}{(s+1)^3}
\]
for an inverse Laplace transform.
How does one setup up partial fractions for a cubic? I know for a square I would do
\[
\frac{A}{s+1} + \frac{Bs+C}{(s+1)^2}
\]
I tried doing
\[
\frac{A+Bs}{(s+1)^2} + \frac{Cs^2+Ds+E}{(s+1)^3}
\]
which led to...
Homework Statement
Use integration by parts to evaluate the integral
∫(7-6x) / (x2-4x+13)The Attempt at a Solution
This is a question from my notes so I already have the solution but I'm not sure what's going on at this one specific step.
∫(7-6x) / (x2-4x+13)
= -∫(6x-7) / (x2-4x+13)
= -∫(...
Homework Statement
Solve the following system of linear equations.
##\begin{array}{cc} \frac{1}{2}x+y-\frac{3}{4}z=1
\\ \frac{2}{3}x-\frac{1}{3}y+z=2
\\ x-\frac{1}{5}y+2z=1 \end{array}##
The Attempt at a Solution
Can I just do elimination by addition? So if I multiple the first equation by...
Homework Statement
Well this is part of an integration process, namely:
\int \frac {sin^2x}{4+3cos^2x}dx
Homework Equations
My attempt involved using a u-substitution, namely t = tan x
The Attempt at a Solution
Using t = tan x, sin^2 x = \frac {t^2}{1+t^2} and cos^2 x = \frac...
*If I turn out to have a wrong answer, please no hints or showing an valid proof. I want to do it on my own !
ad/bd-bc/bd=+-1/bd is neighbor fraction
Now, reduce the common numbers :
a/b-c/d=+-1/bd
We must now prove that the left hand side has irreductible fractions. Let's see what...
Is it necessary for a unit sum composed of unit fractions to include 1/2? Doing maple runs this seems to be the case, but it is not evident to me how this could be
Edit: In fact it seems it could not be, given the Erdos Graham problem Erd?s?Graham problem - Wikipedia, the free encyclopedia
But...
Homework Statement
∫▒√(1+x^2 )/x dx
Homework Equations
The Attempt at a Solution
I don't know how to break this up. I know we break partial fraction problems up based on their denominator, however the denominator i this problem is just 'x'.
Homework Statement
Why when I try to evaluate this with Partial Fractions, why do I end up with the original function?
\int\frac{n}{(n^{2}+1)^{2}}
\frac{n}{(n^{2}+1)(n^{2}+1)}
\frac{Ax+B}{n^{2}+1} + \frac{Cx+D}{(n^{2}+1)^{2}}
1n = (An+B)(n^{2}+1) + Cx + D
0n^{3}+ 0n^{2} + 1n + 0n^{0} =...
EDIT: If you're reading this and are still learning algebra basics, IGNORE this. I made a wrong assumption, thanks to MarkFL for pointing that out!
So far, I was led by my own assumptions to believe that this:
a1
\frac{1}{5} + \frac{y}{2} = 7
could be turned into this:
a2
\frac{2}{10} +...
Let X=\frac{1}{1\cdot2}+\frac{1}{3\cdot4}+\cdots+\frac{1}{2011\cdot2012} and Y=\frac{1}{1007\cdot2012}+\frac{1}{1008\cdot2011}+\cdots+\frac{1}{2012\cdot1007}.
Evaluate \frac{X}{Y}.
Homework Statement
∫(x+1)/(x2+2x+3)dx
The Attempt at a Solution
This problem was under partial fractions in my book. I solved it using u-substitution where u=x2+2x+3, but i can't see how it can be solved using partial fractions?
can it?
Homework Statement
∫ 4x/(x^3+x^2+x+1) dxThe Attempt at a Solution
I really don't know where to start, you can't complete the square, the degree of the numerator is less than the denominator so you can't use long division to simplify it.
I can't really simplify the denominator as well, so I...
Here is the question:
Here is a link to the question:
Integral of ((19x^2)-x+4)/(x(1+4(x^2)))? - Yahoo! Answers
I have posted a link there to this topic so the OP can find my response.
Homework Statement
Write out the form of the partial fraction decomposition of not determine the numerical values of the coefficients.
Homework Equations
x^4 -2x^3 + x^2 +2x -1 / x^2 -2x +1
The Attempt at a Solution
I did the division and I got x^2 + ((2x-1)) / (x^2 -2x +1)...
Hey guys, i have read many posts on physics forums but this would be my first post. I am quite stuck so any help would be much appreciated.
Homework Statement
Use Laplace transforms to solve the initial value problem:
f''(y) + 4f'(y) +8y = u(t-1) where y(0) = 1 and y'(0) = -1
Solve...
Homework Statement
Let x be any positive real number and suppose that ##x^2-ax-b=0## where ##a,b## are positive. I would like to use the equation that I provided in relevant equations which I proved to prove that
$$
\sqrt{\alpha^{2}+\beta}=\alpha+\cfrac{\beta}{2 \alpha+\cfrac{\beta}{2...
Absolute Vales with fractions HELP PLEASE :)
hey everyone,
Homework Statement
find the solution set:
l 1/x -3 l > 6
Homework Equations
none
The Attempt at a Solution
1/x > 9 or 1/x < -3
1 > 9x or 1 < -3x
1/9 > x or -1/3 > x
this answer doesn't seem...
Homework Statement
sqrt((2-sqrt(3))/4)
I tried to split whatever is under the radical sign into two separate parts, 1/2 and (-1/4)sqrt(3)
I realized that 1.5 times 2 is 3 and 2-1.5 = 1/2 so it seemed like it fulfilled the requirements for solving a double radical. so I put down...
Homework Statement
Solve for b. ##\frac{4}{3b-2}-\frac{7}{3b+2}=\frac{1}{9b^2-4}##
The Attempt at a Solution
##\frac{4}{3b-2}-\frac{7}{3b+2}=\frac{1}{(2b-2)(3b+2)}##...
Homework Statement
Find the integral and determine whether Cauchy's Theorem applies. Use partial fractions.
\large \oint \frac{tan \frac{z}{2}}{z^{4} -16} dz C the boundary of the square with vertices ±1, ±i cw
Homework EquationsThe Attempt at a Solution
I just wanted to check if approach is...
i understand the linear case...
example..
#/{(x+5)(x-4)} ----> A/(x+5) + B/(x-4)
but i don't understand this..
example..
#/{(x^2+3)(x^2+9)}------>(Ax+B)/{(x-√3)(x+√3)} + (Cx+D)/(x^2+9)
first of all... (x-√3)(x+√3)= x^2-3, which is nowhere in the original equation.. it's...
Homework Statement
Let w be a primitive n-th root of unity in some finite field. Let 0 < k < n. My question is how to rationalize
[\tex]\dfrac{1}{1 + w^k}[\tex].
That is, can we get rid of the denominator somehow? I know what to do in the case of complex numbers but here I'm at a loss...
How do I turn 1/(x4+1) into partial fractions?
This is what I did. Let me know if this is correct
1/(x4+1) = 1/[(x^2+i)(x^2-i)] = (Ax+B)/(x2+i) + (Cx + D)/(x2-1)
Then I set x = 0
1 = (D-B)i .. My first equation would be D-B = 0.
Is that correct so far?
Homework Statement
I want to factor polynomials. However, I want to know how it is possible to factor polynomials where the linear factors that result do not have integers but rather fractions.
Should we continue factoring if there are fractions, or do we have to stop?
Say we have this...
Determine which value best approximates the area of the region between the x-axis and the graph of ##f(x)=\frac{10}{x(x^2+1)}## over the interval [1,3]. Make your selection on the basis of a sketch of the region and not by performing any calculations. Explain your reasoning.
(a) -6 (b) 6...
1) integral (upper bound:1, lower bound:0) (x^2+1)/(x^3+x^2+4x) dx
2) integral (upper bound:1, lower bound:0) (x^4+x^2+1)/(x^3+x^2+x-3) dx
Now I know how to use Partial Fractions,My question is:
1) For the first part ln(x) is not defined at 0
¼ʃ1/x dx + ¼ʃ(3x-1)/(x²+x+4) dx
= ¼ ln|x| +...
Hello all,
I'm working through old exams for an electrical subject (no solutions given) and I think I've gone wrong somewhere and been left with something I'd like to learn how to work with anyway:
\frac{50}{(s+\frac{1}{s}+1)^2-s^2}
\frac{50}{2s+3+\frac{2}{s}+\frac{1}{s^2}}\times...
Question:
http://gyazo.com/bcb6c97ba462cdf4964121a6a40b0753
Mark scheme:
http://gyazo.com/b0475e7cb980ce98fb443932c28deed2
What I don't understand is the question specifies that x is not equal to 1/3 or -2, so why are you allowed to sub them into get the correct value for A B and C?
Here is what I have written down for describing math from multiplication up to dividing fractions, and as you see I can use pies to help describing the math but for dividing fractions I get stuck with how to use a pie analogy.
Here is what I have wriote to describe math using pie analogy...
Destiny C Sweet on Facebook writes:
Can someone take a look at this mixed number multiplication problem below... and explain to me how they got this answer? When you work the problem out find the product and reduce to the lowest terms...
\[4\frac{3}{4}\times 9\frac{4}{8}=45\frac{1}{8}\]
So the problem is ∫(6x+5)/(2x+1)dx. I know the proper way to solve this is to long divide these two expressions and then solve. However, I tried doing it with substitution.
u = 2x+1
dx = du/2
I then reasoned that 3u + 2 = 6x+5 since 3(2x+1) + 2 = 6x+3+2 = 6x+5 so I substituted it on top...
I realized that some of my problem with Algebra is also that I don't often check to see if a fraction is reduced to its simplest form, and as a general rule, should I? For example: 4/8 = 1/2, I will write the answer as 4/8, and leave it at that. If I am unsure, should I just do it?
Homework Statement
These are the two equations:
(7 + x)/5 - (2x - y)/4 - 3y = -5
(5y - 7)/2 - (4x - 3)/6 - 18 = -5x
Homework Equations
The Attempt at a Solution
When I try to simplify it, I end up with:
6x + 55y = 128
34x + 15y = 129
I'm not sure where to go with...
Hello MHB,
I got stuck on this integrate
\int_0^{\infty}\frac{2x-4}{(x^2+1)(2x+1)}
and my progress
\int_0^{\infty} \frac{2x-4}{(x^2+1)(2x+1)} = \frac{ax+b}{x^2+1}+ \frac{c}{2x+1}
then I get these equation that I can't solve
and I get these equation..
2a+c=0 that is for x^2
2b+a=2 that is for x...
Let's start with:
$$ \int \frac{dx}{1+x^2} = \arctan x + C $$
This is achieved with a basic trig substitution. However, what if one were to perform the following partial fraction decomposition:
$$ \int \frac{dx}{1+x^2} = \int \frac{dx}{(x+i)(x-i)} = \int \left[ \frac{i/2}{x+i} -...
[This item has also been simultaneously posted on MHF]
Polynomial Rings, UFDs and Fields of Fractions In Dummit and Foote Section 9.3 Polynomial Rings that are Unique Factorization Domains, Corollary 6, reads as follows...
I'm working on simple game and am working on a leveling system, using a function to get experience needed. I am using area under a function above y=0.
The first problem, I can't figure out a simple number.
f(x) = x2/5 dx
Then, looking for area, I'm unsure about a really simple thing.
Getting...