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).
Use partial fractions to integrate x^3/(x^3+1)The Attempt at a Solution
\int x^{3}/(x^{3}+1) dx
Homework Statement
Homework Equations
The Attempt at a Solution
\int x^{3}/x^{3}+1 dx
I know that first i have to perform long division but i am at a loss how to do this
THanks
Homework Statement
\int\frac{e^xcos(log_7(e^x+9))}{(e^x+9)ln(7)}dx
Homework Equations
The Attempt at a Solution
Let u= (ex+9)
du= exdx
New integral \int\frac{cos(log_7(u))}{(u)ln(7)}du
This is where I got l little lost. Should I let log7(u)=\frac{ln(u)}{ln(7)}? Or is this...
Is there a free LaTex editor where you can click on a fraction button, and the fraction bar appears along with two empty boxes on top of and below it, representing the numerator and denominator?
The LaTex editors that I've seen, when you click on the fraction button, just produce this...
Homework Statement
Hi folks! :smile:
This is just a conceptual question that has arisen during some reading. At one point the author states that for the reaction:
CO_2 \leftrightharpoons X_{CO}CO + X_{CO_2}CO_2 + X_{O_2}O_2
where X is the mole fraction of each component at equilibrium...
Suppose that q(z) = 1, and p(z) = (1 + z)(1 + 3z).
We wish to express q(z)/p(z) in the form
where A and B are constants. To find them, we multiply through by p(z) =
(1 + z)(1 + 3z) and obtain
1 = A(1 + 3z) + B(1 + z)
= (A + B) + (3A + B)z
Im fine up to this point, But according to...
Hello, I'm wondering what the reason for repeat linear factors in partial fractions is?
I can't find an explanation online, they all just say do it!*
I kind of understand why
\frac{A}{x + 2} + \frac{B}{(x + 2)^2}
can turn into ;
\frac{6x + 7}{(x + 2)^2}
Is there any...
Homework Statement
\int(3x3-4x2-3x+2)/(x4-x2)
Homework Equations
P(x)/Q(x)=A1/(x-r1)+A2/(x-r2)+...
if x-r occurs with multiplicity m, then A/(x-r) must be replaced by a sum of the form:
B1/(x-r)+B2/(x-r)2+...
I think this second equation is the source of my confusion.
The Attempt at a...
Sorry if this is the wrong place to post. Kinda wanted to rant.
I just wanted to express something that's been bugging me for a long time.
Where does mathemaphobia really come from? I guess it's from how easy it is to get stuff wrong. If you're asked to discuss a poem, you can write some poop...
1.\int2x^2+x+9/(9x+1)(x^2+9) dx
2. (A/9x+1) + [(Bx + C ) / (x^2 + 9)]
I get the worst numbers when I solve the system. The question is from an old exam and calculators are not allowed. Am I doing something wrong or is there another way to integrate this?
learning calculus here. got differential calculus, though it is a little foggy, and most of integral calculus, which is a little foggier. also using very unpolished precalc background, though i did give most of it a once-over. i have many questions which i can't think of, but of the top of my...
Hello!
Quick question reagrding partial fractions.
When there is a factor such as (x+2)3 in the denominator, then the fraction is separated into the components (x+2)1+...+(x+2)3.
I am not convinced I understand quite why this is so. Partial fractions guides all offer something on the theme...
Homework Statement
The question asks me to express the integrand in partial fractions to evaluate the integral
\int \frac{13x-4}{6x^{2} -x -2} dx
Homework Equations
The Attempt at a Solution
Well 6x² -x - 2 doesn't factorise (or I can't see it factorised).
So I tried...
Homework Statement
Hi,
\int \frac{1}{x(x^{2}+1)}dx
Homework Equations
The Attempt at a Solution
well I split this into partial fractions
\frac{A}{x} + \frac{Bx + C}{x^{2} + 1}
so 1 \equiv A(x^{2}+1) + (Bx + C)x
when x = 0, A =1
when x = 1, Bx + C = -1 so...
Hello,
I'm don't understand a step in the following integral:
∫(x-1)/(2x+1)dx = ∫(1/2)dx − (3/2)∫1/(2x+1)dx = (1/2)x − (3/4)ln|2x+1| + C
The first step, where you get the 2 integrals ∫(1/2)dx and -(3/2)∫1/(2x+1)dx
Where do (1/2)dx and -(3/2) come from?
And where does (3/4) come...
Homework Statement
Need to refresh my memory :-S
Indefinite integral of x/(1+x^2) ..
Homework Equations
The Attempt at a Solution
Would I use partial fractions on that bad boy?
Homework Statement
\int \sqrt{tanx} dx
The Attempt at a Solution
I used the substitution u = sqrt(tanx), then x = arctan(x^2)
so:
2 \int \frac{u^{2}}{u^{4} + 1} du = 2 \int \frac{u^{2}}{(u^{2} + 1)^{2} - 2u^{2}} =2 \int \frac{u^{2}}{(u^2 -\sqrt{2}u + 1)(u^2 +\sqrt{2}u + 1)}
now...
I have no problem expanding brackets with fractions generally, unless the fraction contains an unknown variable, such as in the following expression:
m/4[6m - 8] + m/2[10m - 2]
I know that the answer is:
12/2m2 - 3m
..but I have no idea how to get to that. Can anyone help?
Alright... What are the rules for when the numerator (Top part) of the fraction has a higher degree than the lower part of the fraction.
Something + {*}
... As in what is that SOMETHING based on the numbers?
I have a midterm in like... 50 minutes and this is the only thing I need to know.
http://www.freeimagehosting.net/image.php?9722bd5444.png
Link: http://www.freeimagehosting.net/image.php?9722bd5444.png"
I've tried to used integration by parts and u substitution and I've also tried just multiplying the fraction by the denominator (6-x)^(1/2) but I am still confused at...
Homework Statement
1.) (a + 3b) /3ab + (a^2 b - 4ab^2) / 5a^2b^2
2.) n/(m^2) + 3/mn + 2/m
3.) Given F(x) = square root of x + 9 determine F(x+h) - F(x)/h
4.) say whether or not {(x,y) l x= y^2} is a function.
Homework Equations
The Attempt at a Solution
im...
Partial fractions of (-2x2+10x+8)/[x2(x+2)]
I initally thought that it was A/x + Bx+C/x2 + D/x+2 but you really just do Ax+B/x2 + C/x+2 ...can anyone explain why the "x2" isn't split?
One way to add two fractions is to multiply the numerators of both fractions with each other's denominator, then adding the two products, gives us the numerator of the final result. Then we multiply together the denominators of each other--this gives us the denominator of the final result...
I have no difficulty with converting decimals to fractions generally (0.125 = 1/8)
However, I am a bit stuck when it comes to converting a recurring decimals. My Math book tells me that 0.7777777r is not expressed as 77/100 as a fraction, but as 7/9. How do I calcuate that? Can anyone talk me...
I'm trying to do a question that requires the expansion of the following using partial fractions:
f(z)=\frac{1}{(1+z^3)^2}.
The fact that the bottom is squared is throwing me off for some reason... I've factorized the bottom, but I'm not sure whether I should use the complex roots or not, or...
Hi, had to learn partial fractions last year for laplace transforms, but have forgotten the general rules, and now i can't work out how to turn this into partial fractions:
\frac{s}{\left(s^{2}+4\right)\left(s^{2}+9\right)}...
Let p be a prime number.
Let A be an integer divisible by p but B be an integer not be divisible by p.
Let A/B be an integer.
How do I prove that A/B is divisible by p?
This sounds like a simple question but I just can't get it. I'm doing it in relation to proving Fermat's little...
Homework Statement
This problem is killing me.
I need to bust this thing up using partial fractions.
Homework EquationsThe Attempt at a Solution
I'm leaning towards it being separated like this. Is this correct?
If it is, I'm not exactly sure what I'm supposed to do next.
Homework Statement
Sorry I don't have equation editor working
1/(z+1)(z2 + 2z + 2)
Homework Equations
The Attempt at a Solution
(z2 + 2z + 2)
z2 + 1) can be factor as (z - i)(z + i) However, I'm having trouble seeing the pattern on what (z2 + 2z + 2) would become, I...
how do you integrate 1/(x2 + 1)2 ?
i have tried integration by partial fractions but when you set 1 equal to (Ax +B)(x2+1) + (Cx + D) this leads to A=B=C=0 and D=1 which just gives you the original equation
Hi
Can anybody help me with these 3 problems?:
1)
Express (3x-1)/(x+3)^2 in the form A/(x+3) + B/((X+3)^2) where A and B are constants.
2)
A curve C has parametric equations:
x=cost and y=2-cos2t (between 0 and pi)
a)prove this can be expressed as the cartesian equation y=3-2x^2
b)...
I'm sure this is a no brainer to someone, but here it is..
what is does the partial fraction of this look like in expanded form? Or how can I make it fit on the table of laplace transforms?
__(2s+1)__
(s-1)^2 + 1
Homework Statement
I need to integrate this using partial fractions. "b/(x^2-a^2)"
Homework Equations
The Attempt at a Solution
I have no idea where to begin.
Integration with partial fractions -- help!
Homework Statement
Here is the problem: http://img130.imageshack.us/img130/1673/integralthing.png
The answer should be 4.
Homework Equations
N/A
The Attempt at a Solution
Here are my steps so far:
(5x^2 - 17x + 10) / ((x-1)^3 *...
I really didn't know how to word the title so sorry if it's a little confusing.
And I didn't know whether to post this in number theory or not but ah well.
The other day, I started thinking about this and I was just wondering if it had been done before or if it's even correct;
Half of all...
Homework Statement
\int \frac{xdx}{x^3-1}
Homework Equations
The Attempt at a Solution
Having difficulties with this one.
I managed to break it down to two partial fractions, being x-1 and x^2+x+1 but couldn't make anything out of it.
Homework Statement
Integral(sinx(x)dx/(cos^2(x)+cos(x)-2)
Homework Equations
The Attempt at a Solution
What I tried to do first was factor the denominator, so i got (cos(x)-1)(cos(x)+2)
from there, I set up my partial fractions equation trying to solve B(cos(x)-1) + A(cos(x)+2) =...
Homework Statement
expand by partial fractions:
Homework Equations
2(s+5)/(1.25*s^2+3s+9)
The Attempt at a Solution
ok I initially used the quadratic formula to get the two roots for the denominator
these being
(s+6/5+12i/5)(s+6/5-12i/5) i.e complex numbers
so now the partial fractions...
Homework Statement
A solution of ethanol (eth) and chloroform (chl) at 45°C with xeth = 0.9900 has a vapor pressure of 177.95 torr. At this high dilution of chloroform, the solution can be assumed to be ideally dilute. The vapor pressure of pure ethanol at 45°C is 172.76 torr.
(a) Find the...
I wrote a program to convert decimals into fractions. It basically puts the decimal over the proper power of ten, and simplifies (at least, for nonreapeating fractions.) However, it uses a brute force method to find the LCD, and as a matter of aesthetics, I never like brute force. Does anyone...
Homework Statement
\frac{dP}{dt}=P-P^{2}
It seems that Partial Fractions should be used to solve this D.E., but I cannot find an example to go by.
I even tried to rewrite the equation as:
\frac{d}{dx}Y\left(x\right)=Y\left(x\right)-Y\left(x\right)^{2}
But, that isn't helping me...
Homework Statement
Compute the integral:
int ((1-x^2)/(x^3+x)) dx
Homework Equations
int ((1-x^2)/(x^3+x)) dx
The Attempt at a Solution
I think I should use the partial fraction method to simplify the fraction
so
(1-x^2)/(x^3+x)= A/x + B/ (1+x^2)
Therefore...