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
Derive a formula for the antiderivative of sec x using the identity that sec x= cos x/ (1-sin^2x). Use a substitution for sin x and then partial fractions. Then multiply the solution by (1+sin x)/ (1+sin x) to obtain the more familiar formula for the antiderivative...
This is probably a "basic" question, but I can't seem to remember how to do partial fractions problems where there is only a 1 in the numerator.
For example (just making this up), let's say I have:
1/s(s+4)(s+5)
So what I'd do is 1/s(s+4)(s+5) = A/s + B/(s+4) + C/(s+5) as one would expect...
I'm trying to solve this integral but I'm not sure if I'm on the right track. My question is: can this integral be solved by partial fractions decomposition? I solved the problem that way but I'm not sure if it is the right answer. thanks!
∫1-x+2x^2-x^3 ÷ x(x^2+1)^2
Homework Statement
\int\frac{dx}{x(1+ln x)}
Homework Equations
Partial Fractions? Maybe I am solving this wrong...
The Attempt at a Solution
\frac{A}{X} + \frac{B}{1+ln x} = 1
A(1+lnx) + Bx =1
A + Alnx + Bx =1
This doesn't seem to work out properly. I have been having a...
is there a way to express any given root of an integer in a continued fraction? i.e. Sqrt[2] = 1 + 1/(2 + Sqrt[2] - 1) and the process can be continued infinitely to get a fraction that defines the radical with only integers.
so my question is can this kind of thing be done with any square...
Homework Statement
\int {\frac{{2s + 2}}
{{(s^2 + 1)(s - 1)^3 }}ds}
The Attempt at a Solution
This is a long one...First, I split the integrand into partial fractions and find the coefficients:
\begin{gathered}
\frac{{2s + 2}}
{{(s^2 + 1)(s - 1)^3 }} = \frac{{As + B}}...
Homework Statement
\int {\frac{{2s + 2}}
{{(s^2 + 1)(s - 1)^3 }}ds}
The Attempt at a Solution
This is a long one...First, I split the integrand into partial fractions and find the coefficients:
\begin{gathered}
\frac{{2s + 2}}
{{(s^2 + 1)(s - 1)^3 }} = \frac{{As + B}}...
Homework Statement
Hi everyone, here is a new partial fractions question I just cannot understand:
\int\frac{x^{3}}{x^{3}+1}dx
Homework Equations
Partial Fractions, difference of perfect cubes, polynomial long division
The Attempt at a Solution
\int\frac{x^{3}}{x^{3}+1} dx...
Homework Statement
\[
\int {\frac{{e^t dt}}
{{e^{2t} + 3e^t + 2}}}
\]
I'm not quite sure how to start this one...Any hints? I tried bringing e^t down to the denominator and multiplying it out which still didn't help. I can't see a way to factor the denominator or split this into a...
I've been working on this one for a while now but just can't figure it out
lim h->0 (1/h) (( 1 / (x + h) ) - ( 1 / x ))
my first thought was to figure out (( 1 / (x + h) ) - ( 1 / x )) first by just putting them togeather and then using the congjigate times by one trick but that just...
please... i need a help in integrating the partial fractions
i can't proceed to the integration part if i don't understand the patter in finding the constant...
that is...
if the given is:
ʃ ( (x^5+1) / ((x^3)(x+1)) )dx
then;
ʃ ( x-2 + ( 4x^3+1 ) / ( x^4 + 2x^3) )
ʃ ( x-2 + (...
How do I solve:
[sin (pi/3)] + [cos (pi/6)]? <--- "pi" is 3.14...
I think that [sin (pi/3)]= (square root 3) divided by 2
AND that [cos (pi/6)]= (square root 3) divided by 2.
Now I can't remember how to add fractions containing square roots.
My textbook...
Hi,
actually, I need to calculate an infinite sum of fractions. The problem is that the Limit of the sum is part of the summands. The formula looks like this:
\lim_{n \to \infty} \sum_{i=1}^{n} \frac{1}{n(1 + \lambda + \sigma^2)-i(1+\lambda)},
where 'itex]\sigma[/itex] and \lambda are...
Homework Statement
∫1/ x^3-1 dx, ok how would i do this
Homework Equations
∫dx/ x^2+a^2= 1/a tan^-1 (x/a) +c
i tried to simplify x^3-1 = (x+1)(x-1)(x+1)
got an exam coming up in a few days and half way through my question i ran into a partial fractions question instead of having the standard (1/(y+c)(y+d))= A/(y+c) + B(y+d) and multiplying out i had a double root so (1/(y+c)(y+c)) does this change the way i go about the question and are there...
Homework Statement
Re-write the following fraction into the sum of fractions:
1/[(n^3)+n]
Homework Equations
None that I can think of. . .
The Attempt at a Solution
I first changed [(n^3)+n] to n[(n^2)+1], so by the rules, the aformentioned fraction should equate to (A/n) +...
.. oy, I'm just not sure how to find 3 constants!
Here is my problem:
5x^2-4/(x-2)(x+2)(x-1) = A/(x-2)+B/(x+2)+C/(x-1)
.. i got a bit of it done, but it's all wrong
OH! and what am i supposed to do if the numerator of the first equation does not have any sort of variable with it??
my...
Ok this is something i learned few years ago and I am a bit rusty.
So i have to find the absolute value of:
\frac{1 - 2i}{3 + 4i} + \frac{i - 4}{6i - 8}
So first i add the two fractions and i get:
\frac{(1 - 2i)(6i - 8) + (i - 4)(3 + 4i)}{(3 + 4i)(6i - 8)}
Next i simplify and then...
For a rational function, (x^2+1)/(x^2-1) = (x^2+1)/[(x+1)(x-1)], if we were to split it into partial fractions so that (x^2+1)/(x^2-1) = A/(x+1) + B/(x-1) = [A(x-1) + B(x+1)]/(x^2-1)...solving for A and B get us A = -1 and B = 1. This would mean that (x^2+1)/(x^2-1) = 2/(x^2-1)...which doesn't...
I'm a little mixed up on the integration for partial fraction decomposition.
I basically have x/ x(x^2 + 1)
I'm wondering for the (x^2 + 1) part, am I to put Ax + B over it because it is a raised power, or since the outside bracket is not squared, it is to only have one variable over it.
Homework Statement
we have 4/((s^2) + 4)(s-1)(s+3)
Homework Equations
The Attempt at a Solution
dividing it up do we get:
A/((s^2) + 4) + B/(s-1) + C/(s+3) = 4
or is it
(As + B)/((s^2) + 4) + C/(s-1) + D/(s+3) = 4
Homework Statement
[e^(-2s)] / (s^2+s-2)
Find the inverse Laplace transform.
Homework Equations
The Attempt at a Solution
I know that I can factor the denominator into (s+2)(s-1). Then I tried to use partial fractions to split up the denominator, but I don't know how to do that...
Is there any way to derive the greatest common divisor from the prime factorizations of the numerator and denominator?
For instance:
\displaystyle{\frac{48}{150} = \frac{ 2 * 2 * 2 * 2 * 3}{2 * 3 * 5 * 5}}
The GCD = 6 in this example, but is there any way to determine that from the...
Homework Statement
Suppose a_n > 0, s_n =a_1 + ... + a_n, and \sum a_n diverges,
a) Prove that
\sum \frac{a_n}{1+a_n}
diverges.
Homework Equations
The Attempt at a Solution
Comparison with a_n fails miserably.
Homework Statement
1/(a-b)(a-c) + 1/(c-a)(c-b) + 1/(b-a)(b-c)
Homework Equations
Is there a way to simplify this? If I start multiplying out everything to get the LCD my final answer will be huge.
The Attempt at a Solution
As I said, without somehow simplifying it at the start...
Telescoping Method & Partial Fractions...PLEASE HELP!
Homework Statement
Find the sum of the series from n=1 to infinity...
2/(4n^2-1)
Homework Equations
The Attempt at a Solution
I want to use the telescoping method...
2/(4n^2) = 2/[(2n-2) * (2n+1)]
I am following an...
Hello all,
I've got an exam tomorrow so any quick responses would be appreciated. I'm following the Boas section on Laurent series... Anyway, here's my problem:
In an example Boas starts with f(z) = 12/(z(2-z)(1+z), and then using partial fractions arrives at f(z) = (4/z)(1/(1+z) +...
(t+1) dx/dt = x^2 + 1 (t > -1), x(0) = pi/4
I have attempted to work this by placing like terms on either side and then integrating.
1/(x^2 + 1) dx = 1/(t + 1) dt
arctan x = ln |t + 1| + C
x = tan (ln |t + 1|) + C
pi/4 = tan(ln |0 + 1|) + C
pi/4 = C
x = tan (ln |t + 1|)...
Homework Statement
Solve y"+4y'=sin 3t subject to y(0)=y'(0)=0 using Laplace Transform
The Attempt at a Solution
So I got:
s^2Y(s)-sy(0)-y'(0)+4[sY(s)-y(0)]=\frac{3}{s^2+9}
\Rightarrow Y(s)=\frac{3}{(s^2+9)(s^2+4)}
Now it looks like two irreducible quadratics, which I...
http://img340.imageshack.us/img340/1967/25616732jw6.jpg
i got answer of 0.4
by trial and error, but i have seem to forgotten the basic of fraction...
please guide me thanks
Homework Statement
Evaluate the indefinite integral.
int (6 x + 7)/(x^2 + 1) dx `
The Attempt at a Solution
A/(x + 1) + B/(x - 1)
6x + 7 = A(x - 1) + B(x + 1)
6x + 7 = (A + B)x + (-A + B)
A + B = 6
-A + B = 7
A + (7 + A) = 6
2A = -1.
A = -.5
B = 3.5
So the...
Homework Statement
Turn this into partial fraction.
k1b1/[((k1+b1*s)(k2+b2*s))-b1^{2}s^{2}]
Homework Equations
n/a
The Attempt at a Solution
original question was to find the transfer function with springs and a damper and I reduced it to this far but I can't get the partial...
This is a calculus equation, but I'm having trouble with the algebra part of it.
http://calcchat.tdlc.com/solutionart/calc8e/02/e/se02e01045.gif
I'm confused about how they simplify from step 4 to 5. Can someone help me?
[SOLVED] integration by partial fractions
Homework Statement
\int((2x^2-1)/(4x-1)(x^2+1))dx
Homework Equations
A1/ax+b + A2/(ax+b)^2 + ... + An/(ax+b)^n
The Attempt at a Solution
(2x^2-1)/(4x-1)(x^2+1) = A/4x-1 + Bx+C/x^2+1
2x^2-1/x^2+1 = A + Bx+C(4x-1)/x^2+1
set x =...
Integrate using partial fractions:
(int) (x^3)/(x^2 -1) dx
I have put into the form (int) (x^3)/((x-1)(x+1)) dx
I thought partial fractions had this property:
'Partial fractions can only be done if the degree of the numerator is strictly less than the degree of the denominator.'...
--Here is an article taken from the USAToday that talks about the teaching and learning of mathematic fractions and some controversal opinions arises with this matter.
Feel free to discuss what you think of this. I'm personally curious of the different points of views on this issue...
I am taking college algebra this semester at my Community College. Prerequisite for pre-calculus . I have been doing good in math, but i have done fractions on calculator. I really don't know how to solve fractions (Addition & Subtractions) Problems without calculator.Please explain me how to...
Homework Statement
Integration of 1/(x^2-5x+6)
Homework Equations
The Attempt at a Solution
I know i cannot do ln|x^2-5x+6|
I've tried some form of substitution or intergration by parts, and they don't work.
Should I factor the bottom?
Homework Statement
The problem is from Stewart, Appendix G, A58, no.45.
Suppose that F, G, and Q are polynomials, and:
F(x)/Q(x) = G(x)/Q(x)
for all x except when Q(x) = 0. Prove that F(x) = G(x) for all x. [Hint: Use Continuity]
The Attempt at a Solution
I thought the statement was...