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).
Here's (what I think is) a step in partial fractions that I don't understand:
http://apthtml.com/images/partialfrac.png
I'm taking the regular partial fractions steps and I keep ending up with 1 / (1 - x) rather than 1 / (x - 1). What am I doing wrong?
Hello :smile:
I've been stuck on this question for almost 3 hours now, and I still have no idea what to do. We haven't done a question like this in class, although we have done integration with partial fractions.
Homework Statement
Evaluate...
Homework Statement
\int^{1}_{0} \frac{ln(x)}{sqrt(x)}
Homework Equations
Integration by Parts
Improper Fractions: Type II Integral (discontinuous at a)
The Attempt at a Solution
Now, this might seem odd, but I'm having a devil of a time understanding how this is converging when the...
Fractions, Exponents, and x :(
I simply just don't get fractions and exponents
i was given this problem:
[6x^(4/5)-3x(2/3)]/3x^(1/3)
i haven't gotten anywhere and I'm stuck
help
thanks in advance
Homework Statement
I need to integrate (2x-5)/(x^2+5x+11)
Homework Equations
The Attempt at a Solution
My problem is just finding a formula for an irreducable quadratic. I know if the denominator was x(x^2+1), I would use A/x+(Bx+C)/(x^2+1). I just don't know the formula in this...
The problem is:
((x^3)+x)/(x-1)
And i need to break it into partial fractions...
I tried long division and got:
((x^2) +x )
But the book gives me the answer of:
(x^2)+x+2+(2/(x-1))
Any help would be very much appreciated, thanks.
Homework Statement
Verify that, for all positive values of n,
1/(n+2)(2n+3) -1/((n+3)(2n+5))=(4n+9)/((n+2)(n+3)(2n+3)(2n+5))
For the series
∑_(n=0)^N▒(4n+9)/((n+2)(n+3)(2n+3)(2n+5))
Find
The sum to N terms,
The sum to infinity.
Homework Equations
no
The Attempt at a...
Okay, I know how to differentiate regular functions. But when it comes to fractions, I'm hopeless. This may be an extremely simple one to some, here is the function; "1/4x-7"
I have to differentiate that using the chain rule.
I think that u=4x-7, but I am not sure. As i said, I am horrible...
Homework Statement
Show that 1/72
cannot be written as the sum of the reciprocals
of the squares of two different positive integers.
Homework Equations
The Attempt at a Solution
Available solutions
1/8²-1/24²
1/9²+1/648
Therefore Proven.
Hey. I have a question. What is the packing fraction of a container full of randomly jammed cylinders? Also, does the packing fraction depend on the ratio of the radius to height of the cylinder? There is some contest of finding certain amount of cylindrical objects inside a container (I can't...
Is it possible to calculate the greatest common divisor of decimals and fractions? As far as I know, the greatest common divisor is a number you can calculate for integers, but I wonder if it's possible to calculate it for decimals and fractions.
Homework Statement
I((3x^2+x+4)/(x^4+3x^2+2),x)
I((3x^2+x+4)/((x^2+1)(x^2+2)),x)
I(3x^2/((x^2+1)(x^2+2)),x)+I((x+4)/((x^2+1)(x^2+2)),x)
from here i have used partial fractions with no luck
Homework Equations
The Attempt at a Solution
Homework Statement
partial fractions
(-x^2+x+3)/((x-2)^2(x+1)) = A/(x-2) + B/((x-2)^2) + C/(x+1)
is my partial fractions set up right
A= -1/9 B = 1/3 C=1/9
I'm having trouble understanding what the numerator needs to be in the partial fractions.
e.g.
\frac{1}{(x-1)(x-2)^2}\equiv \frac{A}{x-1}+\frac{Bx+C}{(x-2)^2}
Notice how the first numerator has a constant A, while the second is linear Bx+C.
Actually... just now I think I may understand...
Hey guys, I have a quick question concerning partial fractions, say i have the values A(..)+B(..)+C(..) and i can only get A b substituting a value for x, but I can not make A or C go zero or A and B go to zero at the same time, if you understand ? How do I get the values B and C ? I remember...
Is it possible to use partial fractions on exponential factors? An expression like this for example...
3^x
______________________
(2^x + 3^x)(5^x + 6^x)
Would it break down into something like this?
A^x/(2^x + 3^x) + B^x/(5^x + 6^x)
What are the rules for repeated factors...
Homework Statement
I need to intergrate the following
(4x2+2x-1)/ (x3+x2)
How to set up problem using linear factors?
The Attempt at a Solution
Factoring the denominator I get:
x2(x+1)
By linear/quadratic factoring I get:
A/x + Bx+C/ x2 + d/(x+1)
Is this right?
Homework Statement
use the method of partial fractions on \int \frac{36}{(x-2)(x-1)^2(x+1)^2} dx
[b]2. Homework Equations
[b]3. The Attempt at a Solution
\frac{A}{(x-2)} + \frac{B}{(x-1)} + \frac{C}{(x-1)^2} + \frac{D}{(x+1)} + \frac{E}{(x+1)^2}
I took the 36 out of the...
Homework Statement
∫1/[(x+a)(x+b)]dx
answer is 1/(a-b) ln[(x+b)/(x+a)] + C
Homework Equations
The Attempt at a Solution
1=A/(x+a) + B/(x+b)
1=B(x+a) + A(x+b)
1=Bx+ Ba + Ax +Ab
so 0=Bx+ Ax, 1=Ba+Ab
A=-B, 1=B(a-b)
∫-1/(x+a) +∫1/(x+b)
Im not sure if I am headed in...
Homework Statement
Integrate: (x-1) / (X^2 - 4x +5)
The attempt at a solution
Normally I would try to factor this into something like (x-1) (x+3) (That's an example completely unrelated to this problem.)
However, as no easy factors quickly occurred to me I did a run through of the...
Homework Statement
F(X)=[tex]\int[/\frac{1}{1+t^3}
Homework Equations
The Attempt at a Solution
I have tried different substitutions to find fog where g(t) = ? But am getting stuck
Homework Statement
Actually i want to ask something actually very easy...
i just don't know the meaning of some words in different questions...
firstly... multiplication of A and B means A*B right?
how about multiplication of A by B means A*B or A/B??
secondly... division of A by B means...
Homework Statement
I would like to know...when trying to take the derivative of a function with a fraction in it ...
should I always turn it into a product and use the product rule, thereby dropping the quotient rule most of the time?
Or is the quotient rule needed more so in some cases...
Homework Statement
Evaluate the integral: integral (-17e^x-36)/(e^(2x)+5e^x+6 dx
Homework Equations
partial fractions
The Attempt at a Solution
Basically, what i did was factored the bottom into (e^x+2) and (e^x+3) because when i expand that, it equals the bottom. From there, i...
Homework Statement
I am supposed to evaluate the integral using partial fractions.
\int \frac{1}{(x+5)^2(x-1)} dx
2. The attempt at a solution
So after doing all the work, I get
(-1/36)ln|x+5| - (13/6)ln|x+5| + (1/36)ln|x-1|
But the answer in the book appears as
(-1/36)ln|x+5| -...
Homework Statement
1/((x^2-1)^2)
Homework Equations
The Attempt at a Solution
so i get (Ax+B)/(x^2-1) + (Cx+D)/((x^2-1)^2)
then i multiply both sides by ((x^2-1)^2)
then i get 1=(Ax+B)(x^2-1)+ (Cx+D)
then i multiply it out Ax^3+Bx^2 -Ax +Cx +D =1
then i equate...
Homework Statement
[(4x-4x2)/x2+2x-3] * [(x2+x-6)/4x]
Homework Equations
The Attempt at a Solution
I'm so lazy, though, I do all this stuff mentally, so... Oby-kaby.
I individualized all groups into smaller units(I don't know how to say that in algebrish), by dividing by least...
Homework Statement
(3x^2-4)/(x^3-4x-6)
Homework Equations
I guess integration by parts... But how do i set this up?
The Attempt at a Solution
The numerator is exponentially lower than the denominator, so no long division.
The denominator seems not to factor out into anything...
Homework Statement
\int\frac{e^x}{(e^x-2)(e^2x +1)} it should be e to the power of 2x
Homework Equations
Using substitution u=e^x, and then using partial fractions
The Attempt at a Solution
I have done this problem two separate ways. One with substitution and then partial...
Homework Statement
Solve the integral x/x^2+4x+13
Homework Equations
I think that you would use partial fractions but I'm not really sure. I know that you need to complete the square on the denominator.
The Attempt at a Solution
The completed square would be (x+2)^2+9. I don't...
Homework Statement
Simplify \frac{x^2 - \sqrt{x}}{\sqrt{x^5}}
Homework Equations
Unsure
The Attempt at a Solution
Tried to factorise the numerator and denominator. Not sure how to proceed given the subtraction in the numerator. Best effort so far...
Homework Statement
From the formula t = \frac{L}{v+c} + \frac{L}{v-c} I've made t = \frac{2L}{c}\left(\frac{1}{1 - \frac{v^2}{c^2}}\right). This is the problem:
\left(\frac{1}{1 - \frac{v^2}{c^2}}\right) = \left(\frac{1 + \frac{v^2}{c^2}}{1 - \left(\frac{v^2}{c^2}\right)^2}\right)
How...
Homework Statement
Integrate
x^3 + 49 / x^2 + 5x + 4
Homework Equations
The Attempt at a Solution
Since the numerator has an x cubed, but the denominator only has an x squared, I know I need to divide the numerator by something.
I'm not sure what, but maybe the...
Homework Statement
The problem asks to evaluate the integral using partial fractions, but I just cannot find out which trick to get this one to work. the equation is
\int\frac{x^3+x^2+2x+1}{(x^2+1)(x^2+2)}
The Attempt at a Solution
I have tried setting it up as a partial fraction...
i'm trying to write the equation dB=log_10(P_2/P_1)
I just need help writing this equation into Fortran.
I know for exponents I use **, but what about subscripts?
Also, my inital thought for a fraction is /. Is this correct?
Homework Statement
Split the function into partial fractions. 1/(w^4-w^3)Homework Equations
1/(w^4-w^3)
The Attempt at a Solution
I started by factoring the denominator to w^3(w-1) and re-writing the original function as
(Aw^2+Bw+C)/w^3 + D/(w-1) and set it = 1/(w^3(w-1))
I end up with...
What do I do when a decimal or a fraction is found only within the parentheses in a linear equation? I know about the general removal process; but my question involves equations like these:
17(2.33 - x) - 35(4 - 30x) = 2
7(4/3 - x) + 24(5x - 60) = 31
Should I distribute first and then get...
Homework Statement
integrate (4x^2 + 3x + 6)/x^2 (x+2) dx
Homework Equations
don't have sorry..
The Attempt at a Solution
firstly = A/x + B/x^2 + C /x+2 , = A(x^2)(x+2) + B(x)(x+2) + C(x)(x^2) equating with the 4x^2 + 3x + 6,then i integrate it,but my ans turn out to be...
I can't seem to make the brackets cover the full fraction when stating a differential, namely
\begin{equation}
(\frac{df}{dt})
\end{equation}
Any tips?
I've been working with Laplace Transforms and integration ALOT lately. Many times I windup having to use partial fractions to solve the problem and frankly my algebra skills just aren't up to the task.
Take this fraction for example;
I know 3 ways to do it... 1 of the ways doesn't work unless...
\frac{s-1}{s(s-2)^2}
How can I expand this fraction?
\frac{A}{s} + \frac{B}{(s-2)} + \frac{C}{(s-2)^2}
right?
This gives me the equation
As^3 - 6As^2 + 12As - 8A Bs^3 - 4Bs^2 + 4Bs + Cs^2 - 2Cs = s-1
so that
(1) A + B =0
(2)- 6A - 4B + C = 0
(3) 12A + 4B - 2C = 1
(4)...
Expressing fractions as powers??
How would you go about expressing this 127/squareroot(5) as a power of 5? I got a copy of last years exams to use for revision and practice and its full of stuff I don't know how to do. This one happens to be a multiple choice and the possible answers are 5 to...
\int e^{ax}cosbx
This one is driving me insane.
So I used e^ax as u and cosbx dx as dv. And then I did it again using e^ax as u and sinbx as dv which left me with \int e^{ax}cosbx = \frac{1}{b}e^{ax}sinbx + \frac{a}{b^{2}}e^{ax}cosbx - \frac{a^{2}}{b^{2}}\int e^{ax}cosbxdx
I have no...
Homework Statement
How does one integrate e.g. \frac{1+x}{(2+x)^{3/2}} by partial fractions?
The Attempt at a Solution
I have no idea about this. I've never seen this technique applied with fractional powers before.