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).
For this problem (b),
The solution is,
However, I don't understand how they got their partial fractions here (Going from step 1 to 2).
My attempt to convert into partial fractions is:
##\frac{2s + 1}{(s - 1)(s - 1)} = \frac{A(s - 1) + B(s - 1)}{(s - 1)(s - 1)}##
Thus,
##2s + 1 = A(s - 1) +...
I know this problem can be done as follows.
P(1 boy and two girls) = (C(2,1)*C(5,2))/C(7,3) = 20/35
My question can this be written as probility fractions? Meaning
Lets say if that (2/5*3/7+1/2*1/7)/(3/7) But that doesn't give same result? what am I doing wrong?
*Kindly note that i created this question (owned by me).
My Approach,
##\dfrac {(x+y)(4x+6y)}{(5x-5y)}##=##-1##
##(x+y)(4x+6y)=-5x+5y##
##\dfrac {4x+6y}{-5x+5y}##=##\dfrac {1}{x+y}##
to get the simultaneous equation,
##4x+6y=1##
##-5x+5y=x+y##
...
##4x+6y=1##
##-6x+4y=0##
giving us...
So the part in italics "an operation performed on one quantity which when performed on unity produces the other." I do not understand. Can anyone help me understand what this means? I know how to multiply fractions, but this explanation is confounding to me.
Let $$y=\frac {1+3x^2}{(1+x)^2(1-x)}= \frac {A}{1-x}+\frac {B}{1+x}+\frac {C}{(1+x)^2}$$
$$⇒1+3x^2=A(1+x)^2+B(1-x^2)+C(1-x)$$
$$⇒A-B=3$$
$$2A-C=0$$
$$A+B+C=1$$
On solving the simultaneous equations, we get ##A=1##, ##B=-2## and ##C=2##
therefore we shall have,
$$y=\frac {1}{1-x}+\frac...
I don't understand the logic behind why derivatives can be treated like fractions in solving equations:
## \frac {du}{dx} = 2 ## simplified to
## du = 2dx ##
I keep seeing this done with the explanation that "even though ## \frac {du}{dx} ## is not a fraction, we can treat it like one". Why...
now a bit confusing here, i always use Bodmas in that case,
##9\frac {1}{5}##-[##3\frac {3}{10}##+##2\frac {2}{3}]##...[1] is this correct and what if i re arrange to
##9\frac {1}{5}##+##2\frac {2}{3}##-##3\frac {3}{10}##...[2]
input guys...cheers
i have seen my mistake [1] is wrong...
A paper out today in Nature might interest some folks in this forum:
https://www.nature.com/articles/s41586-021-03229-4
Permanent citation: Nature volume 590, pages67–73(2021)
The authors used machine learning to generate a large number of continued fraction expressions for fundamental...
So the original question is from Control Theory, and the topic is the inverse z-transform. This is a part from the solution I just can't understand. The reason it has to be in this form (##z^{-1}##) is because that's the form used in the z-transform table. The question essentially is, how do you...
I am trying to get one step further with my search for \sum_{n=1}^{\infty}\frac{1}{n^{2s+1}} . Part of the way is to calculate some algebraic expressions containing fractions with really huge numbers (as in (\frac{1}{5^{9}}+\frac{1}{7^{9}}-\frac{1}{17^{9}}-\frac{1}{19^{9}})\div...
Question 1;
a. sin θ=√3/2
θ=arcsin √3/2
θ=π/3 rad
sin √3/2=60 degrees
60 degrees *π/180=π/3 rad.
To find the other solutions in the range, sin θ=sin(π-θ)
π-π/3=2π/3
The solutions are π/3 and 2π/3 in the range 0 ≤θ ≤2 π
b. cos2θ=0.5
2θ=arccos 0.5
2θ=π/3 rad
Divide both sides by 2;
θ=π/6 rad...
I'm typing up answers to the exercises in my Insight article "A Path to Fractional Integral Representations of Some Special Functions", this problem is from section 1 (Gamma/Beta functions). I need a bit of help with this one:
The problem statement:
1.9) Use partial fraction decomposition to...
I use 2x -4 as the LCD and turn 8/(x - 2) - (13/2) = 3 into 16 - 13x - 4 = 3, I then get 12 - 13x = 3 which leads me to 13x = -9 so x = -9/13 which is the wrong answer.
Where did I make a mistake?
$$c'\begin{pmatrix}u \\ v \end{pmatrix}=\begin{pmatrix}\frac{\partial f^1}{\partial u} & \frac{\partial f^1}{\partial v} \\\frac{\partial f^2}{\partial u} & \frac{\partial f^2}{\partial v}\end{pmatrix}
$$
There is a problem with the first line of the matrix, but I am not too sure what it is.
Determine the value of \frac13+\frac16+\frac1{10}+\frac1{15}+\frac1{21}+...+\frac1{231}
I know that it means \frac1{1+2}+\frac1{1+2+3}+\frac1{1+2+3+4}+\frac1{1+2+3+4+5}+\frac1{1+2+3+4+5+6}+...+\frac1{231}, but how do I answer? It's from a student worksheet for 7th graders, so they haven't...
Evaluate the integral.
$$\displaystyle \int_2^4
\dfrac{x+2}{x^2+3x-4}\, dx$$
W|A returned these partial fractions but I don't know where the the A=2 B=3 and 5 came from
$$
\dfrac{x+2}{(x+4)(x-1)}
=\dfrac{2}{5(x+4)}+\dfrac{3}{5(x-1)}$$
the book...
They are mysterious to me, they do not make any intuitive sense. What is the context for them? I found this website. http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html
It tries to show the history of how they were discovered geometrically. So you start off with a...
I was doing this problem from Griffith's electrodynamics book and can't figure out how to do this integral. The author suggested partial fractions but the denominator has a fractional exponent which I have never seen for partial fractions, and so, I am unsure how to proceed. The integral I am...
Let $a_1,a_2, ... , a_n$ be positive numbers.
Let $i_1,i_2, ... , i_n$ be a permutation of $1,2,...,n$.
Determine the smallest possible value of the sum:
$$\sum_{k=1}^{n}\frac{a_k}{a_{i_k}}$$
Homework Statement
For a commutative ring ##R## with ##1\neq 0## and a nonzerodivisor ##r \in R##, let ##S## be the set
##S=\{r^n\mid n\in \mathbb{Z}, n\geq 0\}## and denote ##S^{-1}R=R\left[\frac{1}{r}\right]##.
Prove that there is a ring isomorphism $$R\left[\frac{1}{r}\right]\cong...
Hello all, So I've been trying to write some basic equations out like 1/2 but would like it to appear in a horizontal fashion (like 1 over 2). I have been reading threads on how to use latex, I've tried to look at others equations, right click and have the latex code shown to me so i can...
Homework Statement
(x+3)(x-2)/x2-2x
Homework EquationsThe Attempt at a Solution
(x+3)(x-2)/x(x-2) = (x+3)/x
What I don't understand is why I can't simplify this further for instance the x's cancel to give 1:
(1+3)/1 = 4/1 = 4
Is it because there is no x next to the 3?
Many thanks :)[/B]
Homework Statement
Prove that ##\forall n \in \mathbb{N}##
$$\frac{n}{2} < 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{2^n - 1} \leq n \text{ .}$$
Homework Equations
Peano axioms and field axioms for real numbers.
The Attempt at a Solution
Okay so my first assumption was that this part...
Dear all,
I am trying to solve this inequality:
\[\frac{2}{x^{2}-1}\leq \frac{1}{x+1}\]
I've tried several things, from multiplying both sides by
\[(x^{2}-1)^{2}\]
finding the common denominator, but didn't get the correct answer, which is:
\[2<x<3\]
or
\[x<-1\]How to you solve this one ?
Meiling baked some cookies. 5/7 of the cookies were butter cookies and the rest were green tea cookies. She gave away 3/4 of the butter cookies and 1/4 of the green tea cookies. After she ate 9 green tea cookies, she had an equal number of butter cookies and green tea cookies left. How many...
Junhao and Bala both collect stamps. 1/3 of Junhao's stamps is equal to 3/5 of Bala's stamps. Junhao has 76 more stamps than Bala. How many stamps does each of them have?
My answer:
Number of stamps Junhao have =J
Number of stamps Bala have = B
We know that Junhao has 76 more stamps than Bala...
Homework Statement
$$\frac{1}{z}+\frac{1}{2-z}=1$$
Homework Equations
Quadratic-formula and algebra
The Attempt at a Solution
Been struggling with this one.. I keep getting the wrong answer, but that isn't the worst part, I can live with a wrong answer as long as the math behind it is...
Homework Statement
For the solution to a given problem, in the second to last step I had:
##-\frac{\sqrt 6}{4} + \frac{\sqrt 2}{4}##
I stated next that the solution was ##-\frac{\sqrt{6}+\sqrt{2}}{4}##
I was told this was incorrect and that the correct solution is...
Homework Statement
y(w)= 3/(iw-1)^2(-4+iw)
Homework Equations
N/A
The Attempt at a Solution
3/(iw-1)^2(-4+iw)
= A/iw-1 + B/(iw-1)^2 + C/-4+iw
for B iw = 1
B=3/-4+1 = -1
for C iw = 4
C= 3/(4-1)^2 = 1/3
I know the answer for A should be -1/3 however I am unsure how to obtain this as if the...
im a bit confused about partial fractions
If we have something like x/((x+1)(x+2)) we could decompose it into a/(x+1) +b/(x+2)
If we had something like x/(x+1)^2 we could decompose it into a/(x+1) + b/(x+1)^2
We use a different procedure when there is a square in part of the polynomial in...
Josh says that 2/3 is always the same as 4/6. Meghan says that 2/3 and 4/6 are equivalent fractions, but they could be different amounts. Which student is correct?
I say Josh.
Hello! I am new to Mathematica and I need some help with the code I attached. Can someone tell me how to pull the variables out of the fraction i.e. instead of ##\frac{7 p_1 p_2}{2 \cdot 5}## I would like ##\frac{7 }{2 \cdot 5} p_1 p_2## (I need this to make it look better for when I import it...
I´m not sure, whether this little challenge has been posted before. I have searched the forum and didn´t find it.
It might still be a duplicate though ...
Find the sum of fractions
$$\frac{2}{3\cdot5}+\frac{2\cdot4}{3\cdot5\cdot7}+\frac{2\cdot4\cdot6}{3\cdot5\cdot7\cdot9}+...$$
Suppose you have the fraction 1/1. If you can make a fraction x/y, you can also make y/(2x). Also, if you can make x/y and a/b where GCD(x,y)=GCD(a,b)=1, you can make (x+a)/(y+b). Which fractions can you make?
Homework Statement
Please see attachment.
Homework Equations
I don't know how to get the final product on the ones with the question marks (textbook answers written next to them). I've gotten to the last step (except for # 29 but don't mind that one, I haven't exhausted all ideas). I've...
Hi PF!
When minimizing some fraction ##f(x)/g(x)## can we use Lagrange multipliers and say we are trying to optimize ##f## subject to the constraint ##g=1##?
Thanks
$\tiny{242 .10.09.8}\\$
$\textsf{Express the integrand as a sum of partial fractions and evaluate integral}$
\begin{align*}\displaystyle
I&=\int f \, dx = \int\frac{\sqrt{16+5x}}{x} \, dx
\end{align*}
\begin{align*}\displaystyle
f&=\frac{\sqrt{16+5x}}{x}...
Hi everyone, I am stuck on a problem. I need to give a partial fraction of 1/N(k-N). I have tried every method so far ( plotting roots, systems of equations). I think I found A=1/k but I have no clue how to find B value. I would really appreciate any help as I am a desperate student trying to...
$\tiny{206.07.05.88}$
\begin{align*}
\displaystyle
I_{88}&=\int\frac{1}{(x+2)\sqrt{x^2+4x+3}} \, dx \\
&=?
\end{align*}
would partial fractions be best for this?
Trouble here in the below partial fraction (Bug)
$\frac{5x^2+1}{(3x+2)(x^2+3)}$
One factor in the denominator is a quadratic expression
Split this into two parts A&B
$\frac{5x^2+1}{(3x+2)(x^2+3)}=\frac{A}{(3x+2)}+\frac{Bx+c}{(x^2+3)}$...
In my physics courses I have seen this kind of notation several times now: $$ \frac {A/B} {C}$$
For instance:
or
To me it doesn't seem intuitive and ## \frac A {BC}## would seem like a neater way of writing it. Therefore I wonder if there's any specific reason to why people write it...