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).
Hello! (Wave)
I want to show that if $c_k=\frac{p_k}{q_k}$ the $k$-th convergents of the continued fraction $[a_0; a_1, a_2, \dots, a_n]$, then $q_k \geq 2^{\frac{k-1}{2}} (1 \leq k \leq n)$.
Could you give me a hint how we could show this? (Thinking)
I'm given the problem 14x+5y=31, 2x-3y=-29
This is a set of linear equations that I am asked to solve. That being said I am getting hung up on how to distribute a fraction. The answer is x=9, however, I am getting x=36 and the step I do not understand is distributing the fraction.
We then get...
Hi,
I would like to expand the following expression:
1/[((a+s)*(1+b/s)^m)], where a, b, and s are real nonnegative values and m is an arbitrary positive integer.
Particularly, according to partial fraction expansion, it becomes:
Sum[A_j/[(1+b/s)^j],{j,1,m}]+B/(a+s). I look for a closed-form...
Homework Statement
Consider solar neutrinos of energy 1 MeV (EDIT: 10 MeV not 1 MeV) which are formed at the center of the sun in the ##\nu_2## eigenstate. What fraction of it do you expect to arrive at Earth as ##\nu_\mu## and what fraction as ##\nu_\tau##? Assume that it evolves adiabaticaly...
When Cern is done with a particle beam, what fraction of a typical beam bunch is left having not scattered out of the bunch? Is it of order 1/2? A very rough estimate would do.
Edit, I should add that I read an accelerated beam might be used for a day and at the end of the day they dispose of...
Homework Statement
Find the partial fraction decomposition for:
##\frac{1}{\left(x^2-1\right)^2}##
Homework EquationsThe Attempt at a Solution
Please see my attached images. I think the image shows my thought process better and it would take me well over an hour to type all that out!
Im...
There is a problem in a PreCalculus book that I'm going over that states:
Express the sum ##\frac{1}{2⋅3}+\frac{1}{3⋅4}+\frac{1}{4⋅5}+...+\frac{1}{2019⋅2020}## as a fraction of whole numbers in lowest terms.
It goes on to state that each term in the sum is of the form...
I’ve had huge difficulty understanding/interpreting the concept of branching fraction. So correct me if I’m wrong please:
Let’s take the decay of Ra-226 with half-life of 1602 years as an example. It decays through alpha 1 chanel to the excited state of Rn-222 with E=0,187 Mev ( branching...
Homework Statement
If mole fraction, pressure fraction and volume fraction are denoted by Xmol , Xp, XV respectively, of a gaseous component, then what is the relation between them?
Homework Equations
mole fraction = mole of component / total moles
pressure fraction = pressure of component /...
Conceptually, at least, this is a simple question, although I recognize that it might be hard to calculate in practice from available data.
The matter-energy budget of the universe is measured (in a model dependent way) to consist of a certain percentage of dark energy, a certain percentage of...
Hi i’m having problems with the following equations:
X(w)=2/(-1+iw)(-2+iw)(-3+iw)
This then becomes the following equation according the the tutorial, although there is no explanation as to how:
X(w)=1/-1+iw, -2/-2+iw, +1/-3+iw
The commas indicated the end of each fraction to make it easier...
Assume a source of EM radiation at wave length λ hits a barrier with a small circular hole of diameter d << λ. What fraction of the radiated power (watts) that hits the hole passes through it? Does it depend on the thickness of the barrier?
I understand that after passing through the hole, the...
Homework Statement
Homework EquationsThe Attempt at a Solution
Honestly speaking , I have absolutely no idea about this problem .This doesn't happen often .In the KTG chapter , only definition and formula of Average , RMS and Most probable speeds is given .
This question is for an...
-
check if right
check if right
Now, 2 seems to be the right answer for A yet when i made x=5 and subtracted new form form the old one I got a difference of ~$\frac{4}{9}$ (should be 0 obviously) I got A=2 B=$\frac{45}{21}$ C=2
How to calculate $\lim_{{x}\to{\infty}}(- e^{-x})$
Homework Statement
\int\frac{x^2}{\sqrt{x^2+4}}dx
Homework Equations
n/a
The Attempt at a Solution
Letting x=2tan\theta and dx=2sec^2\theta d\theta
\int\frac{x^2}{\sqrt{x^2+4}}dx=\int\frac{4tan^2\theta}{\sqrt{4+4tan^2\theta}}2sec^2\theta d\theta=\int\frac{8tan^2\theta...
Homework Statement
When completely hollow. To what fraction of volume should water be poured into it so that it is just enough to cause the sphere to sink. [/B]Homework Equations
Archemedes : Buoyant fource equals weight of water displaced
The Attempt at a Solution
Let's first find the...
Hi
Point kinetics equation with one-group delayed neutrons,
dn/dt= ((ρ-β)/∧)n+λC
dC/dt= (β/Λ)n - λC
When a reactor in steady-state operation, derivative terms would be zero. From second equation,
C/n= β/λΛ is found. For given β=0.0065, λ=0.08 and Λ=0.0001, the ratio of the steady state...
Let me start by saying that this is from a 30 question assessment on Big Oh, Big Theta, and Big Omega. I understood every other question, however, even after being given the correct answer, I do not understand why my answer was wrong for this one. If you could point me in the direction of any...
Homework Statement
Hi,
So I had a pretty long question solving a Linear ODE but now I've gotten stuck at this stage where I can't seem to get it into the right form to carry out partial fraction expansion
Homework EquationsThe Attempt at a Solution
[/B]
I'm quite sure that I what I have at...
Homework Statement
[/B]
When you drink cold water, your body must expend metabolic energy in order to maintain normal body temperature ( 37.0°C ) by warming up the water in your stomach. Could drinking ice water, then, substitute for exercise as a way to “burn calories?” Suppose you expend 430...
Homework Statement
This question is problem 1.3.1 from Herbert Callen's textbook in Thermodynamics, second edition.
In the answers I have the following: Mole fraction of NaCl = 0.057, molar volume =$18\times 10^{-6} m^3/mole$.
How did they arrive at these answers? did they multiplied the...
Hey guys, I am seriously confused by this problem.
3/4 [ 5/6 ( -18/25 ) + 1/2 ]
I would appreciate it if someone shows me the step by step process.
Thanks in advance :)
Homework Statement
The width of a rectangle is increased by one tenth, but the area remains the same. By what fraction has the length of the rectangle been reduced?
The Attempt at a Solution
Length = x
Width = x + 1/10
Set equation:
L * W
x *(10x + x)
10x^2 + x^2
I am stuck... Please help...
Homework Statement
I would like to understand whether reocurring formulas in the example solutions to my homework problems hold in general and ideally where they're derived from. (I'm new here; if there's there a way to embed formulas in f.i. latex code please tell me.)
As an example problem...
I am finding it hard to understand this questions its sounds more like fraction subtraction than multiple but the army test sample questions says its multipication
You have 3 1/4 boxes of paper. You give 1/2 to the paper to a colleague, how many boxes of paper do you have left.
The test says 1...
Homework Statement
If a proton with a kinetic energy of 6 MeV is traveling in a particle accelerator in a circular orbit of radius 0.75m, what fraction of its energy does it radiate per second?
m = 1.67 * 10^-27
epsilon_0 = 8.854 * 10^-12
c = 3 * 10^8
Homework Equations
dE/dt = (q^2 a^2) /...
Homework Statement
See below
Homework EquationsThe Attempt at a Solution
I am looking at a particular integral, and to get started, my text gives the indication that one should use partial fraction decomposition with ##\displaystyle \frac{\cos (ax)}{b^2 - x^2}##. Specifically, it says "then...
I have known the interaction parameter of a certain liquid mixture
which has the phase behaviour can be described by the lattice model.
, is it possible for us to know the temperature at the critical point?
Besides, if we know the temperature at the critical point, can the volume fraction of...
Homework Statement
Find the partial fraction decomposition of ##\displaystyle \frac{1}{x^4 + 2x^2 \cosh (2 \alpha) + 1}##
Homework EquationsThe Attempt at a Solution
Using the identity ##\displaystyle \cosh (2 \alpha) = \frac{e^{2 \alpha} + e^{- 2\alpha}}{2}##, we can get the fraction to the...
Hello Everyone , I need some help in solving this partial fraction $\frac{x^2}{(x-2)(x+3)(x-1)}$
I am using this method in which the partial fraction is broken into 3 parts namely A,B &C
$\frac{x^2}{(x-2)(x+3)(x-1)}=\frac{A}{(x-2)}+\frac{B}{(x+3)}+\frac{C}{(x-1)}$...
I have figured out a nice way to prove that if the complex numbers z_1,z_2,\ldots, z_N\in\mathbb{C} are all distinct, then the equation
\prod_{n=1}^N \frac{1}{z - z_n} = \sum_{n=1}^N \frac{\alpha_n}{z-z_n}
is true for all z\in\mathbb{C}\setminus\{z_1,z_2,\ldots, z_N\}, where the alpha...
$\tiny{206.8.5,42}\\$
$\textsf{partial fraction decomostion}\\$
\begin{align}
\displaystyle
&& I_{42}&=\int\frac{3x^2+x-18}{x^3+9x}\, dx& &(1)&\\
&& \frac{3x^2+x-18}{x^3+9x}
&=\frac{Ax+B}{x^2+9}
+\frac{C}{x}
& &(2)&
\end{align}
$\textit{just seeing if this is set up ok before finding values} $
Hi guys, I have a question that I can't seem to wrap my head around.
This is the question:
A power reactor is fueled with slightly enriched uranium. At the end of core life (i.e., when it is about to be batch refueled), 30% of the power comes from the fissioning of the built up Pu-239...
Hi,
I have the following question:
2 whole number ¼ - 1 whole number 2/3 ------(1)
1 whole number (1/4 -2/3)
I am getting –ve sign which is wrong.
However if I do:
2 whole number ¼ + 1 whole number 2/3
Then:
3whole number (1/4 * 3/3 + 2/3 * 4/4)
= 3 whole number(3/12 + 8/12)
= 3...
Take the fractions 2/5 and 4/10. 2/5 cannot be simplified further without converting the integers into decimals. I need to know the name of the 2/5 fraction. Maybe it's called irreducible form or something like that. Let me give another example, of 1/3, 3/9, or 33/99 only 1/3 is irreducible...
I want to create a binary matrix (${X}_{m*n}$) containing $C$ ones ($||X|| = C$). Additionally I want to have the number of elements of each row ($m$) to have an exponential form. This is: for each row the number elements needs to be equal to $e^{alpha * i}$ or in symbols: ||${X}_{i}$|| =...
Homework Statement
Consider a gas of molecules of mass m at equilibrium at temperature T. Obtain an expression for the fraction of molecules having kinetic energy e = 1/2mC2 in the range e to e + de.
This is problem 5.3, page 48 of Vincenti and Kruger's Intro to Physical Gas Dynamics
Homework...
$\frac{2a}{a^2-4} - \frac{1}{a-2}$
Now using difference of squares in $a^2-4$ = $\left(a+2\right) \left(a-2\right)$
Then the fraction would be $\frac{2a}{\left(a+2\right) \left(a-2\right)} - \frac{1}{a-2}$
Can this fraction be simplified further?
Many Thanks :)
This problem is a little elementary,
If we were to multiply the fraction $\frac{3}{7}$ by two which way should I be using,
$2\left(\frac{3}{7}\right)=\frac{6}{14}$ --------------- 1
or
$\frac{2}{2}\left(\frac{3}{7}\right)=\frac{6}{14}$ --------------- 2
I usually multiply fractions using...
Hey, all! I'm learning partial fraction decomposition from Serge Lang's "A First Course in Calculus." In it, he gives the following example:
\int\frac{x+1}{(x-1)^2(x-2)}dx
He then decomposes this into the following sum:
\frac{x+1}{(x-1)^2(x-2)} =...
Homework Statement I did some homework and wolfram alpha afterwards gave different result than what I have. Apparently I must have done some error, but I couldn't find error after I doublechecked about two times. So I decided to see if you guys can find an error. And what is the correct answer...
Hi,
\frac{3 }{ 2(x+1)} + \frac{1}{x+1} = \frac{5}{6}
Can you help me on this problem. I'm not sure on how to solve this and please be kind enough to explain your steps little bitMany Thanks:)