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).
I would like to confirm my understanding between conjugates and complex conjugates. When rationalizing the denominator (when dealing with square roots) it is shown that we can multiply the fraction by its conjugate. I have seen the conjugate being defined as one of the signs of the binomial...
My first method to simplify the fraction is to to I flip ##\frac{5}{3}## up I get ##2 \times \frac{3}{5} = \frac{6}{5}##
Method 2: if I flip 3 up I get ##\frac{2}{5} \times \frac{1}{3} = \frac{2}{15}##.
Method 3: I could use it multiply ##\frac{3}{3}## since this is the same as mutlipying by...
For this problem,
I don’t understand why the bottom fraction cannot equal zero since we are not plugging in any numbers it just in terms of variables. Would anybody else be able to provide with an explanation?
Many thanks!
When working with fractions and when we have a fraction or equation with fractions like this one for example ##\frac{x}{x-1}+\frac{x}{x+1}=\frac{9}{4}## do we always need to set boundaries? Like, do we always need to write that x can't be a number that would give the denominator 0? In this...
The first part of the question asked me to calculate the mean and standard deviation for the number of remain votes in the simple binomial model consisting of total sample size of 2091 people. I believe this is fairly straightforward, it was simply ##E(X) = \mu = 2091(0.5) = 1045.5## votes and...
I can do the question using brute force. First I multiply both the numerator and denominator by ##\sqrt{5} + \sqrt{3} - \sqrt{2}## then I simplify everything and rationalize again until no more square root in the denominator.
I want to ask if there is a trick to reduce the monstrous calculation...
Say you want to find the following Integrals
$$\int \frac{1}{(x-1)(x+2)} (dx)$$
$$\int \frac{1}{(x-1)(x^2 + 2)} (dx)$$
The easiest way to solve them will be by using partial fraction decomposition on both the given functions.
Decomposing the first function,
$$\frac{1}{(x-1)(x+2)} =...
Hello!
Im having some trouble with solving ODE's using Laplace transformation,specifically ODE's that require partial fraction decomposition.Now I know how to do partial fraction decomposition,and have done it many times on standard polynoms but here some things just are not clear to me.For...
Problem : Let me copy and paste the problem statement as it appears in the text, as shown above.
Attempt : I can sense there is an "elegant" way of doing this, but I don't know how. I show below my attempt using ##\text{Autodesk Sketchbook}##. I hope am not violating anything.
Ok so I have...
So my final equation is:
##\frac {1} {2700} + \frac {1} {3930n^2} + 10^{-5}##
I need to boil this down, the learning materials has the following working, but I can't seem to get it
$$\frac {1} {2700} + \frac {1} {3930n^2} + 10^{-5}$$
$$\frac {3930n^2+2700+2700*3930n^2*10^{-5}}...
Hi,
I have given the following, which I would like to show that this estimation is correct, where ##|\theta| \leq \frac{\pi}{^2}## and ##M \geq 1##:
$$\frac{1}{M^2}\frac{\sin^2(M\theta)}{\sin^2(\theta)} \geq \frac{4}{\pi^2}$$
I would approach an estimation of the denominator via ##\sin(x)...
Problem statement : Let me copy and paste the problem as it appears in the text :
Attempt : I am afraid this looks like a very difficult problem, despite being at the elementary level (high school). My glance through the text shows that the authors have gone about reducing the first set of...
I am a little lost on the last step of this problem. I get that we want to know how much time elapses for the capacitor to reach 2/3 of its final charge. That is why 2/3Qf is equal to Qf(1-e^-t/RC).
I don't understand how we make the jump to e^-t/Tau is equal to 1/3? and then somehow e^-t/Tau...
Can we make sense out of the formula of entropy like we do for density (like "quantity of mass per unit volume")? What's the sense of Q/T? Couldn't it be something else?
Of course it probably is a 'me-problem', but I haven't studied Thermodynamics deeply yet and was wondering what Entropy...
Problem Statement : If ##\dfrac{x}{b+c-a}=\dfrac{y}{c+a-b}=\dfrac{z}{a+b-c},## prove that ##\boxed{\boldsymbol{\dfrac{x+y+z}{a+b+c}=\dfrac{x(y+z)+y(z+x)+z(x+y)}{2(ax+by+cz)}}}##
Attempt : Let the fractions (ratios) ##\dfrac{x}{b+c-a}=\dfrac{y}{c+a-b}=\dfrac{z}{a+b-c} = \boldsymbol{k}##...
good day
I want to study the convergence of this serie and want to check my approch
I want to procede by asymptotic comparison
artgln n ≈pi/2
n+n ln^2 n ≈n ln^2 n
and we know that
1/(n ln^2 n ) converge so the initial serie converge
many thanks in advance!
Greetings.
I'm having a bit of difficulty with getting from the first to the second equation. I know some basic identities, but it all just feels like a mess. My approach was just going to be to write whatever I could, but some of the terms are confusing me...
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...
Hello, I am reading the Selfish gene by Richard Dawkins. I am confused on genes. In some part of the book he gives us clues on how to compute the percentage of genes we have in common with relatives, such as parents and brothers and sisters, in those particular cases it is stated that we share...
I understand how these expressions are derived, but I don't see how to calculate successive resultant values. Should the continued fraction expression simply be replaced by 1?
Hi all!
I have a question ...
I have to calculate the partial pressure of O2
Ccan I calculate the mole fraction from the data of 1 ppm of O2?
The total pressure is 760 torr
Thank you!
I am trying to find the derivative of this problem using the four step process but keep getting stuck when it comes to the third step of f(x+h) - f(x). I do not know what to do once I reach that step. Am I canceling terms out incorrectly? How should I deal with a fraction over a fraction? Any...
Hello,This is actually a piece of a little bigger problem (convergence of a series) - you can see the ratio test ak+1 / ak
That's why the (n) and (n+1) terms
I have lim n->∞ of (n√n) / (n+1)√(n+1) ∞/∞
I have tried L'Hopitals rule (requiring multiple times) and I am not seeing an end...
If f(x)=\frac{3x^2-5}{x+6} then f(0) + f'(0) is ...
A. 2
B. 1
C. 0
D. -1
E. -2
What I did:
If f(x)=\frac{u}{v} then:
u =3x^2-5 → u' = 6x
v = x + 6 → v' = 1
f'(x) =\frac{u'v-uv'}{v^2}=\frac{6x(x+6)-(3x^2-5)(1)}{(x+6)^2}
f(0) + f'(0) = \frac{3(0^2)-5}{0+6} +...
Afternoon all,
Hopefully somebody can help me, I'm doing my final year project and it's looking at the effect of heat treatment on in17, when I run an XRD scan I found that I all the phases sort of hid behind the matrix and so can't really make them out.
So I've been looking at using the SEM...
I am having a problem finding the right order above and below to find the finite expansion of a fraction of usual functions assembled in complicated ways. For instance, a question asked to find the limit as x approaches 0 for the following function
I know that to solve it we must first find...
For the first part, since this is a intrisinc semiconductor, n=p= intrisinc carrier concentration. Hence free electrons and hole = ##(1.5*10^{10})## per cubic centimeter.
As for part 2, here are my steps. But I'm not sure if it's correct.
I first find the number of atoms of one cubic...
Moved from technical math section, so missing the homework template
Summary:: Find a general formula for the nth derivative
Hi everyone!
How would I approach and answer a Q such as this
I began by rewriting the expression in a different form, then used chain rule to each given term
I...
I am looking for the expectation of a fraction of Gauss hypergeometric functions.
$$E_X\left[\frac{{}_2F_1\left(\begin{matrix}x+a+1\\x+a+1\end{matrix},a+1,c\right)}{{}_2F_1\left(\begin{matrix}x+a\\x+a\end{matrix},a,c\right)}\right]=?$$
Are there any identities that could be used to simplify or...
Do anyone know how to find ##1##, ##2x - 5##, and ##2\sqrt{x^2 - 5x + 6}## in the triangle? (please see attached image)
Also, how do you find ##(x - 5/2)^2 - (1/2)^2##?
[Moderator's note: Moved from a technical forum and thus no template.]
Find integration of:
\frac {1}{(x+1)(x^2 + x -1)^\frac{1}{2}}
What I did:
1. Use completing square method for the term inside the square root
2. Use trigonometry substitution (I use secan)
3. After simplifying, use another trigonometry substitution (I use weierstrass substitution)
4. Use...
2. Betty would like your help planning her garden whose rows are all of equal length. She would like to grow carrots, beans, sunflowers, tomatoes and peppers. She tells you she would like to devote twice as many rows to growing peppers as she does to beans. She only has enough tomato plants to...
I just want to clear some confusion I am having with the Fermi-Dirac distribution & density of states (DOS) of a semiconductor, which are given by
Say we have a piece of Silicon in equilibrium and its Fermi level lies 0.25 eV below the conduction band edge, i.e. Ec - EF = 0.25 eV. Let us say...
## \frac{-3x^4}{7}+x=\frac{-3x^4}{7}+10 \\x=10##
I solved the equation and got x=10, My question is how can I solve for x without brute-force method( Multiply all terms by (x-13)(x-7)(x-14)(x-6) and solving for x)
Welders use a logarithmic scale to identify protective eyewear. The shade number n, is given by the
equation n = 1 − 7logT/3 , where T is the fraction of visible light that glass transmits.
a. What shade number should a welder use that only transmits ⅛ of the light entering the glass?
b Viewing...
Although I have over the years seen reports about simulations of the the Earth's moon's origin by a variety of possible mechanisms, none of these reports have ever reported the probability of such events. What is a reasonable estimate of the fraction of solar systems that have a planet of a size...
Well doesn't ##u(x) = 0.4 x## work? Seems too easy, but the phrasing at the end "for all ##x\in I##" makes me think since ##0.4x = x## only at ##x=0##, and not all of ##I##, that this is okay. But am I wrong?
I want to compute the fraction of time both particles spend outside the finite potential well. All I can get is the probability to find them outside. The wavefunction outside the potential is:
$$\frac{d^2\psi}{dr^2} = -L^2 \psi$$
Where:
$$L = \sqrt{\frac{2mE}{\hbar^2}}$$
Solving the...
I have a word equation i cannot figure out! anyone have any ideas?
There is a large bag of red and blue counters. One red and one blue counter are removed from the bag. This process is repeated until 1/4 of the red counter and 3/5 of the blue counters have been removed. What fraction of the...
Homework Statement
Express the sum as a fraction of whole numbers in lowest terms:
##\frac{1}{1⋅2}+\frac{1}{2⋅3}+\frac{1}{3⋅4}+...+\frac{1}{n(n+1)}##
Homework EquationsThe Attempt at a Solution
Please see attached image for my work. The reason I am posting the image rather than typing this...
<Moderator's note: Moved from a technical forum and thus no template.>
Suppose that all stars in this galaxy were born in a single major-merger burst event about 10
Gyr ago. From this original burst, I want to compute the fraction of stellar mass still surviving as stars in the
main sequence ...
Homework Statement
Sorry for so many posts lately, hopefully this is allowed.
What tangent points on ##g(x)=\frac {12} {x+1}## has an instantaneous rate of change of -3?
Homework EquationsThe Attempt at a Solution
[/B]
I know that once I derive ##g(x)=\frac {12} {x+1}## I can set the...
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)