What is Fraction: Definition and 652 Discussions

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:


{\displaystyle {\tfrac {1}{2}}}


{\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



{\textstyle {\frac {\sqrt {2}}{2}}}
(see square root of 2) and π/4 (see proof that π is irrational).

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  1. rstor

    Conjugate vs Complex Conjugate

    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...
  2. C

    Why Does Flipping the Denominator in Complex Fractions Give the Wrong Answer?

    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...
  3. C

    Simplifying this complicated fraction

    How did they get y^7 in the bottom fraction? I got their answer except I had y^6. Would some please be able to help? Many thanks!
  4. C

    I Why Can't the Denominator of a Fraction Be Zero?

    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!
  5. C

    Do We Need Boundaries for Fraction Equations?

    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...
  6. P

    Understanding the meaning of "expected fraction" (Statistics)

    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...
  7. S

    Rationalizing this fraction involving square roots

    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...
  8. Physics Slayer

    A doubt in Partial fraction decomposition

    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)} =...
  9. A

    Partial fraction decomposition with Laplace transformation in ODE

    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...
  10. brotherbobby

    Reducing an algebraic fraction, cyclic in three variables, to another

    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...
  11. Jason-Li

    How Do You Simplify Complex Algebraic Fractions?

    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}}...
  12. P

    I Finding a bound for a fraction

    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)...
  13. brotherbobby

    Reducing one algebraic fraction to another

    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...
  14. guyvsdcsniper

    TIme needed for a capacitor to reach a fraction of its final charge

    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...
  15. Ale_Rodo

    I Why is Entropy defined as a fraction of heat over temperature?

    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...
  16. brotherbobby

    Prove a rational fraction is equal to another

    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}##...
  17. A

    Convergence of a series involving ln() terms in the denominator of a fraction

    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!
  18. WhiteWolf98

    Using Euler's Formula to write a fraction in another form

    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...
  19. WhiteWolf98

    Using Complex Conjugates to Decompose a Fraction

    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...
  20. fluidistic

    What fraction of genes do we share with relatives and with other species?

    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...
  21. S

    I How to calculate successive resultant values of a continued fraction

    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?
  22. il postino

    Chemistry Relationship between ppm and mole fraction

    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!
  23. C

    MHB Trouble finding the derivative of a fraction using four step process

    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...
  24. S

    Limit of a fraction as n-> infinity in the numerator and denomominator

    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...
  25. Monoxdifly

    MHB [ASK] Derivative of an Algebraic Fraction find f(0) + f'(0)

    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} +...
  26. J

    Volume fraction of multiple phases

    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...
  27. C

    I Finite expansion of a fraction of functions

    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...
  28. jisbon

    Fraction of valence electrons free for conduction

    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...
  29. B

    Finding a general formula for the nth derivative of a partial fraction

    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...
  30. rafgger

    A Expectation of a Fraction of Gaussian Hypergeometric Functions

    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...
  31. karush

    MHB -gre.ge.3 Circles Find the shaded area as a fraction

    Ok this is considered a "hard" GRE geometry question... notice there are no dimensions How would you solve this in the fewest steps?
  32. A

    How Do Partial Fractions Relate to Trigonometric Substitution?

    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.]
  33. Sabra_a

    Mass fraction and volume of a gas in a cylinder

    I have attached the full answer in PDF file. I'm not sure about the answers. will really appreciate if they get checked
  34. S

    Simplify this fraction containing imaginary numbers

    i can get to 3i+1/1-3i but no further. I take it this is the correct way to start
  35. S

    Is there a faster way to integrate this fraction?

    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...
  36. S

    MHB Fraction word problem: how many rows are devoted to each plant

    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...
  37. S

    Fraction of occupied states (Fermi-Dirac distribution + DOS)

    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...
  38. S

    Finding the value of a fraction

    I tried to do like this: \frac{((76 - 15)^4 + 324)((76 - 3)^4 + 324)((76 + 9)^4 + 324)((76 + 21)^4 + 324)}{((76 - 21)^4 + 324)((76 - 9)^4 + 324)((76 + 3)^4 + 324)((76 + 15)^4 + 324)} Then stucked Thanks
  39. Manasan3010

    Quickest way to do this calculation

    ## \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)
  40. F

    MHB What fraction of light do these glasses transmit

    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...
  41. Buzz Bloom

    I Fraction of solar systems with an Earth-moon configuration?

    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...
  42. M

    Find a function less than a fraction of itself

    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?
  43. A

    Why is this equation correct? (algebraic fraction simplification)

    Hi all, I cannot figure out how I get to the alternate form. I really do need help here. Not sure how this is done. Thank you
  44. JD_PM

    Calculating the Fraction of Time Particles Spend Outside a Potential Well

    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...
  45. J

    MHB Ratio word problem: What fraction of the original counters remain in the bag

    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...
  46. opus

    Express this sum as a fraction of whole numbers

    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...
  47. F

    From the original burst, fraction of stellar mass still on the Main Sequence

    <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 ...
  48. Specter

    Instantaneous rate of change and deriving a fraction

    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...
  49. evinda

    MHB K-th convergents of continued fraction

    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)