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).
Homework Statement
This is from a Physics GRE practice test:
An 8-cm-diameter by 8-cm-long NaI(Tl) detector detects gamma rays of a specific energy from a point source of radioactivity. When the source is placed just next to the detector at the center of the circular face, 50% of all emitted...
Homework Statement
Write the partial fraction decomposition of the rational expression. Check your result algebraically.
(x2 – 7x + 16)/[(x + 2)(x2 – 4x + 5)]
The Attempt at a Solution
[A/(x+2)] + [(Bx+C)/(x2-4x+5)]
x2-7x+16= A(X2-4x+5)+(Bx+C)(x+2)...
I was reading this website, and I don't understand this last statement.
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/fermi.html
It reads: "Since only a tiny fraction of the electrons in a metal are within the thermal energy kT of the Fermi energy, they are "frozen out" of the heat...
Homework Statement
\int\frac{4y^2-7y-12}{y(y+2)(y-3)} with limits of integration [1, 2]
Homework Equations
The answer given in the book is \frac{27}{5}ln(2)-\frac{9}{5}ln(3)
or \frac{9}{5}ln\frac{8}{3}
The Attempt at a Solution
\int\frac{4y^2-7y-12}{y(y+2)(y-3)}...
Homework Statement
Given the equation for Roaults law and Antoine equation for 2 species I need to recast these into a pair of coupled non linear equations to compute the mole fraction of species one and the temperature for given values of vapor fraction and the total pressure, Antoines...
Homework Statement
How do I simplify the following
\frac{\sqrt[3]{m^2+m} . \sqrt{1+m^2}}{\sqrt{1+m^2}.\sqrt{1+m^2}}
The Attempt at a Solution
I know that the denominator will be 1+m^2 but I don't know how to simplify the numerator. Can anyone show me how?
Homework Statement
Resolve the following into partial fractions.
Homework Equations
The Attempt at a Solution
Is my first step correct?
A/x + B/x^{2} + C/(x+1) + D/(x+1)^{2}
Thanks in advance!
Homework Statement
I basically need to find the integral of (-5x^3-1x^2+4)/(x^4+2x^3)
Homework Equations
Just the basic rules associated with partial fractions when re-writing a polynomial with constants.
The Attempt at a Solution
Since the degree of the denominator is 4 and...
Hi there!
1)It's a very common problem for me .
I don't get when do I have to multiply or divide a fraction in unitary method e.g.
10 men do 1 work in 40 days
10 men does 3/4 work in 40*3/4 days
or should i do 10 men does 3/4 work in 40*4/3 days
I do't undestand!Plz give me a reason...
Homework Statement
If i have 1/2 + 1 = 1,5 (on the calculator), how do i make this into a fraction?
I know this answer would be 1/2 + 1 = 3/2
But how excactly is this done?
Homework Equations
The Attempt at a Solution
Homework Statement
Solve for t:
[-2(t2+1) / 9(t2-1)] < 0
The Attempt at a Solution
I know that the answer is -inf<t<-1 and 1<t<inf, but how do I show the calculation to get that answer? When I tried, I narrowed it down to t<root-1, but that's not possible (without complex...
Ok, so why do some answers contain an A/something + (Bx + C)/something and others contain just an A/something + B/something.
I hope someone knows what I'm talking about!
Homework Statement
integral using partial fraction
Hi everyone ,,
got the following question :
The attempt at a solution
I reached the following ,,
integral ( 4/(X^2+x+1) + (-4x-1)/(X^2+x+1)^2) )
and now I don't know how to integrate .. -_-
help please
1. Homework Statement
The normal boiling points of benzene and toluene aer 80.1C and 110.6C, respectively. Both liquids obey Trouton's rule well. For a benzene-toluene liquid solution at 120C, with the molar fraction of benzene in the solution equalling 0.68, estimate the vapor pressure...
Homework Statement
Very pure liquid water can be subcooled at atmospheric pressure to temperatures well below 0ºC. Assume that 1kg has been cooled as a liquid to -6ºC. A small ice crystal (negligible mass) is added to "seed" the subcooled liquid. If the subsequent change occurs adiabatically...
prove that limit of --- (x^{n}-a^{n})
---------- ---------------- (x-a)
when " x " tends to " a "
equals to ---- n.a^{n-1} ----- ****where " n " is a fraction..
we have been given this and I've...
Homework Statement
Find the convergents to the fraction 57/44.
Homework Equations
Could anyone provide any good internet resuorces about "finding the convergents to the fraction"?
The Attempt at a Solution
Homework Statement
Obviously, we can make rockets to go very fast, but what is a reasonable top speed? Assume that a rocket is fired from rest at a space station in deep space, where gravity is negligible.
If the rocket ejects gas at a relative speed of 1100 and you want the rocket's...
My algebra is really rusty, anyway I'm trying to simplify a Transfer Function so I have
H(jw)=ZL/R+ZL so ZL is the impedance of an inductor and R is the impedance of the Resistor, my question is how would I go about simplifying that transfer function into it's simplest form?
Thanks...
So I've been thinking about this problem for quite some time now any help will be appreciated.
A sample of C_{2}H_{2}(g) has a pressure of 7.4 kPa. After some time a portion of it reacts to form {C}_6{H}_6 (g). The total pressure of the mixture of gases is then 3.6 kPa. Assume the volume and...
I'm looking at this particular splitting of a fraction in a book and I'm not completely sure what rule is being invoked. Here it is:
http://www.apthtml.com/images/split.png
Can anyone identify what this is?
Homework Statement
\frac{(cos60 - isin60)^5 * (cos45 - isin45)^3}{(cos15-isin15)^7}
Homework Equations
The Attempt at a Solution
I have had several tries so far, but simply do not know what to do. Would somebody be so kind and simplify this expression step by step. I...
Homework Statement
What is f'(x) of f(x) 1/sqrt(2x)?
2. The attempt at a solution
In applying the problem to the derivative formula:
(1 / sqrt(2(x + h)) - 1 / sqrt(2x)) / h
I multiplied the problem by a special form of one but that only put the rationals on the bottom of the...
Homework Statement
I need to solve the following integral:
{ x * ((1-x)/3)^(1/7) dx
where { represents the integral symbol b/c I don't know how to type that.
The Attempt at a Solution
I've tried integration by parts and don't get anywhere. None of the integration techiques I...
I am having trouble to find a straight forward method for finding coefficients using partial fraction expansion with repeated or complex roots. My study notes arent too clear so I am finding the differentiation method hard to follow for repeated roots. As for complex roots I can find the roots...
Homework Statement
I just can't understand it I've read plenty of guides online, I just can't figure it out. How do you do partial fraction decomposition the farthest i can get is below
1/[(s^2 + 1)(s+1)]
Homework Equations
The Attempt at a Solution
1/[(s^2 + 1)(s+1)] = A/(s+1)...
Homework Statement
If \lim_{z\rightarrow z_0}f(z)=A and \lim_{z\rightarrow z_0}g(z)=B then prove that \lim_{z\rightarrow z_0}\frac{f(z)}{g(z)}=\frac{A}{B} for B\neq0
The Attempt at a Solution
I write f(z)=A+\epsilon_1(z) and g(z)=B+\epsilon_2(z), where the epsilon-functions tend to zero as z...
Homework Statement
Show that the fraction of electrons within kT of the fermi level is equal to 3kT/2Ef, if D(E) = E^1/2.
Homework Equations
f(E) = 1 / ( exp(E-Ef)/kT + 1 ) fermi distribution
N = integral from 0 to inf of D(E)f(E)dE = total no. of electrons
The Attempt at a...
My TI 83 plus has a function called "Frac" which makes decimal numbers into fractions. If I have 0.33333... on the screen and press Frac the calculator returns 1/3. My question is how does this function work.
Derive the following equation for the fraction of a solute in layer 1 of the 2 layers formed by two immiscible liquids;
Fraction of solute in layer 1 = (KD*V1)/((KD*V1)+V2)
where KD=Distribution coefficient for the solute between chemical in V1 and chemical in V2
V1=Volume of liquid...
Homework Statement
A small piece of aluminum(atomic number 13) contains 1015 . (The atomic number is the number of protons; it determines the (positive) electric charge of the nucleus and, thus, the number of electrons in a neutral atom.) If the piece of aluminum has a net positive charge of...
Homework Statement
hey guys just wondering if anyone could help me simplify this. thansk
[1-(y2/x2)]/[1+(y2/x2)]
Homework Equations
The Attempt at a Solution
I am just coming back to math after a, oh 30 year or so, vacation. In the class I'm taking, we are studying Partial Fraction Decomposition ( Px/Qx:Qx). It doesn't entirely make sense to me, tho like a monkey typing the great American novel, I can solve them given enough time. I am just having...
I am trying to find the integral of the following:
\int\left({\frac{3 + 5x - 6x^2 - 7x^3}{2x^2}}\right)dx
What I did was to split up the fraction like so:
\int\left({\frac{3}{2x^2}}\right)dx + \int\left({\frac{5x}{2x^2}}\right)dx + \int\left({\frac{6x^2}{2x^2}}\right)dx +...
This might be a silly question:
Is there a meaningful way to use L'Hospital's rule to determine
\lim_{\vec{\mathbf{x}} \rightarrow \vec{\mathbf{0}}}\frac{x^3}{x^2+y^2}?
where
\vec{\mathbf{x}}=\left(\begin{array}{c}x\\y\end{array}\right)
Homework Statement
It is a really long word problem, but I only need help with integrating one part of the proble.
Homework Equations
I need to know how to integrate 2+5sin((4*pi*x)/(25))
The Attempt at a Solution
I tried using the chain rule by first integrating the sin into...
Homework Statement
Find the partial fraction decomposition of :
\frac{x^2}{(1-x^4)^2}
The Attempt at a Solution
\frac{x^2}{(1-x^4)^2}=\frac{A}{(1-x^4)}+\frac {B}{(1-x^4)^2}
=A(1-x^4)+B
when x=1
1=A(1-1^4)+B
Hence B=1 and A=0
\frac{x^2}{(1-x^4)^2}=\frac{0}{(1-x^4)}+ \frac{1}{(1-x^4)^2}...
Homework Statement
A "pulse oximeter" operates by using light and a photocell to measure oxygen saturation in arterial blood. The transmission of light energy as it passes through a solution of light-absorbing molecules is described by the Beer-Lambert law, given below, which gives the...
Homework Statement
The coefficient of the restitution of an object is defined as the ratio of its outgoing to incoming speed when the object collides with a rigid surface. For an object with a coefficient of 0.59, what fraction of the object's kinetic energy is lost during a single...
Homework Statement
2 solutions of C12H22O11 are mixed. Solution A is .150M with density of 1.05 g/mL. Solution B is 10%
C12H22O11 by mass with density of 1.03g/mL. 100 mL of solution A is mixed with 100g of B. What is the mole fraction of C12H22O11 in the mixed solution?
Homework...
I wasn't exactly sure where to put this, so I decided to try this section. It may be more appropriate to put this in the Calculus / Analysis section... I don't know.
Can anybody tell me whether or not it's possible to find the value of the continued fraction
0 + 1 / (2 + 3 / (4 + 5 / (6 +...
Homework Statement
I am just looking for a hint on which rule to follow first; what would you tell me if I already know how to deal with an ordinary complex fraction?
I tried using x (and multiplied everything) as the LCD and got the wrong answer.
3 - \frac...
Homework Statement
a - \frac {a}{1-\frac{a}{1-a}}
The Attempt at a Solution
\frac{a}{1-a} This might be the part where I am confused. I always get confused when I need to factor out a -1 from this fraction's denominator in order to put it into the form of \frac{a}{a-1}
So here...