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.

POSITIVE FRACTIONS

We begin with positive common fractions, where 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, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts make up a whole. 

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 102 all equal the fraction 1/100. An integer such as the number 7 can be thought of as having an implicit denominator of one: 7 equals 7/1

Other uses for fractions are to represent ratios and division. Thus the fraction 3/4 is also 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 in the case using a fraction to represent division is an example of the rule that division by zero is undefined.

NEGATIVE FRACTIONS

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 require that, for example, negative divided by positive is negative, 1/2, -1/2

 and 1/-2, all represent the same fraction, negative one-half. 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.

ARITHMETIC WITH FRACTIONS

Multiplying the numerator and denominator of a fraction by the same (non-zero) number results in a fraction that is equivalent to the original fraction. This is true because for any non-zero number n/n=1. Therefore, multiplying by n/n is equivalent to multiplying by one, and any number multiplied by one has the same value as the original number. By way of an example, start with the fraction 1/2. When the numerator and denominator are both multiplied by 2, the result is 2/4, which has the same value (0.5) as 1/2. To picture this visually, imagine cutting a cake into four pieces; two of the pieces together 2/4 make up half the cake 1/2.

EQUIVALENT FRACTIONS

The process for subtracting fractions is, in essence, the same as that of adding them: find a common denominator, and change each fraction to an equivalent fraction with the chosen common denominator. The resulting fraction will have that denominator, and its numerator will be the result of subtracting the numerators of the original fractions. For instance,

Subtraction

COMPARING FRACTIONS

Comparing fractions with the same positive denominator yields the same result as comparing the numerators: 3/4>2/4 because 3 > 2, and the equal denominators 4 are positive. If the equal denominators are negative, then the opposite result of comparing the numerators holds for the fractions:3/-4<2/-4 because a/-b= -a/b and -3<-2.

If two positive fractions have the same numerator, then the fraction with the smaller denominator is the larger number. When a whole is divided into equal pieces, if fewer equal pieces are needed to make up the whole, then each piece must be larger. When two positive fractions have the same numerator, they represent the same number of parts, but in the fraction with the smaller denominator, the parts are larger.

One way to compare fractions with different numerators and denominators is to find a common denominator. To compare a/b and c/d, these are converted to axd/bxd  and bxc/bxd (where the dot signifies multiplication and is an alternative symbol to ×). Then bd is a common denominator and the numerators ad and bc can be compared. It is not necessary to determine the value of the common denominator to compare fractions – one can just compare ad and bc, without evaluating bd, e.g., comparing 2/3 ? 1/2 gives 4/6 > 3/6