Key Points

A fraction represents a part of a whole or a collection. It is formed when a whole is divided into a number of equal parts.

In a fraction, the top number is the numerator, which shows the number of parts considered. The bottom number is the denominator, which shows the total number of equal parts the whole is divided into.

A fraction with 1 in the numerator is called a fractional unit or unit fraction, such as 1/2 or 1/7. It represents one single equal part of the whole.

When comparing fractions with a numerator of 1, the fraction with the larger denominator is smaller. For example, 1/10 is smaller than 1/5.

Fractions can be represented as points on a number line. The segment between two whole numbers, like 0 and 1, can be divided into equal parts to mark fractions.

Proper fractions have a numerator smaller than the denominator and are less than 1. Improper fractions have a numerator greater than or equal to the denominator and are greater than or equal to 1.

A mixed fraction is a combination of a whole number and a proper fraction, like 2 1/4. It is another way to express an improper fraction.

To convert an improper fraction like 7/2 to mixed, divide 7 by 2 to get 3 with a remainder of 1, which is 3 1/2. To convert 3 1/2 to improper, calculate (3 * 2) + 1 = 7, which is 7/2.

Equivalent fractions are fractions that represent the same value, such as 1/2, 2/4, and 3/6. They are created by multiplying or dividing both the numerator and denominator by the same non-zero number.

A fraction is in its simplest form or lowest terms when its numerator and denominator have no common factor other than 1. For example, the simplest form of 16/20 is 4/5.

To compare fractions with different denominators, first convert them to equivalent fractions with a common denominator. Then, compare their numerators; the fraction with the larger numerator is greater.

To add or subtract fractions with the same denominator, simply add or subtract the numerators and keep the common denominator. For example, 5/8 - 2/8 = 3/8.

To add or subtract fractions with different denominators, first find a common denominator and convert them to equivalent fractions. Then, add or subtract the new numerators.

The ancient Indian method for adding or subtracting fractions, formalized by Brahmagupta, involves reducing them to a common denominator before performing the operation. This is the same method used today.