What is a fraction?

The word 'fraction' comes from the Latin 'fractus' which means 'broken.' A fraction is written as one number above another number with a line called the vinculum separating the two. The number on top is called the numerator which tells us the size of the quantity that is is going to be broken up. The denominator which is the number on the bottom tells us into how many pieces we are going to break up the numerator. All the divided pieces will be of the same size, and the size of the remaining pieces is what the fraction is equal to.

You may see the fraction takes the form $\frac{a}{b}$ where $a$ and $b$ are numbers. Every fraction may be expressed this way. Sometimes the numerator, is smaller than denominator, in which case the fraction is called a proper fraction. If the numerator is greater than the denominator then we call the fraction an improper fraction. If $a=b$ then the fraction is an integer.

If we have a whole number along with a fraction, then we call the entire number a mixed fraction such as $1 \frac{1}{2}$.

As an example, consider the improper fraction, $\frac{8}{2}$ which may be written as, 8/2. Here, 8 is the amount we are going to divide, and we are going to break 8 into two pieces of size 4. The size of the remaining pieces is what the fraction is equal to. All the pieces will be of the same size.

Consider $\frac12$. The idea behind this fraction is that something of size 1 is going to be broken up into 2 pieces. Namely, the result will be one unit half the size of 1.

Similar fractions are two or more fractions with the same denominator. The least common denominator of two fractions is the least common multiple of the denominators.

Two fractions are equivalent if both the numerator and denominator of one fraction are multiples of the numerator and denominator the other fraction, respectively. For example consider $\frac{3}{6}$ and $\frac{1}{2}$. 3 is a multiple of 1, and 6 is a multiple of 2 so therefore the fractions are equivalent. The fractions 3/4 and 6/7 are not equivalent because even though 6 is a multiple of 3, 7 is not a multiple of 4.