The set of rational numbers $\mathbb{Q}$ is the set of numbers that can be written as a fraction of two integers: \[ \mathbb{Q} \equiv \left\{ \frac{m}{n}\ \bigg|\ m \in \mathbb{Z}, n \in \mathbb{N}_+ \ \right\}, \] where $\mathbb{Z}$ denotes the set of integers $\mathbb{Z}=\ldots, -1,0,1,2,3,\ldots$ and $\mathbb{N}_+$ denotes the set of positive natural numbers $1,2,3,4,\ldots$. The interpretation is that some whole is cut into a total of $n$ pieces and that we are given $m$ of these pieces.

We read $\frac{1}{4}$ either as one over four or one quarter, which is also equal to $0.25$, but as you can see the notation $\frac{1}{4}$ is more compact and nicer. Why nicer? Well let's take a look at some simple fractions: \[ \begin{align*} \frac{1}{1} &= 1.0 \nl \frac{1}{2} &= 0.5 \nl \frac{1}{3} &= 0.33333\ldots = 0.\overline{3} \nl \frac{1}{4} &= 0.25 \nl \frac{1}{5} &= 0.2 \nl \frac{1}{6} &= 0.166666\ldots = 0.1\overline{6} \nl \frac{1}{7} &= 0.14285714285714285\ldots = 0.\overline{142857} \end{align*} \] Note that a line on top of some numbers means that these numbers are repeated. The fractional notation on the left is preferable, because it shows the underlying structure of the number and it avoids the need to write infinitely long decimals.

Writing down rational numbers as fractions allows us to do precise mathematical calculations easily on pen and paper without the need for a calculator.

Calculate the sum of $\frac{1}{7}$ and $\frac{1}{3}$.

If we use the decimal notation we would have to write our answer as \[ \begin{align*} \textrm{ans} &= 0.\overline{142857} \ + \ 0.\overline{3} \nl &= 0.142\:857\:142\:857\ldots \ + \ 0.333\:333\:333\:333\ldots \nl &= 0.476\:190\:476\:190\:476\ldots \nl & = 0.4\overline{761904}. \end{align*} \] Wow that was complicated! And complicated for nothing too. Let us see how much simpler this calculation is if we use fractions: \[ \frac{1}{7}+\frac{1}{3} = \frac{3\times 1}{3\times 7}+\frac{1 \times 7}{3 \times 7} = \frac{3}{21}+\frac{7}{21} = \frac{3+7}{21} =\frac{10}{21}. \]

The fraction $a$ over $b$ can be written in three different ways: \[ a/b \equiv a \div b \equiv \frac{a}{b}. \] The two constituents of the fraction have special names:

• $b$ is the denominator of the fraction and

tells you how many parts there are in the whole.

• $a$ is the numerator and tells you how

many of these parts are given.

Consider two fractions $\frac{a}{b}$ and $\frac{c}{d}$ that we want to add together. If the denominators the same, then we can simply add the numerators: \[ \frac{1}{5} + \frac{2}{5} = \frac{3}{5}. \] If the denominators are different however, before we can add the fractions we have rewrite the fractions so that they have a common denominator. An easy way to do this is to cross-multiply: \[ \frac{a}{b} + \frac{c}{d} = \frac{ad}{bd} + \frac{bc}{bd} = \frac{ ad + bc }{bd}. \] The common denominator $bd$ is obtained by multiplying the first fraction by $\frac{d}{d}=1$ and the second fraction by $\frac{c}{c}=1$. Because we multiply both the top and the bottom of the fraction by the same number, this operation does not change the fractions.

More generally, in order to add two fractions we need to find the least common multiple ($\textrm{LCM}$) to use in the common denominator. This is a number obtained by myltiplying the numbers together but removing the common factors: \[ \textrm{LCM}(b,d) = \frac{b \times d}{\textrm{GCD}(b,d) }, \] where $\textrm{GCD}$ is the greatest common divisor: the largest number that divides both $b$ and $d$.

For example if we wanted to add $\frac{1}{6}$ and $\frac{1}{15}$, we could put both fractions on the common denominator $6 \times 15$ (the product of the two denominators), but we could also see that $6=3\times 2$ and $15 =3 \times 5$, which means that $3$ is a common divisor of both $6$ and $15$. The least common multiple is then $\frac{6 \times 15}{3} = 30$, and so we write: \[ \frac{1}{6} + \frac{1}{15} = \frac{5\times 1}{5\times 6} + \frac{1 \times 2}{15 \times 2} = \frac{5}{30} + \frac{2}{30} = \frac{7}{30}. \] Note that all this $\textrm{LCM}$ and $\textrm{GCD}$ business is not required: it is simply the most efficient way of adding the fractions so that you don't get excessively large numbers. If you simply use the product denominator $b\times d$ you will get the same answer after simplification: \[ \frac{1}{6} + \frac{1}{15} = \frac{15\times 1}{15\times 6} + \frac{1 \times 6}{15 \times 6} = \frac{15}{90} + \frac{6}{90} = \frac{21}{90}= \frac{7}{30}. \]

Fraction multiplication involves multiplying together of the numerators and the denominators: \[ \frac{a}{b} \times \frac{c}{d} = \frac{a\times c}{b \times d} = \frac{ac}{bd}. \]

To divide two fractions, we compute the product of the first fraction times the second fraction flipped. To illustrate this, consider the following calculation: \[ \frac{ a/b }{ c/d } = \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a\times d}{b \times c} = \frac{ad}{bc}. \]

The multiplicative inverse of something times that something should give $1$ as the answer. We obtain the multiplicative inverse of a fraction by interchaning the roles of the numerator and the denominators: \[ \left( \frac{c}{d} \right)^{-1} = \frac{d}{c}. \] Thus any fraction times its multiplicative inverse gives $1$: \[ \frac{c}{d} \times \left( \frac{c}{d} \right)^{-1} = \frac{c}{d}\times \frac{d}{c} = \frac{cd}{cd} = 1. \] The “flip and multiply” rule for division comes from the fact that division by a numbers $x$ is the same as multiplication by $\frac{1}{x}$.

To indicate a fraction like $\frac{5}{3}$ which are greater than one we sometimes use the notation $1\frac{2}{3}$ which is read as “one and two thirds”. Similarly $\frac{22}{7}=3\frac{1}{7}$.

There is nothing wrong with writing fractions like $\frac{5}{3}$ and $\frac{22}{7}$ but some teachers say that this way of writing fractions is improper and demand that they be written in the whole-and-fraction way like $1\frac{2}{3}$ and $3\frac{1}{7}$. Whatever. Either way it is no big deal to me.

When written as decimal numbers, certain fractions have infinitely long decimal expansions. We use the “overline” notation to indicate the number(s) which repeat infinitely many times in the expansion: \[ \frac{1}{3} = 0.\bar{3} = 0.333\ldots, \quad \frac{1}{7} = 0.\overline{142857} = 0.14285714285714\ldots. \]

[ The Rappin' Mathematician: Fractions ]
http://www.youtube.com/watch?v=VZQDvb5Yjvw