Rational numbers

This leads to the set called $\mathbb{Q}$ of rational numbers.

The set $\mathbb{Q}$ consists of all fractions with the numerator an integer and the denonminator a non-zero integer.

(in the fraction $\frac{a}{b}$, $a$ is called the numerator and $b$ is called denominator.)

The rationals are closed under addition, subtraction, multiplication and division.

A set of numbers in which those 4 operations can be performed is called a field, here we often speak of the field of rational numbers.

Two fractions $\frac{a}{b}$ and $\frac{c}{d}$ are equal if and only if $a \cdot d = b \cdot c$

This definition implies that cancelling common factors in the nominator and denominator does not change the fraction.

$\frac{a \cdot c}{b \cdot c} = \frac{a}{b}$, because $(a \cdot c) \cdot b = a \cdot (b \cdot c)\quad b,c \neq 0$

Two fractions are added or substracted by finding their common denominator (you may want to look for the smallest common denominator):

$\frac{a}{b} \pm \frac{c}{d} = \frac{a \cdot d \pm c \cdot b}{b \cdot d}$

Multiplication is easy: $\frac{a}{c} \cdot \frac{b}{d} = \frac{a \cdot b}{c \dot d}$

Division by a (non-zero) fraction is done by multiplying with its reciprocal.

Some questions cannot be answered within the set of rational numbers, for example what is a solution of $x^2 = 2$ ?