A rational number is a number that can be written as a fraction of two integers.
All integers are also rational numbers, because integer \(x=\frac{x}{1}\), which is a fraction of two integers.
Also: a number is rational if:
because they can be written as a fraction of two integers.
Rational numbers: \(3\), \(\frac{2}{5}\), \(-\frac{3}{8}\), \(5.64\), \(7.123123123123\dotsc\).
Not rational numbers: \(\sqrt{2}\approx 1.41421356\dotsc\), \(\pi \approx 3.14159265\dotsc\), \(e \approx 2.718281828459\dotsc\). Their decimal part does not repeat. They are called irrational numbers.
The “\(\approx\)” sign (above) means “is approx. equal to”, as opposed to the equal sign.
The set of rational numbers contains every number that can be expressed as a fraction of two integers. We label it with \(\mathbb{Q}\) for quotient, another way of saying “fraction”.
The set of integers is a subset of the set of rational numbers, because \(\mathbb{Q}\) contains all integers as well as other numbers.
The set of positive rational numbers can be represented with the symbol \(\mathbb{Q}^+\).
The set of negative rational numbers can be represented with the symbol \(\mathbb{Q}^-\).