Rational numbers

To get a set of numbers, which also includes every result of a division, we need to expand the integers. The rational numbers are used for this. These are formed by fractions of integers. The integers are also included in the rational numbers.

This allows the execution of all four basic operations. The following examples show operations with ratinal numbers.

When adding, the denominators of the two summands must be put on a common main denominator

A subtraction can be performed via the addition.

In multiplication, the numerator is multiplied by the numerator and the denominator by the denominator.

Division by a fraction is equal to multiplying by its reciprocal.

Rational numbers can be written as a decimal fraction. Decimal fractions are comma numbers. Before the comma is the whole part of the number, after the comma the decimal places. Rational numbers in decimal notation have the property that they either have limited decimal places or there is a period in the decimal places in which digit sequences repeat themselves.

Example of rational numbers in decimal notation

\(\displaystyle \frac{1}{2}=0.5 \)

\(\displaystyle \frac{1}{4}=0.25 \)

\(\displaystyle \frac{3}{8}=0.375 \)

\(\displaystyle \frac{888}{100}=8.88 \)

\(\displaystyle \frac{5}{12}=0.41\overline{6} \)

\(\displaystyle \frac{7}{1111}=0.0063\overline{0063} \)

Irrational and real numbers

Numbers with an infinite number of decimal places without period are irrational numbers. Known irrational numbers are the number \(Pi = 3.14159265358 ...\) and the Euler number \(e\).

The rational numbers along with the irrational ones are the real numbers.