Types of Real Numbers

In mathematics, numbers are classified into various types to make them easier to recognize and comprehend. The numbers discussed here fall under the category of real numbers, which can be found on the number line. Real numbers are mainly divided into two groups: rational and irrational numbers. Within rational numbers, there are further subdivisions, including natural numbers, whole numbers, and integers.

Natural or Counting Numbers – The set of natural or counting numbers consists of the numbers we use for counting, beginning with \(1\). These numbers do not include fractions or decimals.

$$1,2,3,4,…$$ 

Whole Numbers – The set of whole numbers is simply the set of the natural numbers plus the number zero \(0\) attached to it.

$$0,1,2,3,4,…$$

Integers – The set of integers is the combination of the set of the whole numbers and the negative natural numbers.

$$…,-4,-3,-2,-1,0,1,2,3,4,…$$

Rational Numbers – Rational numbers are those that can be expressed as a ratio of two integers. This ratio can be visualized as a fraction, provided that the fraction’s denominator is not zero.

$${a \over b}$$

where \(a\) and \(b\) are integers but \(b \ne 0\)

Examples: \({2 \over 7}\), \({-1 \over 5}\)

Rational numbers can be expressed in decimal form. A decimal number qualifies as a rational number if its decimal part either terminates or repeats indefinitely.

Decimal numbers that terminate:

\(0.25\)

\(3.17\)

Decimal numbers that repeat indefinitely:

\(0.5555…\)

\(0.131313…\)

Irrational Numbers – Irrational numbers are numbers that cannot be expressed as fractions. When represented in decimal form, they neither terminate nor exhibit a repeating pattern.

Examples: \(\sqrt 2 \), \(\pi \)