Real Number Definition, Properties and Examples

In the notation,  real numbers are just the sum of rational and irrational numbers. All the arithmetic operations may be done on these values, and they can also be expressed on the number line. Imaginary numbers, on the other hand, are unreal numbers that cannot be represented on a number line and are often employed to represent complex numbers. The summation of both rational and irrational numbers is considered the set of real integers. They are symbolized by the letter "R" and can either be positive or negative. This category includes all-natural integers, decimals, and fractional numbers.

Types of Real Numbers

Since childhood, we have worked on multiple mathematical operations on real numbers; let us learn about these numbers in detail along with suitable examples.

Natural Numbers: A set of all counting numbers which begins with 1 are known as natural numbers. It is denoted by the capital letter N and includes N = {1,2,3,4,5,6,7……infinite}

Whole Numbers: A set of all countable numbers which begins with 0 are known as whole numbers. It is designated by the capital letter W and contains W = {0,1, 2…., 21…. 5551, …. infinite}

Integers: Integer is represented by capital letter Z and is the summation of whole numbers and negative numbers ranging from -1 to infinity.

Rational Numbers: Fractional numbers of the form a/b where b is never a null value or zero, are known as rational numbers. They are a direct subset of real numbers and contain positive as well as negative numbers.

Irrational Numbers: All the left-out numbers that cannot be expressed as rational numbers and have decimal non-repeating values are termed as irrational numbers. For example, π, sqrt (2).

Properties of Real Numbers

Commutative property, associative property, distributive property, and identity property are the four major properties related to real numbers. Consider the three real numbers "k, l, and m." Let us use these numbers to derive all the above-mentioned properties.

Commutative Property

For two numbers, k and l, the commutative property for addition is k + l = l + k. The commutative property for the multiplication of real numbers is k × l = l × k.

Associative Property

Now for three numbers, we define the associative property for addition and multiplication. Let the numbers be i, j, and k. The standard form for addition is given as: i + (j + k) = (i + j) + k and the general form for the multiplication is i * (j * k) = (i * j) * k.

Distributive Property

For the same three numbers k, l, and m, the distributive property is defined as: k (l + m) = kl + km and (k + m) l = kl + ml.

Identity Property

The identity property of real numbers is the numbers which when added or multiplied to the given value, results in the formation of the same value. The additive identity of a real number is 0, because x + 0 = x, and the multiplicative identity of a real number is 1, because 1 * x = x.