Question

What are real numbers?

Answer 1

Hint: Term real means existing in reality. Try to relate the definition with the number line concept and real numbers are not imaginary i.e. we can think about them. So, use these concepts to give definition of real numbers.

Complete Step-by-Step solution:
Any number which can be represented on the number line is called a real number. It consists of rational and irrational both numbers. In other words, the number you can think of is the real number. It includes whole numbers, natural, all integers, fraction everything.
Example: $0,1,2,\dfrac{5}{4},\sqrt{3},\pi ,0.12345.......,6\overline{.7},5.\overline{4}$ etc.
So, we get that real numbers are simply the combination of rational and irrational numbers in the number system. All the arithmetic operations can only be performed on these numbers. At the same time the imaginary numbers are the real numbers, which cannot be expressed on the number line. Real numbers are denoted by the symbol “R”. Hence, all natural numbers, decimal, fractions come under this category. Definition of rational and irrational numbers are given as:
Rational Numbers: Numbers that can be expressed as the fraction of $\dfrac{p}{q}$ , where p and q are integers and value of q will never be 0. It can be a terminating or recurring (non terminating) type. Terminating means the number will end after some digits by the decimal and recurring means some of the digits of it are repeating and it will not end.
Example: \[0.13,\text{ }1.556,\text{ }1.5555555\ldots \ldots ,1.\overline{5}\] etc.
Irrational Number: These are the numbers which cannot be written in the form of $\dfrac{p}{q}$ because we cannot convert them in fraction form. So, these numbers would have non-terminating and non- recurring both types.
Example: $1.73254567..........,\sqrt{3},\sqrt{7},\pi $ etc.

Note: Complex numbers are the top most category of the numbers. It can be divided into two parts: real numbers and imaginary numbers. Hence, real numbers are divided into two more categories rational and irrational and further natural, whole integers etc. are under rational number categories.
Imaginary number do not exist in reality and cannot be used in real life calculations; they are expressed in terms of iota whose value is $\sqrt{-1}$ and represented by ‘I’ i.e. $\sqrt{-1}$ . Any complex number is represented as a + ib where a and b are real numbers. If b of the complex number is 0, then the complex number is known as real number, otherwise it will be termed as imaginary number.