Real Number

Real numbers are the building blocks of mathematics, forming an essential foundation for countless mathematical concepts and applications. From the simplest counting to the most complex scientific calculations, real numbers play a vital role in quantifying, measuring, and understanding the world around us.

This article is about the world of real numbers, exploring their properties, classifications, and significance in mathematics.

Before starting Real Number, there are some basic terminologies which are:

Natural Numbers
The Set of Natural Numbers is given by:
N=\{1,2,3,4, \ldots \}

Whole Numbers
The Set of Whole Numbers is given by:
W=\{ 0,1,2,3,4, \ldots \}

Integers
The Set of Integers is given by:
Z=\{0, \pm 1, \pm 2, \pm 3, \pm 4, \ldots \}
OR
Z =\{ \ldots,-4,-3,-2,-1,0,1,2,3,4, \ldots \}

Decimal Numbers
There are two types of Decimal Numbers.

Terminating Decimal Numbers:
A decimal number that contains a finite number of digits after the decimal point.
Examples:
5.6, 0.5, 19.356

Non-Terminating Decimal Numbers:
A decimal number that has no end after the decimal point.
Examples:
0.35679 \ldots, 23.33333 \ldots , 0.636363 \ldots

Non-Terminating Repeating Decimal Numbers:
In non-terminating decimal Numbers, some digits are repeated in same order after decimal point.
Repeating decimals are called recurring decimals.
Examples:
23.33333 … , 0.636363 \ldots

Non-Terminating Non Repeating Decimal Numbers:
In non-terminating decimal Numbers, the digits are not repeated in same order after decimal point.
Non-repeating decimals are called non-recurring decimals.
Examples:
0.35679 \ldots, 5.34543 \ldots

Rational Number
The word Rational means “Ratio”. A rational number is a number that can be expressed in the form of \frac{p}{q} where p \ and \ q are integers \ \& \ q \neq 0 .
Rational numbers are denoted by Q
Set of Rational Number
The Set of Rational Numbers is given by:
Q= \{ \frac{p}{q} | p,q \in Z,q \neq0 \}

Note:

• Every Natural, Whole and Integer Numbers are rational numbers with a denominator of 1 .
• All terminating decimals are rational numbers.
• Non-terminating recurring (repeating) decimals are rational numbers.

Examples:
\frac{-2}{5}, \frac{6}{11}, \frac{1}{3}, \frac{-44}{11}, \frac{20}{3}
23.33333 \ldots, 0.636363 \ldots,-5, 7

Irrational Numbers
The word Irrational means “Not Ratio”. The irrational number consists of all those numbers which are not rational. Irrational numbers are denoted by Q^/ .
Examples:

\sqrt{2}=1.41421356 \ldots
\pi=\frac{22}{7}=3.1415926 \ldots
3.56884486 \ldots

Definition of Real Number

The set of rational and irrational numbers is called Real Numbers. A real number is denoted by R . Thus QUQ^/=R

Examples:
\frac{-2}{5}, \frac{6}{11}, \frac{1}{3}, \frac{-44}{11}, \frac{20}{3}
23.33333 \ldots, 0.636363 \ldots,-5,7
\sqrt{2}=1.41421356 \ldots
\pi=\frac{22}{7}=3.1415926 \ldots
3.56884486 \ldots

Note:

All the numbers on the number line are real numbers.

Real Number MCQs

Enhance your knowledge of real numbers through our comprehensive Multiple-Choice Questions (MCQs). Test your understanding of rational, irrational, and complex real numbers and explore their fundamental properties through interactive MCQs. Get immediate feedback, analyze your answers, and expand your knowledge with a diverse range of real number MCQs.
Real number MCQs offer an engaging and efficient means of exploring the vast world of real numbers. These MCQs are designed to challenge and engage learners of all levels. So, let the journey of exploration and discovery through real number MCQs begin!