Real numbers - CSEET QT

Real Numbers

Real number is any number that can be found in the real world. We find numbers everywhere around us. Natural numbers are used for counting objects, rational numbers are used for representing fractions, irrational numbers are used for calculating the square root of a number, integers for measuring temperature, and so on. These different types of numbers make a collection of real numbers. In this lesson, let us learn all about what are real numbers, the subsets of real numbers along with real numbers examples.

What are Real Numbers?

Any number that we can think of, except complex numbers, is a real number. For example, 3, 0, 1.5, 3/2, √5, and so on are real numbers.

Real Numbers

The list of real numbers is endless because it includes all kinds of numbers like whole, natural, integers, rational, and irrational numbers. Since it includes integers it has negative numbers too. So, there is no specific number from which the list of real numbers starts or ends. It goes to infinity towards both sides of the number line.

Types of Real Numbers

We know that real numbers include rational numbers and irrational numbers. Thus, there does not exist any real number that is neither rational nor irrational. It simply means that if we pick up any number from R, it is either rational or irrational.

Rational Numbers

Any number which can be defined in the form of a fraction p/q is called a rational number. The numerator in the fraction is represented as 'p' and the denominator as 'q', where 'q' is not equal to zero. A rational number can be a natural number, a whole number, a decimal, or an integer. For example, 1/2, -2/3, 0.5, 0.333 are rational numbers.

Irrational Numbers

Irrational numbers are the set of real numbers that cannot be expressed in the form of a fraction p/q where 'p' and 'q' are integers and the denominator 'q' is not equal to zero (q≠0.). For example, π (pi) is an irrational number. π = 3.14159265...In this case, the decimal value never ends at any point. Therefore, numbers like √2, -√7, and so on are irrational numbers.

Symbol of Real Numbers

Real numbers are represented by the symbol R. Here is a list of the symbols of the other types of numbers.

· N - Natural numbers

· W - Whole numbers

· Z - Integers

· Q - Rational numbers

· ¯¯¯¯Q - Irrational numbers

Subsets of Real Numbers

All numbers except complex numbers are real numbers. Therefore, real numbers have the following five subsets:

· Natural numbers: All positive counting numbers make the set of natural numbers, N = {1, 2, 3, ...}

· Whole numbers: The set of natural numbers along with 0 represents the set of whole numbers. W = {0, 1, 2, 3, ..}

· Integers: All positive counting numbers, negative numbers, and zero make up the set of integers. Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

· Rational numbers: Numbers that can be written in the form of a fraction p/q, where 'p' and 'q' are integers and 'q' is not equal to zero are rational numbers. Q = {-3, 0, -6, 5/6, 3.23}

· Irrational numbers: The numbers that are square roots of positive rational numbers, cube roots of rational numbers, etc., such as √2, come under the set of irrational numbers. ( ¯¯¯¯Q) = {√2, -√6}

What are real numbers?

· Every irrational number is a real number.

· Every rational number is a real number.

· All numbers except complex numbers are real numbers.

· All integers are real numbers.