Number System in Grade VIII

Hello friends, in today's session we are going to discuss about the Number System. The Number System is defined as a set of numbers arranged together in a such a manner so that we can perform different operations like addition, multiplication etc.

The number system is classified as follows:

1. Natural Numbers
2. Integers
3. Rational Numbers
4. Polynomials
5. Real Numbers
6. Complex numbers.

Let's just take them one by one and understand what are these:

Natural numbers is a set of all positive whole number greater than zero.

The set of natural numbers is denoted by N. There are various laws which the set of natural numbers follow:

Commutative axioms: a + b = b + a; a · b =b · a.

Associative axioms: a + (b + c) = (a + b) +c ; a · (b · c) = (a · b) · c.

Distributive axioms: a · (b +c ) = a · b +a · c; (b · c) n = b n · c n.

Identity axioms: a + 0 = a ; a · 1 = a ; a ⁰ = 1.

Now we come to Integers which are defined as for every natural number “a” there exists a Integer “-a” and the set of these numbers is represented by Z.

To perform arithmetic operations on integers we can look at generalizations given below to make our work simple.

−a + −b = −(a + b)

a + 0 = a

a − b = a + −b

−a · b = a · −b = −(a · b)

−a · −b = a · b

√4 can mean either 2 or −2.

The rational number system is a set of numbers used to represent the fractions. It allows the division to be done by all numbers except zero. It is written in the form a / b. where b cannot be equal to zero.

Polynomials are usually not called as numbers but their properties are very much similar to numbers.

Algebraic numbers is the system which includes all the rational numbers and is included in the set of real numbers.

The real numbers are those numbers which really exists. The set of real numbers is usually represented by R. Real numbers can also be defined as those ones which can be written in the decimal form. All the numbers we have studied till date are called as the real numbers, but now what are imaginary numbers. When the root of a  negative number is taken then the result is known as an imaginary number.

For Example – the number 24 is real because it can be written as 24.0. ½ is also a real number because it can be written as 0.5. even the number 1/3 is also real because it can be written as 0.3333.... whereas complex numbers or imaginary numbers are those numbers which are written in the form a + i b. where i is the imaginary unit i.e. a number whose square is minus one.