CBSE Grade 9 Maths Chapter 1- Number Systems -Part 1(Types of Numbers)

Types of Numbers

What are Natural Numbers ?

Set of counting numbers is called the Natural Numbers. $N = 1, 2, 3, 4, 5, . . .$
$N = 1, 2, 3, 4, 5, . . .$

What are Whole numbers ?

Set of Natural numbers plus Zero is called the Whole Numbers $W = 0, 1, 2, 3, 4, 5, . . . .$
$W = 0, 1, 2, 3, 4, 5, . . . .$

Note:
All natural Number are whole number but all whole numbers are not natural numbers

Examples:

2 is Natural Number
-2 is not a Natural number
0 is a Whole number

What are Integers ?

In Number System, Integers is the set of all the whole number plus the negative of Natural Numbers
$Z = . ., - 7, - 6, - 5, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, 5, 6, . .$

Note

Integers contains all the whole number plus negative of all the natural numbers.
The natural numbers without zero are commonly referred to as positive integers.
The negative of a positive integer is defined as a number that produces 0 when it is added to the corresponding positive integer.
Natural numbers with zero are referred to as non-negative integers.
The natural numbers form a subset of the integers.

What are Rational Numbers ?

In Number system, a number is called rational number if it can be expressed in the form $\frac{p}{q}$ where p and q are integers ( q> 0).
Example : $\frac{1}{2}, \frac{4}{3}, \frac{5}{7}, 1$ etc.

Note

In Number system; every integers, natural and whole number is a rational number as they can be expressed in terms of p/q.
Rational Numbers are denoted by the alphabet 'Q'.
‘Rational’ comes from the word ‘ratio’, and Q comes from the word ‘quotient’.
There are infinite rational number between two rational numbers.
They either have termination decimal expression or repeating non terminating decimal expression.So if a number whose decimal expansion is terminating or non-terminating recurring then it is rational.
The sum, difference and the product of two rational numbers is always a rational number.
The quotient of a division of one rational number by a non-zero rational number is a rational number.
Rational numbers satisfy the closure property under addition, subtraction, multiplication and division.

Question: Are the following statements true or false? Give reasons for your answers.

(i) Every whole number is a natural number.

Solution :False, because zero is a whole number but not a natural number.

(ii) Every integer is a rational number.

Solution:True, because every integer m can be expressed in the form m/1
which is a rational number

(iii) Every rational number is an integer.

Solution : False, because
3/5 is not an integer.

What is meant by Standard Form of a Rational Number ?

If p/q is a rational number,and p and q have no common factors other than 1
(that is, p and q are co-prime) then we say that the rational number p/q is in standard form or in its lowest terms.

What are Irrational Numbers ?

In Number system, a number is called Irrational number if it cannot be expressed in the form p/q, where p and q are integers ( q> 0).
There are infinitely many irrational numbers too.

Example : $\sqrt{3}, \sqrt{2}, \sqrt{5}$ etc

If we do the decimal expansion of an irrational number then it would be non –terminating non-recurring and vice-versa. i. e. the remainder does not become zero and also not repeated.

Example:

π = 3.141592653589793238……

What are Real Numbers?

All rational and all irrational number makes the collection of real numbers. It is denoted by the letter 'R'.
Every real number is represented by a unique point on the number line. Also, every point on the number line represents a unique real number.
The square root of any positive real number exists and that also can be represented on number line.
The sum or difference of a rational number and an irrational number is an irrational number.
The product or division of a rational number with an irrational number is an irrational number.
This process of visualization of representing a decimal expansion on the number line is known as the process of successive magnification

Properties of Real Numbers

Real numbers satisfy the

Commutative Property
Associative Property
Distributive laws.

These can be stated as

Commutative Law of Addition

$a + b = b + a$

Commutative Law of Multiplication

$a \times b = b \times a$

Associative Law of Addition

$a + (b + c) = (a + b) + c$

Associative Law of Multiplication

$a \times (b \times c) = (a \times b) \times c$

Distributive Law

$a \times (b + c) = (a \times b) + (a \times c)$
or
$(a + b) \times c = (a \times c) + (b \times c)$

In Number System, if ' r ' is a rational number and ' s ' is an irrational number, then r+s and r-s are irrationals.

Further, if r is a non-zero rational, then r*s and r/s are irrationals

Extra Information

Special properties around number 0 and 1

Addition Property of Zero:Adding zero to a number does not change it. For all real number $x + 0 = x$
Multiplication Property of Zero:Multiplying a number by zero always gives zero. For all real number $x .0 = 0. x = 0$
Powers of Zero:The number zero, raised to any allowable power, equals zero. For n any positive number $0^{n} = 0$ In particular, zero to the zero power ( 0n) is undefined
Zero as a numerator: Zero, divided by any nonzero number, is zero. For all real number except 0 , $\frac{0}{x} = 0$ and for 0, $\frac{0}{0}$ is undefined quantity
Division by zero is not allowed: Any division problem with zero as the denominator is not defined. For example, $\frac{1}{0} . \frac{2}{0}$
Multiplication Property of One:Addition Property of Zero:Multiplication Property of Zero:Powers of Zero:Zero as a numerator:Division by zero is not allowed:Multiplication Property of OneMultiplying a number by one does not change it.For all real number $x .1 = 1. x = x$
Powers of One: The number one, raised to any power, equals one.For all real numbers $1^{n} = 1$ This is true even if the n is a fraction
NegativeNames for the number one:Any nonzero number divided by itself equals one.For all real number except 0 , $\frac{x}{x} = 1$