Number System: Notes, Definition, Types, Solved Examples & FAQs

The number system is divided into two parts:

1. Imaginary numbers
2. Real numbers

Types of Real Numbers (R)

There are two types of real numbers. The types of real numbers are mentioned below.

1. Rational numbers
2. Irrational numbers

A number that can be expressed as a/b is known as a rational number, where a and b are both integers and b is not equal to zero. Example, 5/7, -5/7, etc.

Properties of rational numbers

• The sum of rational numbers is always a rational number.
• The difference of rational numbers is always a rational number.
• The product of rational numbers is always a rational number.
• When you divide a rational number by a non-zero rational number, it gives you a rational number. Irrational numbers (Q)

A number that cannot be expressed as a/b is known as an irrational number, where a and b are both integers and b is not zero. For example, 'a' is irrational if its exact square root does not exist.

Decimal representation of rational numbers

(i) When you divide a rational number, and there is no remainder, the quotients of such divisions are called terminating decimals.

(ii) When dividing a rational number, if the division does not end, the quotients of such divisions are called non-terminating.

(iii) When a digit or a set of digits repeats continually in a non-terminating decimal, it is known as a recurring decimal.

Surds

If “y” is a positive rational integer and “a” is a positive integer, such that y1/a is irrational, y1/a is called a surd or a radical.

Rationalization 

When a surd is rationalized by multiplying it by its rationalizing factor, it is known as rationalization.

There are various types of numbers in the number system. Some of the important numbers are as follows: 

• Natural number 

• Whole number 

• Integers

• Rational number 

• Decimal number 

• Real number 

• Prime number

• Odd number 

The Number System is a basic chapter in mathematics. It is taught in Class 9 and carries eight marks.

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

Solution.
(i). Every whole number is a natural number.
(ii) Every integer is a rational number.
(iii). Every rational number is an integer.

(i) False, because zero is a whole number but not a natural number.
(ii) True, because every integer m can be expressed in the form m/1, so it is a
rational number.
(iii) False, because ⅗ is not an integer.

2. Show that 0.3333... = 0.3 can be expressed in the form p/q, where p and
q are integers and q 0. 

Solution.

Let x= 0.3333…
Now, 10x = 10 * (0.33…) = 3.333…
Now, 3.333.. = 3 + x, ( since x = 0.333…)
Thus, 10x = 3 + x
On solving, you get,
X = ⅓ 

3. Find an irrational number between 1/7 and 2/7.

Solution.

We know that 1/7 = 0.142857.

So we know that 2/7 = 0.285714.

A number that is non-terminating non-recurring that lies between these numbers is the required irrational number between 1/7 and 2/7.
There can be many such numbers that lie between these numbers. An example is 0.150150015000150000...

Q: What makes real numbers? 

Q: Is the negative of an irrational number also irrational?

Q: Is every irrational number a surd?

Q: Is the product of a non-zero rational number and an irrational number rational or irrational?

Q: How important is the chapter?