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Number series: Overview, Questions, Preparation

Number System 2021 ( Number System )

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Rajdeep Das

Rajdeep Das

Updated on Aug 31, 2021 16:29 IST

What is Number System?

The number system is divided into two parts:

  1. Imaginary numbers
  2. Real numbers

Real numbers (R) are of two types:

  1. Rational numbers
  2. Irrational numbers

Rational numbers (Q)

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

The 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 with its rationalizing factor, it is known as rationalization.

Table of Contents
  1. Weightage of Number System
  2. Illustrated examples on Number System
  3. FAQs on Number System

Weightage of Number System

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

Illustrated examples on Number System

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...

FAQs on Number System

Q: What makes real numbers? 

A: Real numbers constitute all rational and irrational numbers.

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

A: Yes, the negative of an irrational number is also irrational.

Q: Is every irrational number a surd?

A: Every surd is an irrational number, but every irrational number is not a surd.

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

A: Always irrational.

Q: How important is the chapter?

A: This chapter can help you score as it carries 8 marks. It is also important as the concept is applied in higher standards algebra. 
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