Number System - Number System

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.

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 arational 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 andq 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 + xOn 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?