Rational Numbers

Understand the Concept of Whole Numbers through the Concept of Natural Numbers

You have studied fractional numbers in your earlier classes. Some examples of fractional numbers are.

These numbers are also known as rational numbers.

What comes first to your mind when you hear the word rational?

Yes,you are right. It is something related to the ratios.

The ratio 4:5 can be written as, which is a rational number. In ratios, the numerator and denominator both are positive numbers while in rational numbers, they can be negative also.

Thus, rational numbers can be defined as follows.

“Any number which can be expressed in the form , where p and q are integers and, is called a rational number.”

For example, is a rational number in which the numerator is 15 and the denominator is 19.

Now, is −34 a rational number?

Yes, it is a rational number. −34 can be written as. It is in the form of and q ≠ 0.

Thus, we can say that every integer is a rational number.

Now, consider the following decimal numbers.

1.6, 3.49, and 2.5

These decimal numbers are also rational numbers as these can be written as

If in a rational number, either the numerator or the denominator is a negative integer, then the rational number is negative.

For example, are negative rational numbers.

If the numerator and the denominator both are either positive integers or negative integers, then the rational number is positive.

For example, are positive rational numbers.

 

Conventions used for writing a rational number:   

We know that in a rational number, the numerator and denominator both can be positive or negative.

Conventionally, rational numbers are written with positive denominators.

For example, –9 can be represented in the form of a rational number as , but generally we do not write the denominator negative and thus, is eliminated. So, according to the convention, –9 can be represented in the form of a rational number as .

 

Equality relation for rational numbers:

For any four non-zero integers p, q, r and s, we have

Order relation for rational numbers:  

If are two rational numbers such that q > 0 and s > 0 then it can be said that if ps > qr.

Absolute Value of a Rational Number: The absolute value of a rational number is its numerical value regardless of its sign. The absolute value of a rational number pq is denoted as pq. Therefore, -32=32, 12-7=127 etc. Note: The absolute value of any rational number is always non-negative.

Now, let us go through the given example.

Example:

Write each of the following rational numbers according to the convention.

i)

ii)

 

Solution:

According to the convention used in rational numbers, the denominator must be a positive number.

Let us now write the given numbers according to the convention.

 

i)

In the number , denominator is negative. 

We have,

 

According to convention, the given number should be written as .

 

ii) 

In the number , denominator is negative. 

We have,

 

According to convention, the given number should be written as . Example: Find the absolute value of the following: (i) -12171 (ii) 1219 Solution: (i) Absolute value = -12171=12171 (ii) Absolute value = 1219=1219  

Can you say what counting numbers are?

The numbers that we use for counting are called counting numbers. They start with 1. They are 1, 2, 3, 4 …. . These counting numbers are also called Natural Numbers. Therefore, we can define the natural numbers as follows:

“The collection of all counting numbers is known as Natural Numbers.”

What will we obtain, if we subtract 1 from 1?

If we subtract 1 from 1, then we obtain

1 − 1 = 0

The number 0 (zero) with all the natural numbers form a system of numbers, which is called Whole Numbers. This means whole numbers are a set of numbers starting from 0 i.e., 0, 1, 2 … and this can be defined as follows:

“If zero is added to the collection of natural numbers, then we obtain the collection of whole numbers, or in other words, we can say that all natural numbers along with zero are called whole numbers.”

Remember: All natural numbers are whole numbers, but all whole numbers are not natural numbers.

Think of any big number, say 20958340. We can write this number using symbols 0, 2, 3, 4, 5, 8 and 9.

Similarly, we can write a natural number using 10 symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Each of such symbols is called a digit or a figure. 

On observing the natural and whole numbers it is found that:

The value of numbers increase as we move from left to right. On moving further to the right, we keep on finding more numbers. Thus, these numbers are endless and it is not possible to tell the highest natural or whole number. 1 is the smallest natural number. 0 is the smallest whole number.  

The number 0 follow certain rules:

a + 0 = 0 + a, for all natural numbers a a. 0 = 0. a, for all natural numbers a 0 + 0 = 0 0.0 = 0

Well ordering property of natural numbers:

Well ordering property of natural numbers states that every non-empty subset of natural numbers of N (or W) has the smallest element. 

For example, let us consider the set of all even natural numbers i.e., {2, 4, 6, ...}. This set is the subset of natural number. 2 is the smallest element of the set of even natural number.

Let us now try and solve the following puzzle to check whether we have understood this concept.

Natural numbers

The counting numbers 1, 2, 3, ... are called natural numbers. 

The set of natural number is denoted by the letter N.

∴ N = {1, 2, 3, ...}

1 is the smallest natural number. The set of natural numbers, N is an infinite set.

Whole numbers

The numbers 0, 1, 2, 3, ... are called whole numbers.

The set of whole numbers is denoted by the letter W.

∴ W = {0, 1, 2, 3, ...}

0 is the smallest whole number. The set of whole numbers, W is an infinite set.

Integers

We had observed that adding any two whole numbers always gives a whole number. We can examine whether this case is true for the operation ‘subtraction’. Let us consider the following examples:

13 − 12 = 1

13 − 13 = 0

12 − 13 …