Number System

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

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.

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.

Do you know how many students are there in your class?

The number of students in your class would be a two-digit number or, at the maximum, a three-digit number. These are smaller numbers.

But if you are asked the number of students in your school, then it would be a bigger number.

The total number of students in a city would be a large number, i.e. at least a five-digit number.

And, if we count the total number of students in the whole country, then we would have to use very large numbers (like eight or nine-digit numbers). Therefore, here, we will learn about large numbers. 

To understand a number, there are two most important things to know, face value and place value of each of its digits. 

In a number, face value of each digit is the actual value of that digit and it never changes whether the number is written according to any numeral system.

Consider the number 3,69,821. In this number, face values of different digits are as follows:

Face value of 3 = 3

Face value of 6 = 6

Face value of 9 = 9

Face value of 8 = 8

Face value of 2 = 2

Face value of 1 = 1

Now, to understand the place value of digits of a large number and expansion of the number, let us go through the following video.

Thus, we can read the number 7,86,790 easily as "Seven lakh eighty six thousand seven hundred and ninety".

Similarly, we can write the numeral value of any given number.

Do you know we also have a relation between different place values?

Let us go through the following video, to understand the relation.

Thus, we must remember the following conversions which will be helpful in reading and writing numbers.

1 hundred = 10 tens

1 thousand = 10 hundreds = 100 tens

1 lakh = 100 thousands = 1000 hundreds

1 crore = 100 lakhs = 10,000 thousands

Let us now have a look at some examples to understand the concept better.

Example 1:

Write 6508927 in expanded form and write the place values and face values of underlined digits.

Solution:

The number 6508927 can be written in expanded form as follows:

6508927 = 6 × 1000000 + 5 × 100000 + 0 × 10000 + 8 × 1000 + 9 × 100 + 2 × 10 + 7 × 1

Place value of 7 = one

Face value of 7 = 7

Place value of 8 = thousand

Face value of 8 = 8

Place value of 5 = lakh

Face value of 5 = 5

Example 2:

Write the following numbers in words and answer the questions given below.

432079, 5601729 and 1794805

(1) Which is the smallest number?

(2) Which is the largest number?

(3) Arrange these numbers in ascending and descending order.

Solution:

4,32,079 = Four lakh thirty two thousand and seventy nine

56,0…