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Natural numbers and whole numbers both are the part of number system.
- Any mathematical value that is used to measure or count some values is defined as a number.
- A number system is defined as a method of writing numbers.
- A whole number, a natural number, an integer, a rational number, an irrational number, a real number, and a complex number are all examples of numbers.
- Natural numbers are a set of positive integers from 1 to ∞.
- Whole numbers include natural numbers and zero(0) i.e. from 0 to ∞.
- Zero is an undefined identity that provides a null result.
- One of the varieties of Number systems is the decimal system, which is widely used in mathematics.
| Table of Content |
Key Terms: Integer, Rational number, Irrational number, Real number, Complex number, Natural number, Whole number, Number system, Counting numbers
Natural Numbers
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Natural numbers are a subset of the number system that consists of only positive integers from 1 to ∞ and can be counted on our hands.
Natural Numbers: 1, 2, 3, 4,………., ∞
- Negative integers, fractions, and decimals are not considered natural numbers.
- 1 is considered the smallest natural number.
- They are also known as Counting numbers.
Types of Natural Numbers
The following are the different types of natural numbers
- Prime Numbers
- Co-prime numbers
- Even and Odd numbers
- Perfect numbers
- Composite numbers
Examples of Natural NumbersThe following are examples of natural numbers
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Whole Numbers
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Whole numbers are a subset of the number system that consists of only positive integers including 0 i.e. from 0 to ∞.
Whole Numbers: 0, 1, 2, 3,……….., ∞
- Whole numbers are shown on the right side of a number line, similar to natural numbers.
- Zero is the smallest whole number.
- Any infinite positive integer value can be the biggest natural number.
- All natural numbers are whole numbers, but all whole numbers are not natural numbers.
- Fractions, decimal values, complex numbers, and so on are not whole numbers.
Examples of Whole NumbersThe following are examples of whole numbers
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Difference between Natural Numbers and Whole Numbers
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The only number that both Natural and Whole numbers have in common is zero, which is the main distinction between them. The difference between natural numbers and whole numbers are as follows:
| Natural Number | Whole Number |
|---|---|
| The natural number or counting number begins with 1. | All-natural numbers including 0. |
| The smallest natural number is 1 | The smallest whole number is 0 |
| The natural numbers set includes all non-zero integers. | All positive integers are included in the whole numbers set. |
| All-natural numbers are the whole numbers. | All whole numbers are not natural numbers. |
| Denoted by 'N' | Denoted by 'W' |
| Natural numbers are closed under addition and multiplication | Whole numbers are closed under addition and multiplication. |
Natural Numbers and Whole Number Properties
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Natural and whole numbers have the same properties when it comes to addition, subtraction, multiplication, and division. They are as follows:
These properties, however, can be defined using the following operator:
Addition Property
The Addition Property of Natural and Whole Numbers is explained as follows:
When two natural numbers are added, only a natural number is produced.
For example, 43 + 45 = 88.
When two whole numbers are added, only a whole number is produced.
For example, 6 + 0 = 6.
Subtraction Property
When two natural numbers are subtracted, the result isn't always a natural number.
For example, 8 + 5 = 3 is a natural number.
5 – 8 = -3, however, is not a natural number.
It is the same for the whole number. The result of subtracting two whole integers does not have to be a whole number.
Multiplication Property
When a natural number is multiplied by another natural number and a whole number is multiplied by another whole number, the result is a natural number and a whole number, respectively.
Example: 4 x 3 = 12 is a natural number
8 x 5 = 40 is a whole number, with 8 and 0 being whole numbers as well.
Division Property
The division of two natural numbers and whole numbers may or may not produce natural and whole numbers. If the result is in a fraction or decimal, it is not considered a natural and whole number.
Example: 10/2 = 5 is a natural number, as is the full number.
7/2 = 3.5, on the other hand, is neither natural nor a whole number.
Additive Identity
For both natural and whole numbers, the number 0 is known as the additive identity. Thus, adding any natural or whole number to 0 produces the number itself.
Example: 6 + 0 = 6
Here, 6 is a natural and a whole number.
Multiplicative Identity
For both natural and whole numbers, 1 is known as the multiplicative identity. Any natural number or whole number multiplied by 1 produces the same natural number or whole number.
Example: 7 × 1 = 7
0 × 1 = 0
Solved Examples
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Ques. Identify whole numbers and natural numbers:
3.5, 107, 57/90, 9, -53, 0, 60
Ans. The whole numbers are: 107, 9, 0, 60
The natural numbers are: 107, 9, 60
Ques. Which of the following is not a natural number?
- 3
- 100
- 0
- 5
Ans. The correct answer is c. 0
Explanation: Natural numbers are the positive integers from 1 to infinity. Hence zero is not a natural number.
Also check:
| Related Concepts | ||
|---|---|---|
| Types of Function | Types of Numbers | Number Lines |
| Whole Numbers: NCERT Solutions | Class 8 Maths: NCERT Solutions | Real Numbers Formula |
Things To Remember
- Natural numbers are also known as counting numbers.
- Natural numbers are positive integers from 1 to ∞.
- Whole numbers are positive integers from 0 to ∞.
- The smallest natural number is 1.
- The smallest whole number is 0.
- 0 is known as the additive identity from both natural and whole numbers.
- 1 is known as the multiplicative identity from both natural and whole numbers.
- When you multiply two or more natural numbers together, you obtain another natural number.
- When two or more natural numbers are multiplied, a natural number is returned.
Sample Questions
Ques. What is a number system? (2 Marks)
Ans. A number system is defined as a method of writing numbers. It is the mathematical notation for consistently expressing numbers from a specific set using digits or other symbols.
Ques. What are natural numbers? (2 Marks)
Ans. Natural numbers, which include all positive integers from 1 to infinity, are a part of the number system. Natural numbers, often known as counting numbers, do not contain zero or negative numbers.
Ques. What are whole numbers? (2 Marks)
Ans. Whole numbers are those that contain both positive and negative integers, as well as zero. They are shown on the right side of a number line, similar to natural numbers.
Ques. Identify whole numbers and natural numbers from the given list 4.5, 121, 56/90, 4, -79, 0, 30. (2 Marks)
Ans. From the given list,
- The whole numbers are 121, 4, 0, 30
- The natural numbers are 121, 4, 30
Ques. Identify whole numbers from the given numbers 7/9, 0, 7, 69.6, 10, -21, -56, 90. (1 Mark)
Ans. From the given list, the whole numbers are 0, 7, 10, and 90
Ques. Which of the following is not a natural number? (1 Mark)
a. 1
b. 0
c. 7
d. 100
Ans. The correct answer is b. 0
Explanation: Natural numbers: 1, 2, 3, 4,…, 100,…
Whole numbers: 0, 1, 2, 3,…
Thus, 0 is not a natural number.
Ques. Choose the whole numbers from the following. 9, -3, 0, 28, 1/2, -1, 14, 9/10. (1 Mark)
Ans. Whole numbers include all the positive integers and 0. So, the whole numbers from the given list are 9, 0, 28, 14.
Ques. Find the value of digits A and B from the expression: BA × 3 = 57A (3 Marks)
Ans. We have BA × 3 = 57A
so A × 3 = last digit = A
so A = 0 or 5
if A = 0
B × 3 = 57
B = 19
if A = 5
B × 3 + 1 = 57
B = 18.667......decimal
so
A = 0
B = 19
Ques. Find three consecutive whole numbers whose sum is more than 45 but less than 54. (2 Marks)
Ans. Let the numbers be (n-1), n, and (n+1)
Sum of three numbers = 3n
hence 45 < 3n <54 or 15 < n < 18
hence numbers 15, 16, and 17 or 16,17 and 18
Ques. Find three consecutive numbers such that the sum of twice the greatest number and thrice the least number is 584. (2 Marks)
Ans. Let the numbers be (n-1), n, and (n+1).
2(n+1)+3(n-1) = 584
5n -1 = 584 or 5n = 585 or n = 117
hence numbers are 116, 117 and 118
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