Difference Between Natural Numbers and Whole Number – Explained along with its Meaning
There are various types of numbers, such as Whole Numbers, Natural numbers, integers, rational numbers, irrational numbers, real numbers, and complex numbers. And more often than not students may find it confusing, that is to say, the student may confuse one with the other. Especially in the natural numbers and the whole numbers, because both of them kind of seem similar. And therefore, it is important for the students to have a clear understanding of the Whole number and the Natural numbers.
Understanding Whole Numbers
Before diving deep into the difference between the Whole numbers and Natural numbers, it is important for the students to first understand, both the terms, Whole Numbers and Natural numbers separately. Because the main difference between the two lies in their meaning. Therefore, let us start with the Whole Numbers.
You may be familiar with the term integers, if not, here is a quick review. Integers are those numbers that can be expressed without the use of fractions, in short numbers which are not a fraction are integers. For example numbers 1, 25, 124, 5624, -54 are all integers because they do not require fractions. But the numbers ¾, ½ are not integers, because they are expressed in a fraction.
Now, coming to the whole numbers, all the positive integers along with the 0 are called whole numbers. It means decimals such as 5.2, 3.2; Negatives, such as -5, -6; and the fractions are not the whole numbers.
If you wish to learn more about the whole numbers you may like to follow this link.
Understanding Natural Numbers
Now, coming to the natural numbers, it is just as simple as Whole numbers. Because all you have to do is to remove the 0 from the whole numbers and you have got natural numbers. Yes, all the positive integers of the number system are called natural numbers. It means integers such as 1, 5, 8, 541, etc are all natural numbers. But just like the whole numbers here again the decimals, negatives, and fractions are not included. You can vaguely say all the numbers that we commonly use for counting are Natural Numbers
Difference Between Natural Numbers and Whole Numbers
Life without numbers is unimaginable. There are numerous properties of numbers that led to their categorization. The most primary thing to note is that all natural numbers are whole numbers. However, there are other categories of numbers such as
Whole Numbers
Natural Numbers
Integers
Rational Numbers
Irrational Numbers
Real Numbers
Complex Numbers
These categories differ from each other by their properties. Let’s explore the differences between Natural and Whole numbers.
0 is the smallest whole number. In mathematics, the most basic set is that of whole numbers. These whole numbers are an integral part of the real number set which comprises various other number sets like integers, rational numbers among others.
Except 0, every whole number has exactly one immediate predecessor that is the number that comes before a whole number. Every whole number has exactly one immediate successor that is the number that comes after a whole number.
The basic difference between natural numbers and whole numbers is that the whole numbers set include 0 Instead natural numbers set doesn't include 0. Let us talk about other differences as shown below.
Difference Between Natural and whole Numbers
Whole Number
All Whole numbers are represented by 'W'
In whole numbers counting starts from '0' ZERO
When 0 is added to a number the answer is just the number you start with: 24+0=24.
For this reason, 0 is called the identity element for addition. The identity element is absent from the natural numbers for additional property.
All whole numbers are also integers. For each whole number, there is a negative number that corresponds with it. For instance, -5 corresponds to the whole number 5, and -120 corresponds to the whole number 120.
Within the set of integers, the sum of two numbers can be 0.
For eg. 20+(−20)=0 and 135+(−135)=0.
20 and -20 will be termed as the additive inverses.
Natural Number
All Natural numbers are represented by 'N'
In natural numbers counting starts from '1'
When you add two or more natural numbers, you get a natural number again.
When you multiply two or more natural numbers, you get a natural number again
The maths way to say it is that “The system of natural numbers is closed under addition and multiplication”. It means that when subtraction or division is performed on natural numbers the result may not always be a natural number.
Examples of Whole Numbers and Natural Numbers
Example 1. Find whole numbers from given numbers. 14,0,8,48,-6,-9,2
Sol. Whole numbers: 14,0,8,48,2
Example 2. Find natural numbers from given numbers.
24,(0.6),6,40,-60,0,-2
Sol. natural numbers: 24,6,40
Explanation of Addition Property of Natural and Whole Numbers
When two natural numbers are added, it results in a natural number only.
Eg: 34+45 = 79
Adding two whole numbers will give you a whole number.
Eg: 6+0= 6
Explanation of Subtraction Property of Natural and Whole Numbers
Subtraction of two natural numbers doesn’t necessarily result in a natural number
Eg: 8 – 5 = 3 is a natural number
But 5 – 8 = -3 is not a natural number
Similar is the condition for whole numbers. Subtracting two whole numbers need not result in a whole number.
Explanation of Multiplication Property of Natural and Whole Numbers
Multiplication of a natural number with a natural number and a whole number with another whole number results in a natural number and whole number respectively.
Eg: 4 x 3 = 12 is a natural number
8 x 5 = 40 is a whole number, where 8 and 0 are also whole numbers.
Explanation of Division Property for Natural and Whole Numbers
Division property also does not hold for the natural numbers and whole numbers for instance.
Eg: 10/2 = 5 is natural as well as the whole number
But 7/2 = 3.5 is neither natural nor the whole number.
Similarly, the difference between natural numbers and whole numbers can be understood by representing them on a number line.
Whole numbers are located on the right side of the number line including zero.
Natural numbers are located on the right side excluding zero.
FAQs on Difference Between Natural and Whole Numbers for JEE Main 2024
1. What is the fundamental difference between the set of Natural Numbers (N) and Whole Numbers (W) for JEE Main 2026?
The fundamental difference lies in a single element: the number zero. The set of Natural Numbers (N) consists of all positive counting numbers, starting from 1 (N = {1, 2, 3, ...}). The set of Whole Numbers (W) includes all natural numbers plus zero (W = {0, 1, 2, 3, ...}). Therefore, every natural number is a whole number, but 0 is a whole number that is not a natural number.
2. Are all natural numbers also whole numbers? And is the reverse true?
Yes, all natural numbers are whole numbers because the set W = {0, 1, 2, 3, ...} contains the entire set N = {1, 2, 3, ...}. However, the reverse statement is false. Not all whole numbers are natural numbers, specifically because the whole number 0 is not included in the set of natural numbers.
3. How do Natural and Whole Number sets behave under the four basic arithmetic operations (addition, subtraction, multiplication, division)?
Understanding the closure properties of these sets is a key concept for JEE. A set is 'closed' under an operation if performing it on any two members of the set results in another member of the same set.
- Addition & Multiplication: Both Natural and Whole number sets are closed. Adding or multiplying any two numbers from either set will always result in a number within that same set.
- Subtraction & Division: Both sets are not closed. For example, 3 - 5 = -2, which is an integer but not a whole or natural number. Similarly, 5 / 2 = 2.5, which is a rational number, not a whole or natural number.
4. What are the identity elements for addition and multiplication within the set of whole numbers?
For the set of whole numbers (W), the identity elements are:
- The additive identity is 0, as adding 0 to any whole number does not change its value (a + 0 = a).
- The multiplicative identity is 1, as multiplying any whole number by 1 does not change its value (a × 1 = a).
5. Why is zero's inclusion in whole numbers a critical concept for JEE Main topics like functions and set theory?
Zero's inclusion makes the set of whole numbers the smallest set (among N, W, Z) that has an additive identity. This property is foundational in algebra. For JEE Main problems involving the domain of a function, specifying the domain as 'whole numbers' versus 'natural numbers' can change the outcome. For instance, a function f(x) defined for x ∈ W includes f(0), while a function defined for x ∈ N would exclude it, potentially affecting the function's range or continuity at the origin.
6. Why are decimals and fractions, like 5.5 or 3/4, excluded from both natural and whole numbers?
Natural and whole numbers are, by definition, types of integers. Natural numbers are positive integers {1, 2, 3, ...}, and whole numbers are non-negative integers {0, 1, 2, 3, ...}. Numbers with decimal parts or fractional forms that cannot be simplified to an integer, like 5.5 or 3/4, belong to the set of Rational Numbers (Q). This distinction is crucial for solving problems in JEE that specify integer solutions, such as in permutations, combinations, or number theory.
7. In the context of the JEE syllabus, how do natural and whole numbers fit into the broader number system?
The number systems form a nested hierarchy that is essential for the JEE syllabus. Natural numbers (N) are a proper subset of Whole numbers (W). Whole numbers are a proper subset of Integers (Z), which also include negative numbers. Integers are a proper subset of Rational numbers (Q), which include all fractions. Finally, Rational and Irrational numbers together form the set of Real Numbers (R). This hierarchy can be expressed as: N ⊂ W ⊂ Z ⊂ Q ⊂ R.
8. How would you determine the number of whole numbers between two integers, for example, 25 and 68?
To find the number of whole numbers strictly between two integers 'a' and 'b' (where b > a), you use the formula (b - a) - 1. For the integers 25 and 68, the number of whole numbers between them is (68 - 25) - 1 = 43 - 1 = 42. Be careful with the phrasing in JEE questions; if it asks for numbers from 'a' to 'b' inclusive, the formula becomes (b - a) + 1.
