Introduction
Let us first start with the meaning of natural numbers
Natural numbers are an important part of the number system, including all the positive integers from 1 to infinity, used for counting purposes. Natural numbers come under real numbers and include the positive integers 1, 2, 3, 4, 5, 6, 7, 8... and so on.
Numbers can be found everywhere around us, used for counting objects, representing or transferring money, calculating temperature, telling time, and so on. "Natural Numbers" refer to the Numbers used to count objects. When counting objects, we might say 5 glasses, 6 books, 1 bottle, and so on.
The number system includes all positive integers from 1 to infinity, which is known as Natural Numbers. Natural Numbers are sometimes known as counting numbers because they do not include zero or negative numbers. They are only positive integers, not zeros, fractions, decimals, or negative Numbers, and they are part of the real Number system.
Natural Numbers
A set of all whole numbers except 0 is referred to as Natural Numbers. These figures play a significant role in our day-to-day activities and communication.
Natural Numbers are those that can be counted and are a portion of real Numbers. The set of Natural Numbers contains only positive integers such as 1, 2, 3, 4, 5, 6, and so on.
Natural Numbers refer to non-negative integers (all positive integers). Examples can be 39, 696, 63, 05110, and so on.
Natural numbers are the positive integers, including numbers from 1 to infinity. Natural numbers are countable numbers and are preferable for calculations. 1 is the smallest natural number and the sum of natural numbers from 1 to 100 is n(n+1)2.
Whole Numbers and Natural Numbers
Natural numbers and whole numbers are different from each other in the matter of including zero. Whole numbers include zero, but all natural numbers are the positive numbers excluding zero.
Every natural number is a whole number, but every whole number is not a natural number.
Set of Natural Numbers
The term "Set" refers to a group of items (Numbers in this context). In mathematics, the Set of Natural Numbers is written as 1,2,3,... The Set of Natural Numbers is symbolised by the symbol N. N = 1,2,3,4,5 and so on. In mathematics, the Set of Natural Numbers is written as 1,2,3,...
N is the natural numbers’ set representation and represents the following:
Statement:
N = Set of numbers starting from 1 and lasting till infinity.
Roster Form:
N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10... and so on}
Set Builder Form:
N = {x: x is a number starting from 1}
Properties of the Natural Number
Natural numbers follow four main properties, which are as follows:
Closure Property
Commutative Property
Associative Property
Distributive Property
Closure Property
A natural number is closed under addition and multiplication. This means that adding or multiplying two natural numbers results in a natural number. However, for subtraction and division, natural numbers do not follow closure property.
Addition
When a and b are two natural numbers, a+b is also a natural number. For example, 2+3=5, 6+7=13, and similarly, all the resultants are natural numbers.
Subtraction
For two natural numbers a and b, a-b might not result in a natural number. E.g. 6-5 = 1 but 5-6=-1.
Multiplication
When a and b are two natural numbers, a*b is also a natural number. Example, 3*5 =15, and similarly all resultants from multiplication are natural numbers.
Division
For the two rational numbers a and b, the division might or might not result in a natural number. E.g. \[\frac{10}{2} =5\] but \[\frac{10}{3} = 3.33.\].
Associative Property
Natural numbers follow associative property for addition and multiplication. For three rational numbers, say, a, b and c, a + (b + c) = (a + b) + c and a * (b * c) = (a * b) * c. Whereas, natural numbers do not follow associative property for multiplication and division.
Addition
For natural numbers a, b and c, addition is associative, i.e. a + (b + c) = (a + b) + c. For example, (15 +3) +1 = 19 = 15 + (3 + 1)
Multiplication
For natural numbers a, b and c, multiplication is associative, which means, a * (b * c) = (a * b) * c. Example: (3 * 1) * 15 = 45 = 3 * (1 * 15).
Subtraction
For three natural numbers a, b, and c, subtraction is not associative, meaning, a – (b – c) is not equal to (a –b) – c. For example: (2 – 15) – 1 = -14 but 2 – (15 – 1) = -12.
Division
For three natural numbers a, b, and c, division is not associative, i.e. \[\frac{a}{(b/c)}\] is not equal to \[\frac{(a/b)}{c}\] . Example: \[\frac{2}{(3/6)} = 4\] but \[\frac{(2/3)}{6} = 0.11\]
Commutative Property
For any two given natural numbers a and b, addition and multiplication are commutative, i.e. a+b = b+a and a*b = b*a. However, division and subtraction are not commutative for the natural number (s), i.e. a-b is not equal to b-a and \[\frac{a}{b}\] is not similar to \[\frac{b}{a}\].
Distributive Property
For the given three natural numbers a, b and c, multiplication is distributive over addition and subtraction. This means that a * (b + c) = ab + ac and a * (b – c) = ab – ac.
Smallest Natural Number
1 is the Smallest Natural Number. We know that the Smallest element in N is 1 and that for each element in N, we may talk about the next element in terms of 1 and N. (which is 1 more than that element). 2 is one greater than one, 3 is one greater than two, and so on.
FAQs on Natural Numbers
1. What are natural numbers and why are they also called 'counting numbers'?
Natural numbers are a fundamental part of the number system, comprising all the positive integers starting from 1 and extending to infinity. They are often called ‘counting numbers’ because they are the numbers we use in our daily lives to count objects, such as 1 book, 5 chairs, or 20 students. The set of natural numbers is represented by the symbol N = {1, 2, 3, 4, ...}.
2. What is the key difference between natural numbers and whole numbers?
The single key difference between natural numbers and whole numbers is the inclusion of the number zero (0).
- Natural Numbers (N): This set includes all positive integers starting from 1. Example: {1, 2, 3, ...}.
- Whole Numbers (W): This set includes all natural numbers plus the number zero. Example: {0, 1, 2, 3, ...}.
Therefore, all natural numbers are whole numbers, but not all whole numbers are natural numbers.
3. Is zero (0) a natural number? Explain why or why not.
No, zero (0) is not a natural number. The definition of natural numbers is based on the concept of counting, which begins with the first object (1). The set of natural numbers starts from 1. Zero is considered a whole number, which represents the absence of quantity and is the starting point for the set of whole numbers (W).
4. What are the four main properties of natural numbers?
Natural numbers follow four main properties, primarily concerning the operations of addition and multiplication:
- Closure Property: Adding or multiplying any two natural numbers always results in another natural number. For example, 2 + 3 = 5 and 2 × 3 = 6.
- Commutative Property: The order in which you add or multiply two natural numbers does not change the result. For example, 5 + 9 = 9 + 5.
- Associative Property: When adding or multiplying three or more natural numbers, the way you group them does not affect the outcome. For example, (2 + 3) + 4 = 2 + (3 + 4).
- Distributive Property: Multiplication of a natural number is distributive over addition. For example, 3 × (4 + 5) = (3 × 4) + (3 × 5).
5. Why don't natural numbers satisfy the closure property for subtraction or division?
Natural numbers are not closed under subtraction and division because these operations can produce results that fall outside the set of natural numbers (which only includes positive integers).
- For Subtraction: If you subtract a larger natural number from a smaller one, the result is a negative integer, not a natural number. For example, 4 - 9 = -5.
- For Division: Dividing two natural numbers can result in a fraction or a decimal, which are not natural numbers. For example, 10 ÷ 3 = 3.33.
6. What is the smallest natural number, and is there a largest one?
The smallest natural number is 1, as it is the first number used for counting. There is no largest natural number. The set of natural numbers is infinite, meaning it goes on forever. For any given natural number, you can always find the next one by adding 1, so there is no end to the sequence.
7. Can a fraction or a decimal be considered a natural number?
No, a fraction or a decimal cannot be a natural number. Natural numbers are exclusively positive whole integers (1, 2, 3, etc.). Fractions like 1/2 or decimals like 2.5 represent parts of a whole, not a complete countable unit, and are therefore part of the set of rational numbers, not natural numbers.
8. Besides counting objects, what is another important use of natural numbers?
Besides their use as cardinal numbers (for counting quantity, like 'ten students'), natural numbers are also essential as ordinal numbers. Ordinal numbers are used to define the position, rank, or order of items in a sequence. For example, when we say someone finished in '1st place', is the '3rd person' in line, or we are on the '5th chapter' of a book, we are using natural numbers to indicate order, not just quantity.
