Types of Number-Natural Number

Numbers are an indispensable part of our daily life. We need to quantify and count everything around us almost every day. Since we use numbers so often, we must know the types of numbers that exist.

A number could be prime or composite, even or odd, fraction, decimal, etc. In mathematics, numbers are classified into natural numbers, whole numbers, objective, complex, rational, and irrational.

Natural Numbers Definition

Natural numbers can be defined as the most basic numbers used for counting and ranking things.

Example of ranking- He is the 2nd best athlete of his college. We can use whole numbers for counting, but we cannot use them for order as there is no 0th rank. A natural number is named so because its occurrence in nature is most common and obvious.

What are Natural Numbers?

We have covered the definition part; now, let’s see what natural numbers are and which mathematical numbers can be regarded as natural numbers. All the positive integers beginning from 1 and stretching up to infinity constitute Natural numbers.

A Natural number is always greater than 0, and a negative number could never be a natural number. 

In mathematical language, a number that belongs to a set {1,2,3,4…. infinity} can be termed as a natural number.

Representation:

For representing a Natural number or set of it, the letter “N” is used. Small or capital N is extensively used in the formulas of natural numbers. 

Relation with Other Types of Numbers:

The natural number happens to be a subset of other types of numbers.

For example, any natural number is also a whole number. As any whole number is an actual number, by extension, all-natural numbers are real numbers too. All natural numbers belong to a set of rational numbers. So, the natural number set is a subset of:

• Whole number set
• Rational number set
• Real number set

Properties of a Natural Number

A natural number exhibits a few mathematical properties. These properties are generally based on determining the result of basic four arithmetic operations performed on natural numbers.

• Closure Property:

The closure property dictates that when two or more numbers have following operations performed on them, the result is also another natural number.

The operations are:

Addition: Let’s take two natural numbers, 3 and 27; now, their addition equals 30, which is also a natural number. The same stands true for all other natural numbers. 

So, we can sum it up as for all a, b ∈ N (a, b belongs to a set of natural numbers), a + b ∈ N holds.

Multiplication: Example: 12×5=60, 7×6=42, and so on. Hence, we can conclude that a × b ∈ N, for all a, b belongs to N.

However, this doesn’t apply to the remaining two arithmetic operations: Subtraction and Division.

Subtraction: 10-6=4, if we were to reverse the order of subtraction, i.e., 6-10 answers will be -4. Hence subtraction of all natural numbers may not always be a natural number.

Division: 6÷4= 1.5, which isn’t a natural number; hence division of natural numbers isn’t always a natural number.

• Associative Property:

The sum or multiplication result of any three natural numbers would always remain the same no matter how they are grouped together or even if the sequence is changed.

Example: The answer of 1+(2+3) and of (1+2) +3 is 6, which is the same. The same can be verified for multiplication.

• Commutative Property:

It states that when two natural numbers are added or multiplied, the result remains the same despite the sequence of those numbers.

• a + b = b + a
• a × b = b × a

• Distributive Property:

An expression containing three natural numbers of below stated form can be resolved as follows:

• a × (b + c) = ab + ac
• a × (b – c) = ab – ac
• a (b – c) = ab – ca

Formulae

While dealing with natural numbers, we have to perform exhausting and tedious calculations. But with the help of some formulas, we can solve them pretty easily.

So here are some shortcut formulas that would come in handy:

• If we want the sum of first “n” natural numbers, we can use: 

sum= [n(n+1)]/2

• In order to calculate the sum of squares of “n” natural numbers:

        sum of squares = [n(n+1) (2n+1)]/6

• The sum of cubes of first “n” natural numbers will be:

Sum of cubes = [n(n+1)/2]2

We require these formulas while solving questions of AP, i.e., arithmetic progression or even aptitude problems.

Conclusion

Numbers are divided into several types based on some properties, out of which one type is Natural numbers which belong to the set {1,2,3, 4…∞}.

A natural number is always a positive number hence lies on the right side of the number line.

A natural number set is a subset of the whole number set, a real number set as well a rational number set.

When arithmetic operations are performed on natural numbers, certain trends are followed. These are called natural numbers properties, including closure property, associative property, commutative property, and distributive property.