Natural Numbers: Definition, Formulas and Examples

Natural numbers are the positive counting numbers 1, 2, 3, and so on. This topic covers their definition, number-line representation, properties, counting methods, and formulas for sums. It also explains the difference between natural numbers, whole numbers, and integers through simple natural numbers examples and aptitude-based calculations.

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What Are Natural Numbers?

Natural numbers are the positive integers used for counting: 1, 2, 3, 4, 5, and so on. They are represented by N = {1, 2, 3, ...}.

Natural numbers do not include fractions, decimals, negative numbers, or 0 under the standard competitive-examination convention. The nth natural number is n. For example, the 15th natural number is 15, and the number of natural numbers from 8 to 20 is 20 - 8 + 1 = 13.

Natural Numbers Video Lessons

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Types of Numbers Classification

Understand the classification of numbers, including natural, whole, integer, rational, irrational, real and complex numbers, and see how these categories are related in the classification tree.

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Natural Numbers Formula & Tricks

Important Formulas

Sum of the first n natural numbers
1 + 2 + 3 + ... + n = n(n + 1)/2

Use this formula to find the sum of all natural numbers from 1 through n.

Sum of consecutive natural numbers
a + (a + 1) + ... + b = (number of terms)(first term + last term)/2

The number of terms is b - a + 1, so the sum is (b - a + 1)(a + b)/2.

Sum of squares
1² + 2² + ... + n² = n(n + 1)(2n + 1)/6

This gives the sum of the squares of the first n natural numbers.

Sum of cubes
1³ + 2³ + ... + n³ = [n(n + 1)/2]²

The sum of the first n cubes equals the square of the sum of the first n natural numbers.

Quick Tricks

Count inclusive terms correctly

When both endpoints are included, subtract the smaller number from the larger number and add 1.

Example: The numbers from 17 to 29 contain 29 - 17 + 1 = 13 terms.
Use pairing for consecutive sums

For a sequence with equally spaced terms, pair the first and last terms. Each pair has the same sum.

Example: 1 + 2 + ... + 10 = 10(1 + 10)/2 = 55.
Separate even and odd natural numbers

Among the first 2n natural numbers, there are n even and n odd numbers. Among the first 2n + 1 numbers, there are n + 1 odd and n even numbers.

Example: The first 9 natural numbers contain 5 odd numbers and 4 even numbers.

Natural Numbers Concepts

Natural Numbers on the Number Line

Natural numbers are the positive integer points to the right of 0 on the number line.

They are arranged in increasing order as 1, 2, 3, 4, and so on. The difference between two consecutive natural numbers is 1. Every natural number has a successor, obtained by adding 1, but 1 has no predecessor within the set of natural numbers.

Example: The successor of 38 is 39. The predecessor of 38 is 37.

Counting Natural Numbers in an Interval

The number of natural numbers from a to b, including both endpoints, is b - a + 1, where a and b are natural numbers and a ≤ b.

If the interval excludes one endpoint, subtract 1 from the inclusive count. For numbers strictly between a and b, the count is b - a - 1 when b > a.

Example: From 12 to 25 inclusive, there are 25 - 12 + 1 = 14 numbers. Strictly between 12 and 25, there are 25 - 12 - 1 = 12 numbers.

Properties of Natural Numbers

Natural numbers are closed under addition and multiplication, but they are not closed under subtraction and division.

The sum or product of two natural numbers is always a natural number. Subtracting a larger natural number from a smaller one can give a negative integer, and dividing two natural numbers can give a fraction or a non-natural result. Natural numbers are also commutative under addition and multiplication: a + b = b + a and ab = ba.

Example: 5 + 7 = 12 and 5 × 7 = 35 are natural numbers. However, 5 - 7 = -2 and 5 ÷ 2 = 2.5 are not natural numbers.

Natural, Whole and Integer Numbers

Natural numbers are {1, 2, 3, ...}, whole numbers are {0, 1, 2, 3, ...}, and integers are {..., -2, -1, 0, 1, 2, ...}.

The set of natural numbers is contained in the set of whole numbers, and the set of whole numbers is contained in the set of integers. In some mathematical conventions, 0 is included in natural numbers, but competitive aptitude questions generally treat 0 as a whole number, not a natural number.

Example: 7 is a natural number, whole number, and integer. 0 is a whole number and integer but is usually not counted as a natural number in aptitude questions. -4 is an integer only among these three sets.

Natural Numbers Data Table

Use the structured data below while following the explanation and examples.

Comparison of Number Sets

Definitions and examples of natural numbers, whole numbers, and integers.

Number set Definition Examples
Natural numbers Positive counting integers 1, 2, 3, 4, ...
Whole numbers Non-negative integers 0, 1, 2, 3, ...
Integers Positive, negative, and zero integers ..., -2, -1, 0, 1, 2, ...

Note: For quantitative aptitude, natural numbers normally begin with 1.

Practice Natural Numbers Questions

Practise published questions related to this topic.

1The HCF of three numbers is 5. If these numbers are in the ratio of 105 : 165 : 231, find the sum of the three numbers.→ 2The Highest Common Factor (HCF) of two numbers is 48 and their Least Common Multiple (LCM) is 3456. If one of the numbers is 768, find the other number.→ 3√((0.064 × 0.256)/(0.004 × 0.16)) = ?→ 4Six bells commence ringing together and ring at intervals of 4, 8, 12, 16, 20 and 24 seconds respectively. In 18 minutes, how many times do they ring together? (including the first ringing together at commencement)→ 5If x = √484 ÷ √121+√64, then what is the value of x2?→ 6What is the largest five-digit number that is divisible by 9, 15, 16, 21, and 5?→ 7Hari, a printer, is printing a book. He numbers the pages beginning with 1 and uses 3,189 digits in all. Can you tell the total number of pages in the book Hari is printing?→ 8What is the least number that should be added to 443 so that the number obtained is exactly divisible by 12, 5 and 6?→ 9If two positive integers m and n are written as m = x²y and n = x³y³ and x and y are prime numbers, then HCF of m and n is→ 10The product of Highest Common Factor and Least Common Multiple of 2 numbers is 6930. One of the numbers is 154. What is the other number?→

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Quick Revision Notes

Natural Numbers: Quick Revision

Use these rules and formulas for natural numbers questions and number system problems.

  • Natural numbers are 1, 2, 3, ...; 0 is generally excluded in aptitude.
  • The nth natural number is n.
  • The count from a to b, including both endpoints, is b - a + 1.
  • The sum of the first n natural numbers is n(n + 1)/2.
  • The sum of the first n squares is n(n + 1)(2n + 1)/6.
  • The sum of the first n cubes is [n(n + 1)/2]².
  • Natural numbers are closed under addition and multiplication, but not generally under subtraction and division.
  • There are n odd and n even natural numbers among the first 2n natural numbers.

Related Topic Explainers

Number System → Prime Numbers → Divisibility Rules → Divisibility Rule of 2 → Divisibility Rule of 3 → Divisibility Rule of 4 → Divisibility Rule of 6 → Divisibility Rule of 8 →

Natural Numbers FAQs

Is 0 a natural number?

In competitive aptitude, natural numbers usually begin with 1, so 0 is not included. Some mathematical texts include 0 by convention, so use the definition specified in the question.

What is the formula for the sum of the first n natural numbers?

The formula is n(n + 1)/2. For n = 20, the sum is 20 × 21/2 = 210.

How many natural numbers are there from 25 to 60?

Including both endpoints, the count is 60 - 25 + 1 = 36.

What is the sum of the natural numbers from 11 to 20?

There are 20 - 11 + 1 = 10 terms. Their sum is 10(11 + 20)/2 = 155.

Are natural numbers closed under subtraction?

No. For example, 4 - 9 = -5, which is not a natural number. Subtraction remains within natural numbers only when the result is positive.

What is the sum of the first 10 squares of natural numbers?

Use n(n + 1)(2n + 1)/6. For n = 10, the sum is 10 × 11 × 21/6 = 385.

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