Number System Important Formulae – Complete Guide with Definitions, Rules, Properties & Examples

The Number System is one of the most fundamental topics in mathematics. Every calculation in arithmetic, algebra, geometry, statistics, and advanced mathematics begins with numbers.

Understanding number system formulae helps students solve problems faster and build a strong mathematical foundation for school exams and competitive exams.

This complete guide covers all important number system formulas, concepts, rules, examples, shortcuts, and frequently asked questions.

What is Number System?

A Number System is a way of representing numbers and performing mathematical operations on them.

Numbers are classified into different categories based on their properties.

Examples:

1,\ 2,\ 3,\ -5,\ \frac34,\ \sqrt2,\ 0.25

All belong to different parts of the number system.

Classification of Numbers

1. Natural Numbers (N)

Natural numbers are counting numbers.

Set

N={1,2,3,4,5,\dots}

Properties

Smallest natural number = 1
No largest natural number
Infinite numbers

Examples

10,\ 25,\ 100

Properties of Natural Numbers

2. Whole Numbers (W)

Whole numbers include zero.

Set

W={0,1,2,3,4,\dots}

Formula

W=N+0

Example:

0,\ 5,\ 100

3. Integers (Z)

Integers include positive, negative numbers and zero.

Set

Z={\dots,-2,-1,0,1,2,\dots}

Types

Positive Integers: 1,2,3

Negative Integers: -1,-2,-3

4. Rational Numbers (Q)

Numbers written in fraction form.

Expression

\frac{p}{q}

Where:

q\neq0

Examples:

\frac34,\ \frac78,\ 0.25

Conversion Formula

Decimal=\frac{Numerator}{Denominator}

Example:

\frac34=0.75

5. Irrational Numbers

Numbers that cannot be expressed as:

\frac{p}{q}

Examples:

\sqrt2,\ \sqrt3,\ \pi

Properties:

Non-terminating
Non-repeating

6. Real Numbers (R)

Combination of rational and irrational numbers.

Formula

R=Q+I

Examples:

2,\ -7,\ \sqrt5

7. Imaginary Numbers

\begin{aligned} Numbers \ involving:\sqrt{-1} \\ \\ Formula: i=\sqrt{-1} \\ \\ Example: 3+2i \end{aligned}

Even and Odd Number Formulae

Even Number : Numbers divisible by 2 or we can say multiples of 2.

Divisible by 2.
Multiples of 2
Numbers having 2,4,6,8 or 0 on its unit place.

Formula

Even Numbers are generally represented by 2n, where n is any natural number.

Examples:

2,\ 4,\ 6,\ 100

Odd Number : Numbers not divisible by 2.

Formula

Odd Numbers : 2n+1

Examples:

1,\ 3,\ 5

Prime Number Formulae

Prime numbers have exactly two factors.

Examples: 2,3,5,7,11

Number of Prime Factors

N=p^a\times q^b

Total factors:

(a+1)(b+1)

Example:

12=2^2\times3^1

Factors:

(2+1)(1+1)=6

Composite Number Formula

Composite numbers have more than two factors.

Examples:

4,\ 8,\ 12

Divisibility Rules Formulae

1. Divisibility Rule of 2

Statement:
A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.

Example:
248 → Last digit = 8
∴ 248 ÷ 2 = 124
So, 248 is divisible by 2

2. Divisibility Rule of 3

Statement:
A number is divisible by 3 if the sum of its digits is divisible by 3.

Example:
357 → 3+5+7=15
15 is divisible by 3
∴ 357 is divisible by 3

3. Divisibility Rule of 4

Statement:
A number is divisible by 4 if its last two digits form a number divisible by 4.

Example:
916 → Last two digits = 16
16 ÷ 4 = 4
∴ 916 is divisible by 4

4. Divisibility Rule of 5

Statement:
A number is divisible by 5 if the last digit is 0 or 5.

Example:
725 → Last digit = 5
∴ 725 is divisible by 5

5. Divisibility Rule of 6

Statement:
A number is divisible by 6 if it is divisible by both 2 and 3.

Example:
432
Last digit = 2 → divisible by 2
4+3+2=9 → divisible by 3
∴ 432 is divisible by 6

6. Divisibility Rule of 7

Statement:
Double the last digit and subtract it from the remaining number.
If the result is divisible by 7, the original number is divisible by 7.

Example:
203
20−(3×2)=14
14 is divisible by 7
∴ 203 is divisible by 7

7. Divisibility Rule of 8

Statement:
A number is divisible by 8 if its last three digits form a number divisible by 8.

Example:
7128 → Last three digits = 128
128 ÷ 8 = 16
∴ 7128 is divisible by 8

8. Divisibility Rule of 9

Statement:
A number is divisible by 9 if the sum of its digits is divisible by 9.

Example:
729
7+2+9=18
18 ÷ 9 = 2
∴ 729 is divisible by 9

9. Divisibility Rule of 10

Statement:
A number is divisible by 10 if its last digit is 0.

Example:
540
Last digit = 0
∴ 540 is divisible by 10

10. Divisibility Rule of 11

Statement:
Find the difference between the sum of alternate digits.
If the difference is 0 or divisible by 11, the number is divisible by 11.

Example:
121
(1+1)−2=0
∴ 121 is divisible by 11

11. Divisibility Rule of 12

Statement:
A number is divisible by 12 if it is divisible by both 3 and 4.

Example:
384
3+8+4=15 → divisible by 3
Last two digits = 84 → divisible by 4
∴ 384 is divisible by 12

12. Divisibility Rule of 13

Statement:
Multiply the last digit by 4 and add it to the remaining number. Repeat if needed.
If the result is divisible by 13, the original number is divisible by 13.

Example:
351
35+(1×4)=39
39 ÷ 13 = 3
∴ 351 is divisible by 13

13. Divisibility Rule of 15

Statement:
A number is divisible by 15 if it is divisible by both 3 and 5.

Example:
345
3+4+5=12 → divisible by 3
Last digit = 5 → divisible by 5
∴ 345 is divisible by 15

14. Divisibility Rule of 25

Statement:
A number is divisible by 25 if the last two digits are
00, 25, 50, or 75

Example:
1475 → Last two digits = 75
∴ 1475 is divisible by 25

15. Divisibility Rule of 100

Statement:
A number is divisible by 100 if the last two digits are 00.

Example:
2500
∴ 2500 is divisible by 100

16. Divisibility Rule of 1000

Statement:
A number is divisible by 1000 if the last three digits are 000.

Example:
56000
∴ 56000 is divisible by 1000

Divisibility Rules : Quick Summary Table

Number	Divisibility Rule
2	Last digit even
3	Sum of digits divisible by 3
4	Last 2 digits divisible by 4
5	Last digit 0 or 5
6	Divisible by 2 and 3
7	Double last digit and subtract
8	Last 3 digits divisible by 8
9	Sum of digits divisible by 9
10	Last digit 0
11	Difference of alternate sums
12	Divisible by 3 and 4
13	Multiply last digit by 4 and add
15	Divisible by 3 and 5
25	Ends with 00, 25, 50, 75
100	Ends with 00
1000	Ends with 000

Factor Formulae

Number of Factors

If:

N=p^a\times q^b\times r^c

Then: Total Number Factors:

(a+1)(b+1)(c+1)

Example:

72=2^3\times3^2

Total Number Factors:

(3+1)(2+1)=12

Sum of Factors Formula

(1+p+p^2+\dots)

General formula:

\frac{p^{a+1}-1}{p-1}

Multiply all terms.

Example:

12=2^2\times3

Sum:

(1+2+4)(1+3)

7\times4=28

Greatest Integer Function

Formula:

[x]

Meaning: Largest integer ≤ x

Examples:

[4.9]=4

[-2.3]=-3

Fractional Part Formula

{x}=x-[x]

Example: {5.8}=0.8

Place Value Formula

Number: 5738

Place\ Value=Digit\times Position

Example:

7\times100=700

Face Value Formula

Face value = Actual digit

Example:

Digit: 7

Face value: 7

Number of Digits Formula

For: N

Digits:

\lfloor\log_{10}N\rfloor+1

Example: 5000

digits = 3+1=4

Sum of Important Series :

\begin{aligned} \text{Sum of First }n\text{ Natural Numbers:}\qquad 1+2+3+\cdots+n &=\frac{n(n+1)}{2} \\[10pt] \text{Example:}\qquad 1+2+3+4+5 &=\frac{5(5+1)}{2} =\frac{30}{2} =15 \\[15pt] \text{Sum of Squares:}\qquad 1^2+2^2+3^2+\cdots+n^2 &=\frac{n(n+1)(2n+1)}{6} \\[10pt] \text{Example:}\qquad 1^2+2^2+3^2 &=1+4+9 =14 \\[15pt] \text{Sum of Cubes:}\qquad 1^3+2^3+3^3+\cdots+n^3 &=\left(\frac{n(n+1)}{2}\right)^2 \\[10pt] \text{Example:}\qquad 1^3+2^3+3^3 &=1+8+27 =36 \end{aligned} “`

Binary Number Formulae

Decimal to Binary

Repeated division by 2.

Example: 13

Binary: 1101

Binary to Decimal

Formula:

\sum digit\times2^n

Example: 101

1(2^2)+0(2^1)+1(2^0)

Important Number System Formula Summary

Topic	Formula
Even Number	(2n)
Odd Number	(2n+1)
Rational	(p/q)
Digits	(\log_{10}N+1)
Natural Sum	(n(n+1)/2)
Square Sum	(n(n+1)(2n+1)/6)
Cube Sum	([n(n+1)/2]^2)
Factors	((a+1)(b+1))

Common Mistakes Students Make

Confusing integers with whole numbers
Considering 1 as prime
Using wrong divisibility rules
Forgetting irrational numbers
Errors in factor counting
Ignoring place values

Practice Questions

Fill in the blanks

Smallest whole number = ______
Answer: 0
Prime numbers have exactly ______ factors
Answer: 2
Sum of first n numbers = ______
Answer:

\frac{n(n+1)}2

Solve

1+2+3+\dots+20

Answer: 210

Total factors of: 48

Answer: 10

FAQs

Is 0 a natural number?

Generally, no. It belongs to whole numbers.

Is √2 rational?

No, it is irrational.

Is every whole number an integer?

Yes.

Which formula is most important?

Sum formula, divisibility rules, factors, and number classification.

Why study Number System?

Because every branch of mathematics starts with numbers.

Conclusion

Number System Formulae create the foundation of mathematics. From counting to advanced calculations, understanding number classifications, factor formulas, divisibility rules, and numerical properties makes solving mathematics easier and faster.

Learn formulas → Practice daily → Build mathematical speed and accuracy.