Number System

NUMBER SYSTEM :

The number system or the numeral system is the system of naming or representing numbers.

Types of Number System -

There are various types of the number system in mathematics.

The four most common number system types are:

1. Decimal number system (0 to 9) -  [Base - 10]
2. Binary number system (0, 1) - [Base - 2]
3. Octal number system (0 to 7) - [Base - 8]
4. Hexadecimal number system (0 to 9 and A to F) - (Base - 16)

Decimal Number System (Base 10 Number System) :

Decimal number system has base 10 because it uses ten digits from 0 to 9. In the decimal number system, the positions successive to the left of the decimal point represent units, tens, hundreds, thousands and so on. This system is expressed in DECIMAL NUMBERS. The base of the decimal is 10. This shows that there are ten symbols, 0 to 9. 

Every position shows a particular power of the base (10).

For example :

The decimal number 1567 consists of the digit 7 in the units position, 6 in the tens place, 5 in the hundreds position, and 1 in the thousands place whose value can be written as -

(1×1000) + (5×100) + (6×10) + (7×1)

(1×10³) + (5×10²) + (6×10¹) + (7×1)

1000 + 500 + 60 + 7

1567

Binary Number System (Base 2 Number System) :

The base 2 number system is known as the BINARY NUMBER SYSTEM, wherein, only two binary digits exist, i.e., 0 and 1. Specifically, the usual base-2 is a radix of 2. The figures described under this system are known as binary numbers which are the combination of 0 and 1. For example, 110101 is a binary number.

We can convert any system into binary and vice versa.

For Example, to write (24)₁₀ as a binary number.

Solution:

2	24
2	12	0
2	6	0
2	3	1
	1	1

Base 2 Number System Example

∴ (24)₁₀ = (11100₎₂

Octal Number System (Base 8 Number System)

Octal numbers are commonly used in computer applications. In the OCTAL NUMBER SYSTEM, the base is 8 and it uses numbers from 0 to 7 to represent numbers. Converting an octal number to decimal is same as decimal conversion and is explained below using an example.

Example: Convert (259)₈ into decimal.

Solution:

  2    1      0

(2     5      9)_{8 =}     

= 2 × 8²  +  5 × 8¹  +  9 × 8⁰

= 2 × 64  +  5 × 8   +  9 × 1

= (177) ₁₀

Hexadecimal Number System (Base 16 Number System)

In the hexadecimal system, numbers are written or represented with base 16. In the hex system, the numbers are first represented just like in decimal system i.e. from 0 to 9. Then, the numbers are represented using the alphabets from A to F. The below-given table shows the representation of numbers in the Hexadecimal Number System.

Hexadecimal	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
Decimal	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15

Conversions

1) Decimal Number to Binary Number

Q. (65)10 = (?)2

Q. (205)10 = (?)2

Ans. - 

(65)10 = (1000001)2
(205)10 = (11001101)2

                                 

2) Binary Number to  Decimal Number

Q. (1000001)2 = (?)10

Q. (11001101)2 = (?)10

Ans. -

(1000001)2 = (65)10

(11001101)2 = (205)10

            

             

3) Decimal Number  to Octal Number

Q. (98)10 = (?)8

Q. (112)10 = (?)8

Ans. -

(98)10 = (142)8

(112)10 = (160)8

                              

4) Octal Number to Decimal Number :

Q. (142)8 = (?)10

Q. (160)8 = (?)10

Ans. -

(142)8 = (98)10

(160)8 = (112)10

             

5) Decimal Number to Hexadecimal Number :

Q. (234)10 = (?)16

Q. (945)10 = (?)16

Ans. -

(234)10 = (EA)16

(945)10 = (3B1)16

 

6) Hexadecimal  Number to Decimal  Number :

Q. (EA)16 = (?)10

Q. (3B1)16 = (?)10

Ans. -

(EA)16 = (234)10

(3B1)16 = (945)10

   

Note: 10=A, 11=B, 12=C, 13=D, 14=E, 15=F

BINARY  ADDITION:

BINARY  SUBSTRACTION:

OTHER TOPICS :

• Computer Hardware