Base Conversions for Number Systems

BASE CONVERSIONS FOR NUMBER SYSTEMS

Electronic and digital systems work with different number systems such as Decimal, Binary, Octal and Hexadecimal. These number systems are important in computing and digital electronics.

• Decimal (base‒10) is the number system we use in everyday life for counting and calculations.

• Binary (base‒2) is the main number system used in digital circuits and the computers work using 0s and 1s.

• Octal (base‒8) and Hexadecimal (base‒16) are used to make long binary numbers easier to read and write.

Number Systems:

Decimal Number: Base 10 (0‒9)

Binary Number: Base 2 (0‒1)

Octal Number: Base 8 (0‒7)

Hexa Decimal Number: Base 16 (0‒9, A‒F)

Decimal Number System (Base‒10)

The decimal system has a base of 10.

It uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Each digit's position is based on powers of 10 (10⁰,10¹,10²,...).

For Example, the Decimal Number 4315.7 can be expanded according to the place value of each digit, as follows:

 4000 + 300 + 10 + 5 + 0.7 = 4315.7

This can also be written using powers of 10, as follows:

4×10³ + 3×10² + 1×10¹ + 5×10⁰ + 7×10^‒1

The same number may also be written as (4315.7)₁₀, where the subscript 10 indicates that the number is in the decimal (base‒10) system.

Binary Number System (Base‒2)

The binary system has a base of 2.

It uses only two digits: 0 and 1.

Each digit's position is based on powers of 2 (2⁰, 2¹, 2², ... ).

For example, the binary number 1101.101 shall be represented in powers of 2 and its decimal value shall be computed as follows.

Computation of decimal value shall be done using the following steps, viz.,

Step 1: The number can be expanded as:

 N = (1×2³)+(1×2²)+(0×2¹)+(1×2⁰)+(1×2^‒1)+(0×2^‒2)+(1×2^‒3)

Step 2: Now calculate each term:

1 × 2³ = 8

1 × 2² = 4

0 × 2¹ = 0

1 × 2⁰ = 1

1 × 2^‒1 = 0.5

0 × 2^‒2 = 0

1 × 2^‒3 = 0.125

Step 3:

Add them all: 8+4+0+1+0.5+0+0.125 = 13.625

The decimal equivalent is N = (13.625)₁₀

Octal Number System (Base‒8)

The octal system has a base of 8.

It uses eight digits: 0, 1, 2, 3, 4, 5, 6, 7.

Each digit's position is based on powers of 8 (8⁰, 8¹, 8², ...).

Hexadecimal Number System (Base‒16)

The hexadecimal system has a base of 16.

It uses sixteen digits symbols: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F.

Each digit's position is based on powers of 16 (16⁰,16¹,16², ...).

Example 1: Convert decimal (68)₁₀ to binary number.

Solution

(68)₁₀ = (1 0 0 0 1 0 0)₂

Example 2: Convert decimal (13.125)₁₀ to binary number.

Solution

 (13.125)₁₀ = (1101. 001)₂

Example 3: Convert binary 110101 to decimal.

Solution

(110101)₂ = (53)₁₀

Example 4: Convert 110.11₂ into decimal.

Solution

 (110.11)₂ = (6.75)₁₀