## Solved Example on Number System Set-II (Binary to Decimal, Octal, Hexadecial)

#### 1. How to convert binary to decimal ?

1. Find the position of every binary digit. We should count the position from the right direction of the number. And the position count starts from 0.

#### Example

0001 - position of 1 = 0, 0 = 1, 0 = 2, 0 = 3.

2. Multiply every digit with 2 to the power of their corresponding position. (2 position)

3. Finally, calculate the sum of all the multiples.

Example: Convert (1111)2 to decimal

= 1 x 2 3 + 1 x 2 2 + 1 x 2 1 + 1 x 2 0

= 8 + 4 + 2 + 1

= (15)10

Example: Convert (1001)2 to decimal

= 1 x 2 3 + 0 x 2 2 + 0 x 2 1 + 1 x 2 0

= 8 + 0 + 0 + 1

= (9)10

#### 2. How to convert binary to octal ?

Group every 3 binary bits from right to left and construct the octal number system.

Example: (101010101)2 to octal

= (101010101)2

= (101)(010)(101)

= (525)8

Example: (11111111)2 to octal

= (11111111)2

= (11)(111)(111)

= (377)8

#### 3. How to convert binary to hexadecimal ?

Group every 4 binary bits from right to left and construct the hexadecimal number system.

= (101010101)2

= (1)(0101)(0101)

= (155)16

= (11111111)2

= (1111)(1111)

= (15)(15)

= (ff)16

#### Converting Decimal Fraction to Binary, Octal, Hexadecimal

A fractional number is a number less than 1. It may be .5, .00453, .564, etc. We use the multiplication operation to convert decimal fraction to any other base.
To convert a decimal fraction to—
• binary - multiply by 2
• octal - multiply by 8
• hexadecimal - multiply by 16

#### Steps for conversion of a decimal fraction to any other base are—

1. Multiply the fractional number with the to Base, to get a resulting number.
2. The resulting number has two parts, non-fractional part and fractional part.
3. Record the non-fractional part of the resulting number.
4. Repeat the above steps at least four times.
5. Write the digits in the non-fractional part starting from upwards to downwards.

#### Example-1: Convert 0.2345 from Base 10 to Base 2

The binary equivalent of (0.2345)10 is (0.001111)2

#### Example-2: Convert 0.865 from Base 10 to Base 2,8 and 16

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