Inter conversion of various types of number systems

# In this section you cover the following conversions:

**Decimal to
other conversions :-**

**Binary to
other conversions :-**

**Octal to
other conversion :-**

**Hexadecimal
to other conversion :-**

**Decimal to other conversions**

### 1. Decimal to binary conversion :-

#### For example
we need to convert (250)_{10 }to
(?)_{2}

**Step 1: Divide 250 by 2 and write all reminders.**

250/2 =125, Reminder will be 0

125/2 =62, Reminder will be 1

62/2 =31, Reminder will be 0

31/2 =15, Reminder will be 1

15/2 =7, Reminder will be 1

7/2 =3, Reminder will be 1

3/2 =1, Reminder will be 1

1/2 =0, Reminder will be 1

**Step 2: Write the value from downward (MSB) to upward (LSB) as 11111010 which is the binary equivalent of 250**

*Result = (11111010)*_{2}

#### Let us see
the second example (20.250)_{10 }=(?)_{2}

First we solve the part 1 left side of decimal(.) then after other side of decimal(.)

**Step 1: Divide 20 by 2 and write all reminders.**

20/2 =10, Reminder will be 0

10/2 =5, Reminder will be 0

5/2 =2, Reminder will be 1

2/2 =1, Reminder will be 0

1/2 =0, Reminder will be 1

**Step 2: Write the value from downward (MSB) to upward (LSB) as 10100 which is the binary equivalent of 20**

Now solving part 2 right side of decimal (.)

.250 x 2 =0.500, real number will be 0

.500 x 2 =1.000, real number will be 1

**Step 3: Write the values from upward to downward as 01, so that .01 is binary equivalent of .250**

**Step 4: Combining both for final answer we get 10100.01 as the binary equivalent of 20.250**

*Result = (10100.01)*_{2}

### 2. Decimal to octal conversion :-

#### For example
we need to convert (66.38)_{10} = (?)_{8}

First we solve the part 1 left side of decimal(.) then after other side of decimal(.)

**Step 1: Divide 66 by 8 and write all reminders.**

66/8 =8, Reminder will be 2

8/8 = 1, Reminder will be 0

1/8 =0, Reminder will be 1

**Step 2: Write the value from downward (MSB) to upward (LSB) as 102 which is the octal equivalent of 66**

Now solving part 2 right side of decimal (.)

.38 x 8 =3.04, real number will be 3

.04 x 8 =0.32, real number will be 0

.32 x 8 =2.56, real number will be 2

.56 x 8 =4.48, real number will be 4

.48 x 8 =3.84, real number will be 3

.84 x 8 =6.72, real number will be 6

*Note that if the whole number or close whole number to 8 in left side of decimal is not coming in solving (after decimal portion) then do only six steps in the calculation above.*

**Step 3: Write the values from upward to downward as 302436, so that .302436 is octal equivalent of .38**

**Step 4: Combining both for final answer we get 102.302436 as the octal equivalent of 66.38**

*Result = (102.302436)*_{8}

### 3. Decimal to hexadecimal conversion :-

#### For example
we need to convert (214.356)_{10} = (?)_{16}

First we solve the part 1 left side of decimal(.) then after other side of decimal(.)

**Step 1: Divide 214 by 16 and write all reminders.**

214/16 =13, Reminder will be 6

13/16 =0, Reminder will be 13 which is D

**Step 2: Write the value from downward (MSB) to upward (LSB) as D6 which is the hexadecimal equivalent of 214**

Now solving part 2 right side of decimal (.)

.356 x 16 =5.696, real number will be 5

.696 x 16 =11.136, real number will be 11 which is B

.136 x 16 =2.176, real number will be 2

.176 x 16 =2.816, real number will be 2

**Step 3: Write the values from upward to downward as 5B22, so that .5B22 is hexadecimal equivalent of .356**

**Step 4: Combining both for final answer we get D6.5B22 as the hexadecimal equivalent of 214.356**

*Result = (D6.5B22)*_{16}

## Binary to other conversions

### 1. Binary to decimal conversion :-

#### For example we need to convert (10111.0110)_{2 }to (?)_{10}

**Step 1: Multiplying the numbers to 2**^{…,-1,-2,0,1,2…}** as given below **

= 1×2^{4 }+
0x2^{3 }+ 1×2^{2 }+ 1×2^{1 }+ 1×2^{0 }+ 0x2^{-1
}+ 1×2^{-2 }+ 1×2^{-3 }+ 0x2^{-4}

= 16 + 0 + 4 + 2 + 1 + 0 + 0.25 + 0.125 + 0

= 23.375

*Result = (23.375)*_{10}

### 2. Binary to octal conversion :-

#### For example
we need to convert (1001110.0110101)_{2 }to (?)_{8}

**Step 1: Grouping each 3 bits and then writing their octal equivalent .**

*Result = (116.324)*_{8}

*Note that if 3 bit group is not forming (before decimal) then put 0’s before the number to fit it in 3 bit group as given in green colour{left side} , and if 3 bit group are not forming (after decimal) then put 0’s after the number to fit into 3 bit group given in green colour {right side} .*

### 3. Binary to hexadecimal conversion :-

#### For example we need to convert (10101011001.1010100)_{2 }to (?)_{16}

**Step 1: Grouping each 4 bits and then writing their hexadecimal equivalent .**

*Result =(AD9.A8)*_{16}

**Note that if ****4 bit**** group is not forming then put 0’s before or after the number as (before and after decimal) to fit it in ****4 bit**** group as given in green colour.**

## Octal to other conversion

### 1. Octal to decimal conversion :-

#### For example
we need to convert (762.231)_{8 }to
(?)_{16}

**Step 1: Multiplying the numbers to 8**^{…,-1,-2,0,1,2…}** as given below **

= 7×8^{2 }+
6×8^{1 }+ 2×8^{0 }+ 2×8^{-1 }+ 3×8^{-2 }+ 1×8^{-3}

= 448 + 48 + 2 + 2/8 +2/64 +3/512

= 498 + (128+24+1)/512

= 498 + 0.2988

= 498.2988

*Result = (498.2988)*_{16}

### 2.Octal to binary conversion :-

For example
we need to convert (723.301)_{8 }to
(?)_{2}

Step 1: Expend it to 3 bit groups of binary equivalent.

*Result = (111010011.011000001)*_{2}

### 3. Octal to hexadecimal conversion :-

#### For example we
need to convert (224.21)_{8} = (?)_{16}

First we need to convert it to binary and then in hexadecimal

**Step 1: Expend it to 3 bit groups of binary equivalent.**

= (010010100.010001)_{2}

**Step 2: Now make 4 bit groups and convert binary to hexadecimal**

*Result = (0A4.44)*_{16}

*Note that if 4 bit group is not forming (before decimal) then put 0’s before the number to fit it in 4 bit group as given in green colour{left side} , and if 4 bit group are not forming (after decimal) then put 0’s after the number to fit into 4 bit group given in green colour {right side} .*

**Hexadecimal to other conversions**

### 1. Hexadecimal to decimal conversion :-

#### For example
we need to convert (A23.351)_{16 }to
(?)_{10}

**Step 1: Multiplying the numbers to 16**^{…,-1,-2,0,1,2…}** as given below **

= 10×16^{2}
+ 2×16^{1} + 3×16^{0} + 3×16^{-1} + 5×16^{-2} +
1×16^{-3}

= 2560 + 32 + 3 + 3/16 + 5/256 + 1/4096

= 2595 + (768+80+1)/4096

= 2595 + 849/4096

= 2595 + 0.20727

= 2595.20727

*Result = (2595.20727)*_{10}

### 2. Hexadecimal to Binary conversion :-

#### For example we need to convert (F23.6A1)_{16 }to (?)_{2}

**Step 1: Expand it to 4 bit groups of its binary equivalent.**

*Result = (111100100011.011010100001)*_{2}

### 3. Hexadecimal to octal conversion :-

#### For example we
need to convert (123.31)_{16} = (?)_{8}

First we need to convert it to binary and then in octal

**Step 1: First expend it in 4 bit groups to its binary equivalent.**

= (000100100011.00110001)_{2}

**Step 2: Now group it in 3 bit sets and write their octal equivalent.**

*Result = (0443.142)*_{8}

*Note that if **3 bit** group is not forming then put 0’s before or after the number as (before and after decimal) to fit it in **3 bit** group as given in green colour.*

Quick Conversion Table