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# Inter_Conversion of Number systems

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 :-

## 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

= 1×24 + 0x23 + 1×22 + 1×21 + 1×20 + 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 .

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

= 7×82 + 6×81 + 2×80 + 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} .

### 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

= 10×162 + 2×161 + 3×160 + 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

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