Question

With a detailed step-by-step process, convert the following decimal number into binary, Hexadecimal and IEEE 754 formats :

72.nn ( where nn is 80)

Answer 1

Part 1: convert 72.80 to binary:

Step 1: convert 72(integral part ) to binary:

Division by 2	Quotient	Remainder
72/2	36	0
36/2	18	0
18/2	9	0
9/2	4	1
4/2	2	0
2/2	1	0
1/2	0	1

Now write remainder column from down to up it is a binary for 72

(72)₁₀=(1001000)₂

Step 2: Convert .80 (fractional part) to binary:

1. Multiply the fractional decimal number by 2.
2. Integral part is a binary number.
3. repeat process

Multiply by 2	Result	fractional part	Integral part(Binary)
.80*2	1.60	.60	1
.60*2	1.20	.20	1
.20*2	0.40	.40	0
.40*2	.80	.80(Same as first stop)	0

(Note: You may stop multiplication when you get similar number)

Write integral part(Binary) column from up to down. it is a binary equivalent of decimal number

(.80)₁₀ =(1100)₂

So Binary equivalent of decimal is:

(72.80)₁₀ =(1001000.1100)₂

Part 2: convert 72.80 to Hexadecimal:

Step 1: convert 72(integral part ) to hexadecimal

Division by 16	Quotient	Remainder->(hexadecimal)
72/16	4	8->8
4/16	0	4->4

Now write remainder column from down to up it is a hexadecimal for 72

(72)₁₀=(48)16

Step 2 :Convert .80 (fractional part) to hexadecimal:

1. Multiply the fractional decimal number by 16.
2. Integral part is a hexadecimal number.
3. repeat process

Multiply by 16	Result	fractional part	Integral part(hexadecimal)
.80*16	12.8	.8	12 -> C
.8*16	12.8	.8(Same as first stop)	12 ->C

(Note: You may stop multiplication when you get similar number)

Write integral part(hexadecimal) column from up to down. it is a hexadecimal equivalent of decimal number

(.80)₁₀ =(CC)16

So Hexadecimal equivalent of decimal is:

(72.80)₁₀ =(48.CC)₁₆

Part 3: convert 72.80 to IEEE 754 format:

In IEEE 754 we have mainly 3 part

Single precision:

1. Sign of Mantissa ( 0 represents positive 1 represents negative) (1bit)
2. exponent (127 actual for Single precision) (8 bits)
3. Normalised Mantissa (23 bits)

Step 1: Write binary of a decimal mumber

(72.80)₁₀ =(1001000.1100)₂

Step 2: Write binary number in a power of 2

1.0010001100 * 2⁶ (2⁶ because we shift decimal by 6)

(0010001100) is calles a normalised mantisa

Sign bit is 0 because number is positive.

Step 3: add 6(power of 2) in actual exponent (127)

127+6=133

Step 4: Conver 133 into binary

(133)₁₀ =(10000101)₂

Normalized mantisa is:

0010001100

Step 5 : make 23 bit Normalized mantisa by adding extra 0's

  00100011000000000000000

Step 6 :Sign bit is 0

Step 7 : Write Sign, Exponent and mantisa in their format which is the IEEE 754 format

Format is:

Sign(1bit)

Exponent(8bits)

Mantisa(23 bits)

(0  10000101   00100011000000000000000)

So,

(72.80)₁₀ = (0  10000101   00100011000000000000000) in IEEE 754 format