Mumbai University > Information Technology> sem 4> computer organization and architecture

Marks: 10M

Year: Dec16

Express (4.50)10 in the IEEE single precision standard of floating point representation:

Step 1: Convert the decimal number to its binary fractional form.

Convert this format is in base2 format For this, first convert 4 in to binary format

4=100

Convert 0.50 in to binary format

0.50*2=1.0 1

Binary format of 4.50=100.1

Shifting this binary number

1.0012 *2 Normalized

1.001 is mantissa

2 **2 is exponent

add exponent 127 +2=129

Step 2: Normalize the binary fractional number.

Move the decimal point left or right so that only a single binary

digit "1" is to the left of the binary decimal point. Compensate by

adjusting the exponent in the opposite direction.

1.001 times 2*2

Moving the decimal left seven decreases the size of the number; so,

we use an exponent of 6 to compensate and keep the number the same size.

Step 3: Convert the exponent to 8-bit excess-127 notation

Add 127 to the exponent and convert it to 8-bit binary:

2+ 127 = 129--> 10000001

Step 4: Convert the mantissa/significand to "hidden bit" format.

Since every binary floating-point number (except zero!) is normalized

with "1." at the start, there is no need to store that leftmost "1".

Remove the leading "1." from the mantissa/significand:

1.001--> 001

Step 5: Write down the 1+8+23 = 32 bits.

4.50 is positive - the sign bit is zero: 0

The next eight bits are the exponent: 10000001

The next 23 bits are the mantissa: 00100000000000000000000

Binary result (32 bits): 01000000100100000000000000000000

Express (4.50)10 in the IEEE double precision standard of floating point representation:

Step 1: Convert the decimal number to its binary fractional form.

Convert this format is in base2 format For this, first convert 4 in to binary format

4=100

Convert 0.50 in to binary format

0.50*2=1.00 1

Binary format of 4.50=100.1

Shifting this binary number

1.0012 *2 Normalized

1.001 is mantissa

2 **2 is exponent

add exponent 1023+2=1025

Step 2: Normalize the binary fractional number.

Move the decimal point left or right so that only a single binary

digit "1" is to the left of the binary decimal point. Compensate by

adjusting the exponent in the opposite direction.

1.001 times 2*2

Moving the decimal left seven decreases the size of the number; so,

we use an exponent of 6 to compensate and keep the number the same size.

Step 3: Convert the exponent to 11-bit excess-1023 notation

Add 1023 to the exponent and convert it to 11-bit binary:

2+1023= 1025--> 10000000001

Step 4: Convert the mantissa/significand to "hidden bit" format.

Since every binary floating-point number (except zero!) is normalized

with "1." at the start, there is no need to store that leftmost "1".

Remove the leading "1." from the mantissa/significand:

1.001--> 001

Step 5: Write down the 1+11+52 = 64bits.

4.50 is positive - the sign bit is zero: 0

The next 11 bits are the exponent: 10000000001

The next 52 bits are the mantissa:
00100000000000000000000000000000000000000000000000

Binary result (64 bits): 010000000001 00100000000000000000000000000000000000000000000000