Express (28.75)10 in the IEEE single and double precision standard of floating point representation

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teamques10 ★ 69k

Express (28.75)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 127 in to binary format

28=11100

Convert 0.75 in to binary format

0.75*2=1.50 1

Binary format of 28.75=11100.1

Shifting this binary number

1.110012 *4 Normalized

1.11001 is mantissa

2 **4 is exponent

add exponent 127 +4=131

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.

1.11001 times 2*4

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:

4+ 127 = 131-> 10000011

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:

1.11001 --> 11001

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

28.75 is positive - the sign bit is zero: 0

The next eight bits are the exponent: 10000011

The next 23 bits are the mantissa: 11001000000000000000000

Binary result (32 bits): 01000001111001000000000000000000

Express (127.125)10 in the IEEE double precision standard of floating point representation: