Question

What is the biggest positive FP number that can be represented in 16-bit format using 1-bit sign, 4-bit biased exponent, and 11-bit fraction, where bias is 7? (Show All Steps)

Answer 1

16-bit format

sign = 0(+ve)
exponent bits = 1110
    because 1111 is reserved/used for representing infinite value.
    so, 1110 is the exponent for largest positive number
frac bits = 11111111111

Now, let's convert this number to decimal and see what it is

0 1110 11111111111
sign bit is 0(+ve)
exp bits are 1110
   => 1110
   => 1x2^3+1x2^2+1x2^1+0x2^0
   => 1x8+1x4+1x2+0x1
   => 8+4+2+0
   => 14
in decimal it is 14
so, exponent/bias is 14-7 = 7
frac bits are 11111111111

IEEE-754 Decimal value is 1.frac * 2^exponent
IEEE-754 Decimal value is 1.11111111111 * 2^7
1.11111111111 in decimal is 1.99951171875
   => 1.11111111111
   => 1x2^0+1x2^-1+1x2^-2+1x2^-3+1x2^-4+1x2^-5+1x2^-6+1x2^-7+1x2^-8+1x2^-9+1x2^-10+1x2^-11
   => 1x1+1x0.5+1x0.25+1x0.125+1x0.0625+1x0.03125+1x0.015625+1x0.0078125+1x0.00390625+1x0.001953125+1x0.0009765625+1x0.00048828125
   => 1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125+0.00390625+0.001953125+0.0009765625+0.00048828125
   => 1.99951171875
so, 1.99951171875 * 2^7 in decimal is 255.9375
so, 0111011111111111 in 16-bit format is 255.9375
Answer: 255.9375