Question

Binary math:

Convert 0xc996b4d7 from IEEE 745 to scientific notation (X * 10^Y) (show your work)

Answer 1

Hexadecimal     Binary
    0           0000
    1           0001
    2           0010
    3           0011
    4           0100
    5           0101
    6           0110
    7           0111
    8           1000
    9           1001
    A           1010
    B           1011
    C           1100
    D           1101
    E           1110
    F           1111
Use this table to convert from hexadecimal to binary
Converting C996B4D7 to binary
C => 1100
9 => 1001
9 => 1001
6 => 0110
B => 1011
4 => 0100
D => 1101
7 => 0111
So, in binary C996B4D7 is 11001001100101101011010011010111
11001001100101101011010011010111
1 10010011 00101101011010011010111
sign bit is 1(-ve)
exp bits are 10010011
   => 10010011
   => 1x2^7+0x2^6+0x2^5+1x2^4+0x2^3+0x2^2+1x2^1+1x2^0
   => 1x128+0x64+0x32+1x16+0x8+0x4+1x2+1x1
   => 128+0+0+16+0+0+2+1
   => 147
in decimal it is 147
so, exponent/bias is 147-127 = 20
frac bits are 00101101011010011010111

IEEE-754 Decimal value is 1.frac * 2^exponent
IEEE-754 Decimal value is 1.00101101011010011010111 * 2^20
1.00101101011010011010111 in decimal is 1.1773937940597534
   => 1.00101101011010011010111
   => 1x2^0+0x2^-1+0x2^-2+1x2^-3+0x2^-4+1x2^-5+1x2^-6+0x2^-7+1x2^-8+0x2^-9+1x2^-10+1x2^-11+0x2^-12+1x2^-13+0x2^-14+0x2^-15+1x2^-16+1x2^-17+0x2^-18+1x2^-19+0x2^-20+1x2^-21+1x2^-22+1x2^-23
   => 1x1+0x0.5+0x0.25+1x0.125+0x0.0625+1x0.03125+1x0.015625+0x0.0078125+1x0.00390625+0x0.001953125+1x0.0009765625+1x0.00048828125+0x0.000244140625+1x0.0001220703125+0x6.103515625e-05+0x3.0517578125e-05+1x1.52587890625e-05+1x7.62939453125e-06+0x3.814697265625e-06+1x1.9073486328125e-06+0x9.5367431640625e-07+1x4.76837158203125e-07+1x2.384185791015625e-07+1x1.1920928955078125e-07
   => 1+0.0+0.0+0.125+0.0+0.03125+0.015625+0.0+0.00390625+0.0+0.0009765625+0.00048828125+0.0+0.0001220703125+0.0+0.0+1.52587890625e-05+7.62939453125e-06+0.0+1.9073486328125e-06+0.0+4.76837158203125e-07+2.384185791015625e-07+1.1920928955078125e-07
   => 1.1773937940597534
so, 1.1773937940597534 * 2^20 in decimal is 1234586.875
so, 11001001100101101011010011010111 in IEEE-754 single precision format is -1234586.875
Answer: -1.234586875 * 10^6