Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 0110 0011 - 100 0100 0010 0001 0000 1010 Converted and Written as a Base Ten Decimal System Number (as a Float)
0 - 0110 0011 - 100 0100 0010 0001 0000 1010: 32 bit single precision IEEE 754 binary floating point standard representation number converted to decimal system (base ten)
1. Identify the elements that make up the binary representation of the number:
The first bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.
0
The next 8 bits contain the exponent:
0110 0011
The last 23 bits contain the mantissa:
100 0100 0010 0001 0000 1010
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
0110 0011(2) =
0 × 27 + 1 × 26 + 1 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 1 × 20 =
0 + 64 + 32 + 0 + 0 + 0 + 2 + 1 =
64 + 32 + 2 + 1 =
99(10)
3. Adjust the exponent.
Subtract the excess bits: 2(8 - 1) - 1 = 127,
that is due to the 8 bit excess/bias notation.
The exponent, adjusted = 99 - 127 = -28
4. Convert the mantissa from binary (from base 2) to decimal (in base 10).
The mantissa represents the fractional part of the number (what comes after the whole part of the number, separated from it by a comma).
100 0100 0010 0001 0000 1010(2) =
1 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 1 × 2-5 + 0 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 1 × 2-10 + 0 × 2-11 + 0 × 2-12 + 0 × 2-13 + 0 × 2-14 + 1 × 2-15 + 0 × 2-16 + 0 × 2-17 + 0 × 2-18 + 0 × 2-19 + 1 × 2-20 + 0 × 2-21 + 1 × 2-22 + 0 × 2-23 =
0.5 + 0 + 0 + 0 + 0.031 25 + 0 + 0 + 0 + 0 + 0.000 976 562 5 + 0 + 0 + 0 + 0 + 0.000 030 517 578 125 + 0 + 0 + 0 + 0 + 0.000 000 953 674 316 406 25 + 0 + 0.000 000 238 418 579 101 562 5 + 0 =
0.5 + 0.031 25 + 0.000 976 562 5 + 0.000 030 517 578 125 + 0.000 000 953 674 316 406 25 + 0.000 000 238 418 579 101 562 5 =
0.532 258 272 171 020 507 812 5(10)
5. Put all the numbers into expression to calculate the single precision floating point decimal value:
(-1)Sign × (1 + Mantissa) × 2(Adjusted exponent) =
(-1)0 × (1 + 0.532 258 272 171 020 507 812 5) × 2-28 =
1.532 258 272 171 020 507 812 5 × 2-28 =
0.000 000 005 708 106 876 056 717 737 810 686 230 65
0 - 0110 0011 - 100 0100 0010 0001 0000 1010 converted from a 32 bit single precision IEEE 754 binary floating point standard representation number to a decimal system number, written in base ten (float) = 0.000 000 005 708 106 876 056 717 737 810 686 230 65(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.
More operations with 32 bit single precision IEEE 754 binary floating point standard representation numbers: