Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 0110 1011 - 001 1110 0101 0011 0010 1010 Converted and Written as a Base Ten Decimal System Number (as a Float)
1 - 0110 1011 - 001 1110 0101 0011 0010 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.
1
The next 8 bits contain the exponent:
0110 1011
The last 23 bits contain the mantissa:
001 1110 0101 0011 0010 1010
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
0110 1011(2) =
0 × 27 + 1 × 26 + 1 × 25 + 0 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 1 × 20 =
0 + 64 + 32 + 0 + 8 + 0 + 2 + 1 =
64 + 32 + 8 + 2 + 1 =
107(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 = 107 - 127 = -20
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).
001 1110 0101 0011 0010 1010(2) =
0 × 2-1 + 0 × 2-2 + 1 × 2-3 + 1 × 2-4 + 1 × 2-5 + 1 × 2-6 + 0 × 2-7 + 0 × 2-8 + 1 × 2-9 + 0 × 2-10 + 1 × 2-11 + 0 × 2-12 + 0 × 2-13 + 1 × 2-14 + 1 × 2-15 + 0 × 2-16 + 0 × 2-17 + 1 × 2-18 + 0 × 2-19 + 1 × 2-20 + 0 × 2-21 + 1 × 2-22 + 0 × 2-23 =
0 + 0 + 0.125 + 0.062 5 + 0.031 25 + 0.015 625 + 0 + 0 + 0.001 953 125 + 0 + 0.000 488 281 25 + 0 + 0 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0 + 0 + 0.000 003 814 697 265 625 + 0 + 0.000 000 953 674 316 406 25 + 0 + 0.000 000 238 418 579 101 562 5 + 0 =
0.125 + 0.062 5 + 0.031 25 + 0.015 625 + 0.001 953 125 + 0.000 488 281 25 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 003 814 697 265 625 + 0.000 000 953 674 316 406 25 + 0.000 000 238 418 579 101 562 5 =
0.236 912 965 774 536 132 812 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)1 × (1 + 0.236 912 965 774 536 132 812 5) × 2-20 =
-1.236 912 965 774 536 132 812 5 × 2-20 =
-0.000 001 179 612 127 089 058 049 023 151 397 705 07
1 - 0110 1011 - 001 1110 0101 0011 0010 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) = -0.000 001 179 612 127 089 058 049 023 151 397 705 07(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: