Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 1000 0010 - 000 0001 1100 1100 0110 1110 Converted and Written as a Base Ten Decimal System Number (as a Float)
1 - 1000 0010 - 000 0001 1100 1100 0110 1110: 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:
1000 0010
The last 23 bits contain the mantissa:
000 0001 1100 1100 0110 1110
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
1000 0010(2) =
1 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 0 × 20 =
128 + 0 + 0 + 0 + 0 + 0 + 2 + 0 =
128 + 2 =
130(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 = 130 - 127 = 3
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).
000 0001 1100 1100 0110 1110(2) =
0 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 0 × 2-5 + 0 × 2-6 + 1 × 2-7 + 1 × 2-8 + 1 × 2-9 + 0 × 2-10 + 0 × 2-11 + 1 × 2-12 + 1 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 2-16 + 1 × 2-17 + 1 × 2-18 + 0 × 2-19 + 1 × 2-20 + 1 × 2-21 + 1 × 2-22 + 0 × 2-23 =
0 + 0 + 0 + 0 + 0 + 0 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0 + 0 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0 + 0 + 0 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0.000 000 238 418 579 101 562 5 + 0 =
0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0.000 000 238 418 579 101 562 5 =
0.014 051 198 959 350 585 937 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 + 0.014 051 198 959 350 585 937 5) × 23 =
-1.014 051 198 959 350 585 937 5 × 23 =
-8.112 409 591 674 804 687 5
1 - 1000 0010 - 000 0001 1100 1100 0110 1110 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) = -8.112 409 591 674 804 687 5(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: