Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 1101 1001 - 000 1100 1010 0001 0110 1101 Converted and Written as a Base Ten Decimal System Number (as a Float)
1 - 1101 1001 - 000 1100 1010 0001 0110 1101: 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:
1101 1001
The last 23 bits contain the mantissa:
000 1100 1010 0001 0110 1101
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
1101 1001(2) =
1 × 27 + 1 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 1 × 20 =
128 + 64 + 0 + 16 + 8 + 0 + 0 + 1 =
128 + 64 + 16 + 8 + 1 =
217(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 = 217 - 127 = 90
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 1100 1010 0001 0110 1101(2) =
0 × 2-1 + 0 × 2-2 + 0 × 2-3 + 1 × 2-4 + 1 × 2-5 + 0 × 2-6 + 0 × 2-7 + 1 × 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 + 1 × 2-17 + 1 × 2-18 + 0 × 2-19 + 1 × 2-20 + 1 × 2-21 + 0 × 2-22 + 1 × 2-23 =
0 + 0 + 0 + 0.062 5 + 0.031 25 + 0 + 0 + 0.003 906 25 + 0 + 0.000 976 562 5 + 0 + 0 + 0 + 0 + 0.000 030 517 578 125 + 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 + 0.000 000 119 209 289 550 781 25 =
0.062 5 + 0.031 25 + 0.003 906 25 + 0.000 976 562 5 + 0.000 030 517 578 125 + 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 + 0.000 000 119 209 289 550 781 25 =
0.098 676 323 890 686 035 156 25(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 + 0.098 676 323 890 686 035 156 25) × 290 =
-1.098 676 323 890 686 035 156 25 × 290 =
-1 360 095 411 559 153 053 435 166 720
1 - 1101 1001 - 000 1100 1010 0001 0110 1101 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) = -1 360 095 411 559 153 053 435 166 720(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: