Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 0111 1010 - 000 1100 1110 0110 1010 0101 Converted and Written as a Base Ten Decimal System Number (as a Float)
0 - 0111 1010 - 000 1100 1110 0110 1010 0101: 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:
0111 1010
The last 23 bits contain the mantissa:
000 1100 1110 0110 1010 0101
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
0111 1010(2) =
0 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 0 × 20 =
0 + 64 + 32 + 16 + 8 + 0 + 2 + 0 =
64 + 32 + 16 + 8 + 2 =
122(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 = 122 - 127 = -5
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 1110 0110 1010 0101(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 + 1 × 2-9 + 1 × 2-10 + 0 × 2-11 + 0 × 2-12 + 1 × 2-13 + 1 × 2-14 + 0 × 2-15 + 1 × 2-16 + 0 × 2-17 + 1 × 2-18 + 0 × 2-19 + 0 × 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 + 0.001 953 125 + 0.000 976 562 5 + 0 + 0 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0 + 0.000 015 258 789 062 5 + 0 + 0.000 003 814 697 265 625 + 0 + 0 + 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 + 0.001 953 125 + 0.000 976 562 5 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 015 258 789 062 5 + 0.000 003 814 697 265 625 + 0.000 000 476 837 158 203 125 + 0.000 000 119 209 289 550 781 25 =
0.100 788 712 501 525 878 906 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)0 × (1 + 0.100 788 712 501 525 878 906 25) × 2-5 =
1.100 788 712 501 525 878 906 25 × 2-5 =
0.034 399 647 265 672 683 715 820 312 5
0 - 0111 1010 - 000 1100 1110 0110 1010 0101 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) = 0.034 399 647 265 672 683 715 820 312 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: