Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 0001 1110 - 000 1011 1010 1000 0110 0010 Converted and Written as a Base Ten Decimal System Number (as a Float)
0 - 0001 1110 - 000 1011 1010 1000 0110 0010: 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:
0001 1110
The last 23 bits contain the mantissa:
000 1011 1010 1000 0110 0010
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
0001 1110(2) =
0 × 27 + 0 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 0 × 20 =
0 + 0 + 0 + 16 + 8 + 4 + 2 + 0 =
16 + 8 + 4 + 2 =
30(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 = 30 - 127 = -97
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 1011 1010 1000 0110 0010(2) =
0 × 2-1 + 0 × 2-2 + 0 × 2-3 + 1 × 2-4 + 0 × 2-5 + 1 × 2-6 + 1 × 2-7 + 1 × 2-8 + 0 × 2-9 + 1 × 2-10 + 0 × 2-11 + 1 × 2-12 + 0 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 2-16 + 1 × 2-17 + 1 × 2-18 + 0 × 2-19 + 0 × 2-20 + 0 × 2-21 + 1 × 2-22 + 0 × 2-23 =
0 + 0 + 0 + 0.062 5 + 0 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0 + 0.000 976 562 5 + 0 + 0.000 244 140 625 + 0 + 0 + 0 + 0 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0 + 0 + 0 + 0.000 000 238 418 579 101 562 5 + 0 =
0.062 5 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.000 976 562 5 + 0.000 244 140 625 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0.000 000 238 418 579 101 562 5 =
0.091 076 135 635 375 976 562 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 + 0.091 076 135 635 375 976 562 5) × 2-97 =
1.091 076 135 635 375 976 562 5 × 2-97 =
0.000 000 000 000 000 000 000 000 000 006 885 658 46
0 - 0001 1110 - 000 1011 1010 1000 0110 0010 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) = 0.000 000 000 000 000 000 000 000 000 006 885 658 46(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: