Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 0010 0000 - 010 0100 0010 1000 0011 0010 Converted and Written as a Base Ten Decimal System Number (as a Float)
0 - 0010 0000 - 010 0100 0010 1000 0011 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:
0010 0000
The last 23 bits contain the mantissa:
010 0100 0010 1000 0011 0010
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
0010 0000(2) =
0 × 27 + 0 × 26 + 1 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =
0 + 0 + 32 + 0 + 0 + 0 + 0 + 0 =
32 =
32(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 = 32 - 127 = -95
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).
010 0100 0010 1000 0011 0010(2) =
0 × 2-1 + 1 × 2-2 + 0 × 2-3 + 0 × 2-4 + 1 × 2-5 + 0 × 2-6 + 0 × 2-7 + 0 × 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 + 0 × 2-17 + 1 × 2-18 + 1 × 2-19 + 0 × 2-20 + 0 × 2-21 + 1 × 2-22 + 0 × 2-23 =
0 + 0.25 + 0 + 0 + 0.031 25 + 0 + 0 + 0 + 0 + 0.000 976 562 5 + 0 + 0.000 244 140 625 + 0 + 0 + 0 + 0 + 0 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0 + 0 + 0.000 000 238 418 579 101 562 5 + 0 =
0.25 + 0.031 25 + 0.000 976 562 5 + 0.000 244 140 625 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 238 418 579 101 562 5 =
0.282 476 663 589 477 539 062 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 + 0.282 476 663 589 477 539 062 5) × 2-95 =
1.282 476 663 589 477 539 062 5 × 2-95 =
0.000 000 000 000 000 000 000 000 000 032 374 262 44
0 - 0010 0000 - 010 0100 0010 1000 0011 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) = 0.000 000 000 000 000 000 000 000 000 032 374 262 44(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: