Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 1000 0011 - 111 0101 0000 0000 0001 0010 Converted and Written as a Base Ten Decimal System Number (as a Float)
1 - 1000 0011 - 111 0101 0000 0000 0001 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.
1
The next 8 bits contain the exponent:
1000 0011
The last 23 bits contain the mantissa:
111 0101 0000 0000 0001 0010
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
1000 0011(2) =
1 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 1 × 20 =
128 + 0 + 0 + 0 + 0 + 0 + 2 + 1 =
128 + 2 + 1 =
131(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 = 131 - 127 = 4
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).
111 0101 0000 0000 0001 0010(2) =
1 × 2-1 + 1 × 2-2 + 1 × 2-3 + 0 × 2-4 + 1 × 2-5 + 0 × 2-6 + 1 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 0 × 2-12 + 0 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 0 × 2-18 + 1 × 2-19 + 0 × 2-20 + 0 × 2-21 + 1 × 2-22 + 0 × 2-23 =
0.5 + 0.25 + 0.125 + 0 + 0.031 25 + 0 + 0.007 812 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 001 907 348 632 812 5 + 0 + 0 + 0.000 000 238 418 579 101 562 5 + 0 =
0.5 + 0.25 + 0.125 + 0.031 25 + 0.007 812 5 + 0.000 001 907 348 632 812 5 + 0.000 000 238 418 579 101 562 5 =
0.914 064 645 767 211 914 062 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)1 × (1 + 0.914 064 645 767 211 914 062 5) × 24 =
-1.914 064 645 767 211 914 062 5 × 24 =
-30.625 034 332 275 390 625
1 - 1000 0011 - 111 0101 0000 0000 0001 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) = -30.625 034 332 275 390 625(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: