32 bit single precision IEEE 754 binary floating point number 0 - 0011 1111 - 001 0000 0000 0000 0000 0000 converted to decimal base ten (float)

How to convert 32 bit single precision IEEE 754 binary floating point:
0 - 0011 1111 - 001 0000 0000 0000 0000 0000
to decimal system (base ten)

1. Identify the elements that make up the binary representation of the number:

First bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.


The next 8 bits contain the exponent:
0011 1111


The last 23 bits contain the mantissa:
001 0000 0000 0000 0000 0000

2. Convert the exponent from binary (base 2) to decimal (base 10):

The exponent is allways a positive integer.

0011 1111(2) =


0 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20 =


0 + 0 + 32 + 16 + 8 + 4 + 2 + 1 =


32 + 16 + 8 + 4 + 2 + 1 =


63(10)

3. Adjust the exponent.

Subtract the excess bits: 2(8 - 1) - 1 = 127, that is due to the 8 bit excess/bias notation:

Exponent adjusted = 63 - 127 = -64


4. Convert the mantissa from binary (base 2) to decimal (base 10):

Mantissa represents the number's fractional part (the excess beyond the number's integer part, comma delimited)

001 0000 0000 0000 0000 0000(2) =

0 × 2-1 + 0 × 2-2 + 1 × 2-3 + 0 × 2-4 + 0 × 2-5 + 0 × 2-6 + 0 × 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 + 0 × 2-19 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 0 × 2-23 =


0 + 0 + 0.125 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =


0.125 =


0.125(10)

5. Put all the numbers into expression to calculate the single precision floating point decimal value:

(-1)Sign × (1 + Mantissa) × 2(Exponent adjusted) =


(-1)0 × (1 + 0.125) × 2-64 =


1.125 × 2-64 =


0.000 000 000 000 000 000 060 986 372 202 309 624 41

Conclusion:

0 - 0011 1111 - 001 0000 0000 0000 0000 0000
converted from
32 bit single precision IEEE 754 binary floating point
to
base ten decimal system (float) =

0.000 000 000 000 000 000 060 986 372 202 309 624 41(10)

More operations of this kind:

0 - 0011 1111 - 000 1111 1111 1111 1111 1111 = ?

0 - 0011 1111 - 001 0000 0000 0000 0000 0001 = ?


Convert 32 bit single precision IEEE 754 floating point standard binary numbers to base ten decimal system (float)

32 bit single precision IEEE 754 binary floating point standard representation of numbers requires three building blocks: sign (it takes 1 bit and it's either 0 for positive or 1 for negative numbers), exponent (8 bits), mantissa (23 bits)

Latest 32 bit single precision IEEE 754 floating point binary standard numbers converted to decimal base ten (float)

0 - 0011 1111 - 001 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 060 986 372 202 309 624 41 Dec 05 22:57 UTC (GMT)
0 - 1011 1110 - 010 0011 1101 1000 1001 1111 = 11 806 361 245 500 571 648 Dec 05 22:57 UTC (GMT)
0 - 0110 0001 - 100 0000 0000 0000 0000 0010 = 0.000 000 001 396 984 083 967 822 698 468 808 084 72 Dec 05 22:56 UTC (GMT)
1 - 1000 0100 - 001 0011 1111 1111 1111 1111 = -36.999 996 185 302 734 375 Dec 05 22:54 UTC (GMT)
1 - 1000 0001 - 100 0001 0000 0010 0000 1000 = -6.031 497 955 322 265 625 Dec 05 22:53 UTC (GMT)
0 - 1110 0111 - 000 0000 0000 0000 0000 0000 = 20 282 409 603 651 670 423 947 251 286 016 Dec 05 22:53 UTC (GMT)
0 - 1000 0100 - 011 1010 1000 0000 0000 0000 = 46.625 Dec 05 22:52 UTC (GMT)
1 - 0100 0101 - 111 0000 0000 0000 0000 0011 = -0.000 000 000 000 000 006 505 214 275 683 945 433 57 Dec 05 22:51 UTC (GMT)
0 - 1000 1001 - 001 0100 0111 1111 1111 1111 = 1 187.999 877 929 687 5 Dec 05 22:51 UTC (GMT)
0 - 0011 1110 - 001 1111 0010 0000 0000 0000 = 0.000 000 000 000 000 000 033 696 029 432 960 135 36 Dec 05 22:50 UTC (GMT)
0 - 1000 0010 - 101 0111 0000 1010 1000 1111 = 13.440 077 781 677 246 093 75 Dec 05 22:50 UTC (GMT)
1 - 1111 1110 - 100 0001 1000 0000 0000 0000 = -257 205 617 184 381 221 406 886 666 164 246 675 456 Dec 05 22:49 UTC (GMT)
0 - 1111 0110 - 101 0000 0001 0001 1011 0001 = 1 080 356 570 923 271 249 919 304 397 028 851 712 Dec 05 22:49 UTC (GMT)
All base ten decimal numbers converted to 32 bit single precision IEEE 754 binary floating point

How to convert numbers from 32 bit single precision IEEE 754 binary floating point standard to decimal system in base 10

Follow the steps below to convert a number from 32 bit single precision IEEE 754 binary floating point representation to base 10 decimal system:

  • 1. Identify the three elements that make up the binary representation of the number:
    First bit (leftmost) indicates the sign, 1 = negative, 0 = pozitive.
    The next 8 bits contain the exponent.
    The last 23 bits contain the mantissa.
  • 2. Convert the exponent, that is allways a positive integer, from binary (base 2) to decimal (base 10).
  • 3. Adjust the exponent, subtract the excess bits, 2(8 - 1) - 1 = 127, that is due to the 8 bit excess/bias notation.
  • 4. Convert the mantissa, that represents the number's fractional part (the excess beyond the number's integer part, comma delimited), from binary (base 2) to decimal (base 10).
  • 5. Put all the numbers into expression to calculate the single precision floating point decimal value:
    (-1)Sign × (1 + Mantissa) × 2(Exponent adjusted)

Example: convert the number 1 - 1000 0001 - 100 0001 0000 0010 0000 0000 from 32 bit single precision IEEE 754 binary floating point system to base 10 decimal system (float):

  • 1. Identify the elements that make up the binary representation of the number:
    First bit (leftmost) indicates the sign, 1 = negative, 0 = pozitive.
    The next 8 bits contain the exponent: 1000 0001
    The last 23 bits contain the mantissa: 100 0001 0000 0010 0000 0000
  • 2. Convert the exponent, that is allways a positive integer, from binary (base 2) to decimal (base 10):
    1000 0001(2) =
    1 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 1 × 20 =
    128 + 0 + 0 + 0 + 0 + 0 + 0 + 1 =
    128 + 1 =
    129(10)
  • 3. Adjust the exponent, subtract the excess bits, 2(8 - 1) - 1 = 127, that is due to the 8 bit excess/bias notation:
    Exponent adjusted = 129 - 127 = 2
  • 4. Convert the mantissa, that represents the number's fractional part (the excess beyond the number's integer part, comma delimited), from binary (base 2) to decimal (base 10):
    100 0001 0000 0010 0000 0000(2) =
    1 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 0 × 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 + 1 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 0 × 2-18 + 0 × 2-19 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 0 × 2-23 =
    0.5 + 0 + 0 + 0 + 0 + 0 + 0.007 812 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 061 035 156 25 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =
    0.5 + 0.007 812 5 + 0.000 061 035 156 25 =
    0.507 873 535 156 25(10)
  • 5. Put all the numbers into expression to calculate the single precision floating point decimal value:
    (-1)Sign × (1 + Mantissa) × 2(Exponent adjusted) =
    (-1)1 × (1 + 0.507 873 535 156 25) × 22 =
    -1.507 873 535 156 25 × 22 =
    -6.031 494 140 625
  • 1 - 1000 0001 - 100 0001 0000 0010 0000 0000 converted from 32 bit single precision IEEE 754 binary floating point representation to decimal number (float) in decimal system (in base 10) = -6.031 494 140 625(10)