32 bit single precision IEEE 754 binary floating point number 0 - 0111 1011 - 110 0011 1000 1110 0011 1001 converted to decimal base ten (float)

How to convert 32 bit single precision IEEE 754 binary floating point:
0 - 0111 1011 - 110 0011 1000 1110 0011 1001
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:
0111 1011


The last 23 bits contain the mantissa:
110 0011 1000 1110 0011 1001

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

The exponent is allways a positive integer.

0111 1011(2) =


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


0 + 64 + 32 + 16 + 8 + 0 + 2 + 1 =


64 + 32 + 16 + 8 + 2 + 1 =


123(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 = 123 - 127 = -4


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)

110 0011 1000 1110 0011 1001(2) =

1 × 2-1 + 1 × 2-2 + 0 × 2-3 + 0 × 2-4 + 0 × 2-5 + 1 × 2-6 + 1 × 2-7 + 1 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 1 × 2-12 + 1 × 2-13 + 1 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 1 × 2-18 + 1 × 2-19 + 1 × 2-20 + 0 × 2-21 + 0 × 2-22 + 1 × 2-23 =


0.5 + 0.25 + 0 + 0 + 0 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0 + 0 + 0 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0 + 0 + 0 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 0 + 0 + 0.000 000 119 209 289 550 781 25 =


0.5 + 0.25 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 0.000 000 119 209 289 550 781 25 =


0.777 777 791 023 254 394 531 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)0 × (1 + 0.777 777 791 023 254 394 531 25) × 2-4 =


1.777 777 791 023 254 394 531 25 × 2-4 =


0.111 111 111 938 953 399 658 203 125

Conclusion:

0 - 0111 1011 - 110 0011 1000 1110 0011 1001
converted from
32 bit single precision IEEE 754 binary floating point
to
base ten decimal system (float) =

0.111 111 111 938 953 399 658 203 125(10)

More operations of this kind:

0 - 0111 1011 - 110 0011 1000 1110 0011 1000 = ?

0 - 0111 1011 - 110 0011 1000 1110 0011 1010 = ?


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 - 0111 1011 - 110 0011 1000 1110 0011 1001 = ? Jan 24 12:51 UTC (GMT)
0 - 0000 1111 - 110 1000 0000 0000 0000 0000 = ? Jan 24 12:50 UTC (GMT)
1 - 0011 0101 - 110 0010 1000 1001 1011 1111 = ? Jan 24 12:49 UTC (GMT)
1 - 1000 1011 - 000 0000 0000 0110 0000 0010 = ? Jan 24 12:49 UTC (GMT)
0 - 0111 1101 - 011 0101 1111 1111 1111 1111 = ? Jan 24 12:49 UTC (GMT)
1 - 1111 1111 - 111 1111 1111 1111 1111 1111 = ? Jan 24 12:49 UTC (GMT)
0 - 1011 1100 - 110 0000 0000 0000 0000 0001 = ? Jan 24 12:49 UTC (GMT)
1 - 1000 0100 - 101 0000 1111 0000 0000 0001 = ? Jan 24 12:48 UTC (GMT)
1 - 1000 0011 - 100 1000 0000 0000 0000 0000 = ? Jan 24 12:47 UTC (GMT)
0 - 1000 0010 - 001 0010 0111 1101 0111 0010 = ? Jan 24 12:47 UTC (GMT)
0 - 0111 1011 - 111 1100 1100 1100 1100 1111 = ? Jan 24 12:47 UTC (GMT)
0 - 0111 1010 - 111 1000 0000 0000 0000 0000 = ? Jan 24 12:46 UTC (GMT)
0 - 0111 1011 - 111 0001 1111 1111 1111 1001 = ? Jan 24 12:46 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)