32 bit single precision IEEE 754 binary floating point number 1 - 1000 0101 - 001 0010 0100 1001 0010 0100 converted to decimal base ten (float)

How to convert 32 bit single precision IEEE 754 binary floating point:
1 - 1000 0101 - 001 0010 0100 1001 0010 0100.

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:
1000 0101


The last 23 bits contain the mantissa:
001 0010 0100 1001 0010 0100

2. Convert the exponent, that is allways a positive integer, from binary (base 2) to decimal (base 10):

1000 0101(2) =


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


128 + 0 + 0 + 0 + 0 + 4 + 0 + 1 =


128 + 4 + 1 =


133(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 = 133 - 127 = 6

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):

001 0010 0100 1001 0010 0100(2) =

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


0 + 0 + 0.125 + 0 + 0 + 0.015 625 + 0 + 0 + 0.001 953 125 + 0 + 0 + 0.000 244 140 625 + 0 + 0 + 0.000 030 517 578 125 + 0 + 0 + 0.000 003 814 697 265 625 + 0 + 0 + 0.000 000 476 837 158 203 125 + 0 + 0 =


0.125 + 0.015 625 + 0.001 953 125 + 0.000 244 140 625 + 0.000 030 517 578 125 + 0.000 003 814 697 265 625 + 0.000 000 476 837 158 203 125 =


0.142 857 074 737 548 828 125(10)

Conclusion:

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 + 0.142 857 074 737 548 828 125) × 26 =


-1.142 857 074 737 548 828 125 × 26 =


-73.142 852 783 203 125

1 - 1000 0101 - 001 0010 0100 1001 0010 0100
converted from
32 bit single precision IEEE 754 binary floating point
to
base ten decimal system (float) =


-73.142 852 783 203 125(10)

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)

1 - 1000 0101 - 001 0010 0100 1001 0010 0100 = -73.142 852 783 203 125 Feb 23 11:09 UTC (GMT)
1 - 0001 0000 - 100 1000 0000 0000 0000 0000 = -0.000 000 000 000 000 000 000 000 000 000 000 601 84 Feb 23 11:08 UTC (GMT)
1 - 0011 1100 - 001 0000 0000 0000 0000 0000 = -0.000 000 000 000 000 000 007 623 296 525 288 703 04 Feb 23 11:08 UTC (GMT)
0 - 0101 1100 - 111 0111 0110 0011 1101 1111 = 0.000 000 000 056 250 001 184 698 916 745 219 321 45 Feb 23 11:08 UTC (GMT)
1 - 0001 0010 - 101 0000 0000 0000 0000 0000 = -0.000 000 000 000 000 000 000 000 000 000 002 503 7 Feb 23 11:06 UTC (GMT)
1 - 0000 1100 - 010 1000 0000 0000 0000 0000 = -0.000 000 000 000 000 000 000 000 000 000 000 031 59 Feb 23 11:05 UTC (GMT)
1 - 1101 0101 - 110 1000 0000 0000 0000 0000 = -140 235 395 075 296 984 265 916 416 Feb 23 11:05 UTC (GMT)
0 - 1000 0011 - 101 1100 0111 1010 1110 0001 = 27.559 999 465 942 382 812 5 Feb 23 11:05 UTC (GMT)
0 - 0111 0011 - 101 1011 0100 0000 0000 0000 = 0.000 418 186 187 744 140 625 Feb 23 11:05 UTC (GMT)
1 - 1000 0110 - 000 1001 0010 0100 1001 0000 = -137.142 822 265 625 Feb 23 11:05 UTC (GMT)
0 - 1111 1110 - 100 1100 1100 1100 1100 1100 = 272 225 877 310 823 087 849 363 346 787 613 540 352 Feb 23 11:05 UTC (GMT)
0 - 0111 1110 - 001 0011 1010 1011 1110 1101 = 0.576 842 129 230 499 267 578 125 Feb 23 11:05 UTC (GMT)
0 - 1111 1101 - 000 0000 0000 0000 0000 1000 = 85 070 672 859 873 030 472 525 347 646 947 196 928 Feb 23 11:05 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)