32 bit single precision IEEE 754 binary floating point number 0 - 1000 0101 - 111 0011 1101 0001 1110 1010 converted to decimal base ten (float)

How to convert 32 bit single precision IEEE 754 binary floating point:
0 - 1000 0101 - 111 0011 1101 0001 1110 1010
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:
1000 0101


The last 23 bits contain the mantissa:
111 0011 1101 0001 1110 1010

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

The exponent is allways a positive integer.

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 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)

111 0011 1101 0001 1110 1010(2) =

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


0.5 + 0.25 + 0.125 + 0 + 0 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0 + 0.000 488 281 25 + 0 + 0 + 0 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0 + 0.000 000 953 674 316 406 25 + 0 + 0.000 000 238 418 579 101 562 5 + 0 =


0.5 + 0.25 + 0.125 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 488 281 25 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0.000 000 953 674 316 406 25 + 0.000 000 238 418 579 101 562 5 =


0.904 843 568 801 879 882 812 5(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 + 0.904 843 568 801 879 882 812 5) × 26 =


1.904 843 568 801 879 882 812 5 × 26 =


121.909 988 403 320 312 5

Conclusion:

0 - 1000 0101 - 111 0011 1101 0001 1110 1010
converted from
32 bit single precision IEEE 754 binary floating point
to
base ten decimal system (float) =

121.909 988 403 320 312 5(10)

More operations of this kind:

0 - 1000 0101 - 111 0011 1101 0001 1110 1001 = ?

0 - 1000 0101 - 111 0011 1101 0001 1110 1011 = ?


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 - 1000 0101 - 111 0011 1101 0001 1110 1010 = 121.909 988 403 320 312 5 Oct 30 03:49 UTC (GMT)
0 - 0010 0001 - 000 0000 0000 0010 0000 0000 = 0.000 000 000 000 000 000 000 000 000 050 490 179 41 Oct 30 03:49 UTC (GMT)
1 - 1000 0101 - 111 0000 1001 1111 1111 1111 = -120.312 492 370 605 468 75 Oct 30 03:49 UTC (GMT)
0 - 0111 1101 - 100 1000 0000 0000 0000 0001 = 0.390 625 029 802 322 387 695 312 5 Oct 30 03:49 UTC (GMT)
0 - 1111 1110 - 111 1111 1111 1111 1011 0010 = 340 280 784 892 989 378 633 081 539 546 167 902 208 Oct 30 03:48 UTC (GMT)
0 - 0101 1010 - 010 1010 0000 0000 0000 0001 = 0.000 000 000 009 663 382 073 699 100 516 250 837 24 Oct 30 03:48 UTC (GMT)
0 - 1000 0011 - 010 1010 1000 0000 0000 0011 = 21.312 505 722 045 898 437 5 Oct 30 03:48 UTC (GMT)
0 - 1001 0100 - 100 0001 1111 0011 1111 1110 = 3 177 727.5 Oct 30 03:48 UTC (GMT)
0 - 0100 1000 - 100 0010 0100 0111 1110 1010 = 0.000 000 000 000 000 042 127 957 872 745 976 997 46 Oct 30 03:47 UTC (GMT)
1 - 1111 1000 - 111 1011 1111 1111 1111 1110 = -5 233 834 599 577 806 135 444 039 551 502 254 080 Oct 30 03:47 UTC (GMT)
0 - 0001 0111 - 010 1111 1100 1011 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 067 712 97 Oct 30 03:47 UTC (GMT)
0 - 1000 0010 - 100 0011 1111 1111 1111 1111 = 12.249 999 046 325 683 593 75 Oct 30 03:47 UTC (GMT)
1 - 1000 0001 - 100 0101 0001 1110 1011 1000 = -6.159 999 847 412 109 375 Oct 30 03:46 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)