Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 1000 0010 - 001 0000 0000 0000 0001 0001 Converted and Written as a Base Ten Decimal System Number (as a Float)

1 - 1000 0010 - 001 0000 0000 0000 0001 0001: 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 0010

The last 23 bits contain the mantissa:
001 0000 0000 0000 0001 0001


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

The exponent is allways a positive integer.

1000 0010(2) =

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

128 + 0 + 0 + 0 + 0 + 0 + 2 + 0 =

128 + 2 =

130(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 = 130 - 127 = 3


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).


001 0000 0000 0000 0001 0001(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 + 1 × 2-19 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 1 × 2-23 =

0 + 0 + 0.125 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 001 907 348 632 812 5 + 0 + 0 + 0 + 0.000 000 119 209 289 550 781 25 =

0.125 + 0.000 001 907 348 632 812 5 + 0.000 000 119 209 289 550 781 25 =

0.125 002 026 557 922 363 281 25(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 + 0.125 002 026 557 922 363 281 25) × 23 =

-1.125 002 026 557 922 363 281 25 × 23 =

-9.000 016 212 463 378 906 25

1 - 1000 0010 - 001 0000 0000 0000 0001 0001 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) = -9.000 016 212 463 378 906 25(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:

» Number 1 - 1000 0010 - 001 0000 0000 0000 0001 0000 converted from 32 bit single precision IEEE 754 binary floating point standard representation to decimal system written in base ten (float) = ?

» Number 1 - 1000 0010 - 001 0000 0000 0000 0001 0010 converted from 32 bit single precision IEEE 754 binary floating point standard representation to decimal system written in base ten (float) = ?

The latest 32 bit single precision IEEE 754 floating point binary standard numbers converted and written as decimal system numbers (in base ten, float)

The number 0 - 0000 0001 - 000 0001 0000 0000 0000 0100 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Sep 14 01:25 UTC (GMT)
The number 1 - 0000 1101 - 010 1101 0100 0000 0100 0011 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Sep 14 01:25 UTC (GMT)
The number 0 - 0000 0000 - 000 0000 0000 1100 1100 0100 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Sep 14 01:25 UTC (GMT)
The number 0 - 1000 0000 - 000 0001 0100 1000 0110 1001 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Sep 14 01:25 UTC (GMT)
The number 0 - 1000 0001 - 101 1101 0101 1100 0010 1000 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Sep 14 01:25 UTC (GMT)
The number 1 - 1000 0011 - 100 1001 1001 0110 0100 0111 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Sep 14 01:25 UTC (GMT)
The number 1 - 1000 0101 - 110 0010 1111 1111 1001 1110 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Sep 14 01:25 UTC (GMT)
The number 1 - 0111 1110 - 101 1111 1001 1111 1100 0110 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Sep 14 01:24 UTC (GMT)
The number 1 - 1000 0100 - 110 0010 0111 0000 0111 0100 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Sep 14 01:24 UTC (GMT)
The number 0 - 1010 1010 - 101 0101 0101 0101 0100 1110 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Sep 14 01:24 UTC (GMT)
All 32 bit single precision IEEE 754 binary floating point representation numbers converted to base ten decimal numbers (float)

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)