Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 1111 1111 - 011 1111 1111 1111 1111 1111 Converted and Written as a Base Ten Decimal System Number (as a Float)

0 - 1111 1111 - 011 1111 1111 1111 1111 1111: 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.
0

The next 8 bits contain the exponent:
1111 1111

The last 23 bits contain the mantissa:
011 1111 1111 1111 1111 1111


2. Reserved bitpattern.

We notice that all the bits that make up the exponent are set on 1 (set) and the first bit of the mantissa is 0 (clear) while there is at least one bit of the mantissa set on 1.

This is one of the reserved bitpatterns of the special values of: SNaN (Signalling Not a Number).

A SNaN signals an exception when used in operations. A SNaN is a category of NaN (Not A Number).

In general, NaN is used to represent a value that is not a number.


0 - 1111 1111 - 011 1111 1111 1111 1111 1111 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) = SNaN, Signalling Not a Number

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 0 - 1111 1111 - 011 1111 1111 1111 1111 1110 converted from 32 bit single precision IEEE 754 binary floating point standard representation to decimal system written in base ten (float) = ?

» Number 0 - 1111 1111 - 100 0000 0000 0000 0000 0000 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 - 1000 1101 - 111 0100 0100 0100 0000 0000 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Apr 19 19:59 UTC (GMT)
The number 0 - 1111 0111 - 100 0111 1111 1111 1111 1011 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Apr 19 19:58 UTC (GMT)
The number 1 - 1111 1011 - 000 1010 0101 1111 1110 0100 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Apr 19 19:58 UTC (GMT)
The number 0 - 1110 0000 - 110 1111 1111 1111 1100 1110 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Apr 19 19:58 UTC (GMT)
The number 0 - 1111 0110 - 101 0000 0000 0000 0001 1011 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Apr 19 19:58 UTC (GMT)
The number 0 - 0100 1000 - 010 0100 1101 0010 0010 0000 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Apr 19 19:58 UTC (GMT)
The number 0 - 1000 0101 - 010 1010 0111 1111 1011 0101 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Apr 19 19:57 UTC (GMT)
The number 0 - 1000 1011 - 001 1100 0011 0100 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 = ? Apr 19 19:57 UTC (GMT)
The number 0 - 1010 0011 - 011 1010 0100 0011 1100 0010 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Apr 19 19:56 UTC (GMT)
The number 0 - 0011 0100 - 001 1001 1111 1111 1110 0111 converted from 32 bit single precision IEEE 754 binary floating point system and written as a decimal number (float) written in base ten = ? Apr 19 19:56 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)