The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 1111 1110 - 000 0000 0000 0000 0000 0000 Converted and Written as a Base Ten Decimal System Number (Float)

Number 1 - 1111 1110 - 000 0000 0000 0000 0000 0001 converted from 32 bit single precision IEEE 754 binary floating point standard representation to decimal system written in base ten (float) = ?

A number in 32 bit single precision IEEE 754 binary floating point standard representation...

... requires three building elements: the sign (it takes 1 bit and it's either 0 for positive or 1 for negative numbers), the exponent (8 bits) and the mantissa (23 bits)

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₎

Conversions Between Decimal System Numbers (Written in Base Ten) and Binary System Numbers (Base Two and Computer Representation):

1. Integer -> Binary

• 1.1. Unsigned integer -> Unsigned binary

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  1.2. Signed integer -> Signed binary

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  1.3. Signed integer -> Signed binary one's complement

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  1.4. Signed integer -> Signed binary two's complement

2. Decimal -> Binary

• 2.1. Decimal -> 32bit single precision IEEE 754 binary floating point

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  2.2. Decimal -> 64bit double precision IEEE 754 binary floating point

3. Binary -> Integer

• 3.1. Unsigned binary -> Unsigned integer

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  3.2. Signed binary -> Signed integer

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  3.3. Signed binary one's complement -> Signed integer

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  3.4. Signed binary two's complement -> Signed integer

4. Binary -> Decimal

• 4.1. 32bit single precision IEEE 754 binary floating point -> Float

  ---

  4.2. 64bit double precision IEEE 754 binary floating point -> Double