32bit IEEE 754: Decimal ↗ Single Precision Floating Point Binary: 0.694 205 Convert the Number to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard, From a Base 10 Decimal System Number

Number 0.694 205(10) converted and written in 32 bit single precision IEEE 754 binary floating point representation (1 bit for sign, 8 bits for exponent, 23 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 0.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 0 ÷ 2 = 0 + 0;

2. Construct the base 2 representation of the integer part of the number.

Take all the remainders starting from the bottom of the list constructed above.


0(10) =


0(2)


3. Convert to binary (base 2) the fractional part: 0.694 205.

Multiply it repeatedly by 2.


Keep track of each integer part of the results.


Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.694 205 × 2 = 1 + 0.388 41;
  • 2) 0.388 41 × 2 = 0 + 0.776 82;
  • 3) 0.776 82 × 2 = 1 + 0.553 64;
  • 4) 0.553 64 × 2 = 1 + 0.107 28;
  • 5) 0.107 28 × 2 = 0 + 0.214 56;
  • 6) 0.214 56 × 2 = 0 + 0.429 12;
  • 7) 0.429 12 × 2 = 0 + 0.858 24;
  • 8) 0.858 24 × 2 = 1 + 0.716 48;
  • 9) 0.716 48 × 2 = 1 + 0.432 96;
  • 10) 0.432 96 × 2 = 0 + 0.865 92;
  • 11) 0.865 92 × 2 = 1 + 0.731 84;
  • 12) 0.731 84 × 2 = 1 + 0.463 68;
  • 13) 0.463 68 × 2 = 0 + 0.927 36;
  • 14) 0.927 36 × 2 = 1 + 0.854 72;
  • 15) 0.854 72 × 2 = 1 + 0.709 44;
  • 16) 0.709 44 × 2 = 1 + 0.418 88;
  • 17) 0.418 88 × 2 = 0 + 0.837 76;
  • 18) 0.837 76 × 2 = 1 + 0.675 52;
  • 19) 0.675 52 × 2 = 1 + 0.351 04;
  • 20) 0.351 04 × 2 = 0 + 0.702 08;
  • 21) 0.702 08 × 2 = 1 + 0.404 16;
  • 22) 0.404 16 × 2 = 0 + 0.808 32;
  • 23) 0.808 32 × 2 = 1 + 0.616 64;
  • 24) 0.616 64 × 2 = 1 + 0.233 28;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (losing precision...)


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.694 205(10) =


0.1011 0001 1011 0111 0110 1011(2)


5. Positive number before normalization:

0.694 205(10) =


0.1011 0001 1011 0111 0110 1011(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 1 positions to the right, so that only one non zero digit remains to the left of it:


0.694 205(10) =


0.1011 0001 1011 0111 0110 1011(2) =


0.1011 0001 1011 0111 0110 1011(2) × 20 =


1.0110 0011 0110 1110 1101 011(2) × 2-1


7. Up to this moment, there are the following elements that would feed into the 32 bit single precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)


Exponent (unadjusted): -1


Mantissa (not normalized):
1.0110 0011 0110 1110 1101 011


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =


Exponent (unadjusted) + 2(8-1) - 1 =


-1 + 2(8-1) - 1 =


(-1 + 127)(10) =


126(10)


9. Convert the adjusted exponent from the decimal (base 10) to 8 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =


126(10) =


0111 1110(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.


b) Adjust its length to 23 bits, only if necessary (not the case here).


Mantissa (normalized) =


1. 011 0001 1011 0111 0110 1011 =


011 0001 1011 0111 0110 1011


12. The three elements that make up the number's 32 bit single precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)


Exponent (8 bits) =
0111 1110


Mantissa (23 bits) =
011 0001 1011 0111 0110 1011


The base ten decimal number 0.694 205 converted and written in 32 bit single precision IEEE 754 binary floating point representation:
0 - 0111 1110 - 011 0001 1011 0111 0110 1011

The latest decimal numbers converted from base ten to 32 bit single precision IEEE 754 floating point binary standard representation