32bit IEEE 754: Decimal ↗ Single Precision Floating Point Binary: 886 179 999 999 Convert the Number to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard, From a Base 10 Decimal System Number

Number 886 179 999 999(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. 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;
  • 886 179 999 999 ÷ 2 = 443 089 999 999 + 1;
  • 443 089 999 999 ÷ 2 = 221 544 999 999 + 1;
  • 221 544 999 999 ÷ 2 = 110 772 499 999 + 1;
  • 110 772 499 999 ÷ 2 = 55 386 249 999 + 1;
  • 55 386 249 999 ÷ 2 = 27 693 124 999 + 1;
  • 27 693 124 999 ÷ 2 = 13 846 562 499 + 1;
  • 13 846 562 499 ÷ 2 = 6 923 281 249 + 1;
  • 6 923 281 249 ÷ 2 = 3 461 640 624 + 1;
  • 3 461 640 624 ÷ 2 = 1 730 820 312 + 0;
  • 1 730 820 312 ÷ 2 = 865 410 156 + 0;
  • 865 410 156 ÷ 2 = 432 705 078 + 0;
  • 432 705 078 ÷ 2 = 216 352 539 + 0;
  • 216 352 539 ÷ 2 = 108 176 269 + 1;
  • 108 176 269 ÷ 2 = 54 088 134 + 1;
  • 54 088 134 ÷ 2 = 27 044 067 + 0;
  • 27 044 067 ÷ 2 = 13 522 033 + 1;
  • 13 522 033 ÷ 2 = 6 761 016 + 1;
  • 6 761 016 ÷ 2 = 3 380 508 + 0;
  • 3 380 508 ÷ 2 = 1 690 254 + 0;
  • 1 690 254 ÷ 2 = 845 127 + 0;
  • 845 127 ÷ 2 = 422 563 + 1;
  • 422 563 ÷ 2 = 211 281 + 1;
  • 211 281 ÷ 2 = 105 640 + 1;
  • 105 640 ÷ 2 = 52 820 + 0;
  • 52 820 ÷ 2 = 26 410 + 0;
  • 26 410 ÷ 2 = 13 205 + 0;
  • 13 205 ÷ 2 = 6 602 + 1;
  • 6 602 ÷ 2 = 3 301 + 0;
  • 3 301 ÷ 2 = 1 650 + 1;
  • 1 650 ÷ 2 = 825 + 0;
  • 825 ÷ 2 = 412 + 1;
  • 412 ÷ 2 = 206 + 0;
  • 206 ÷ 2 = 103 + 0;
  • 103 ÷ 2 = 51 + 1;
  • 51 ÷ 2 = 25 + 1;
  • 25 ÷ 2 = 12 + 1;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the positive number.

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


886 179 999 999(10) =


1100 1110 0101 0100 0111 0001 1011 0000 1111 1111(2)


3. Normalize the binary representation of the number.

Shift the decimal mark 39 positions to the left, so that only one non zero digit remains to the left of it:


886 179 999 999(10) =


1100 1110 0101 0100 0111 0001 1011 0000 1111 1111(2) =


1100 1110 0101 0100 0111 0001 1011 0000 1111 1111(2) × 20 =


1.1001 1100 1010 1000 1110 0011 0110 0001 1111 111(2) × 239


4. 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): 39


Mantissa (not normalized):
1.1001 1100 1010 1000 1110 0011 0110 0001 1111 111


5. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =


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


39 + 2(8-1) - 1 =


(39 + 127)(10) =


166(10)


6. 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;
  • 166 ÷ 2 = 83 + 0;
  • 83 ÷ 2 = 41 + 1;
  • 41 ÷ 2 = 20 + 1;
  • 20 ÷ 2 = 10 + 0;
  • 10 ÷ 2 = 5 + 0;
  • 5 ÷ 2 = 2 + 1;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

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

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


Exponent (adjusted) =


166(10) =


1010 0110(2)


8. 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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =


1. 100 1110 0101 0100 0111 0001 1011 0000 1111 1111 =


100 1110 0101 0100 0111 0001


9. 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) =
1010 0110


Mantissa (23 bits) =
100 1110 0101 0100 0111 0001


The base ten decimal number 886 179 999 999 converted and written in 32 bit single precision IEEE 754 binary floating point representation:
0 - 1010 0110 - 100 1110 0101 0100 0111 0001

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