24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 768 6 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 768 6(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 768 6(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 24.
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;
  • 24 ÷ 2 = 12 + 0;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

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.

24(10) =

1 1000(2)


3. Convert to binary (base 2) the fractional part: 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 768 6.

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) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 768 6 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 537 2;
  • 2) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 537 2 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 074 4;
  • 3) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 074 4 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 148 8;
  • 4) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 148 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 297 6;
  • 5) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 297 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 595 2;
  • 6) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 595 2 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 190 4;
  • 7) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 190 4 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 554 380 8;
  • 8) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 554 380 8 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 108 761 6;
  • 9) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 108 761 6 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 217 523 2;
  • 10) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 217 523 2 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 435 046 4;
  • 11) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 435 046 4 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 870 092 8;
  • 12) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 870 092 8 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 740 185 6;
  • 13) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 740 185 6 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 480 371 2;
  • 14) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 480 371 2 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 110 960 742 4;
  • 15) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 110 960 742 4 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 221 921 484 8;
  • 16) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 221 921 484 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 443 842 969 6;
  • 17) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 443 842 969 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 887 685 939 2;
  • 18) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 887 685 939 2 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 775 371 878 4;
  • 19) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 775 371 878 4 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 550 743 756 8;
  • 20) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 550 743 756 8 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 101 487 513 6;
  • 21) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 101 487 513 6 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 202 975 027 2;
  • 22) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 202 975 027 2 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 405 950 054 4;
  • 23) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 405 950 054 4 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 811 900 108 8;
  • 24) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 811 900 108 8 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 623 800 217 6;
  • 25) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 623 800 217 6 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 247 600 435 2;
  • 26) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 247 600 435 2 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 110 495 200 870 4;
  • 27) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 110 495 200 870 4 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 220 990 401 740 8;
  • 28) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 220 990 401 740 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 441 980 803 481 6;
  • 29) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 441 980 803 481 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 883 961 606 963 2;
  • 30) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 883 961 606 963 2 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 767 923 213 926 4;
  • 31) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 767 923 213 926 4 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 535 846 427 852 8;
  • 32) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 535 846 427 852 8 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 071 692 855 705 6;
  • 33) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 071 692 855 705 6 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 143 385 711 411 2;
  • 34) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 143 385 711 411 2 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 286 771 422 822 4;
  • 35) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 286 771 422 822 4 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 573 542 845 644 8;
  • 36) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 573 542 845 644 8 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 147 085 691 289 6;
  • 37) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 147 085 691 289 6 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 554 294 171 382 579 2;
  • 38) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 554 294 171 382 579 2 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 108 588 342 765 158 4;
  • 39) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 108 588 342 765 158 4 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 217 176 685 530 316 8;
  • 40) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 217 176 685 530 316 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 434 353 371 060 633 6;
  • 41) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 434 353 371 060 633 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 868 706 742 121 267 2;
  • 42) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 868 706 742 121 267 2 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 737 413 484 242 534 4;
  • 43) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 737 413 484 242 534 4 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 474 826 968 485 068 8;
  • 44) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 474 826 968 485 068 8 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 110 949 653 936 970 137 6;
  • 45) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 110 949 653 936 970 137 6 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 221 899 307 873 940 275 2;
  • 46) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 221 899 307 873 940 275 2 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 443 798 615 747 880 550 4;
  • 47) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 443 798 615 747 880 550 4 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 887 597 231 495 761 100 8;
  • 48) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 887 597 231 495 761 100 8 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 775 194 462 991 522 201 6;
  • 49) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 775 194 462 991 522 201 6 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 550 388 925 983 044 403 2;
  • 50) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 550 388 925 983 044 403 2 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 100 777 851 966 088 806 4;
  • 51) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 100 777 851 966 088 806 4 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 201 555 703 932 177 612 8;
  • 52) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 201 555 703 932 177 612 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 403 111 407 864 355 225 6;
  • 53) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 403 111 407 864 355 225 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 806 222 815 728 710 451 2;

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 - the converted number we get in the end will be just a very good approximation of the initial one).


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:


0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 768 6(10) =

0.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0(2)

5. Positive number before normalization:

24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 768 6(10) =

1 1000.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 4 positions to the left, so that only one non zero digit remains to the left of it:


24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 768 6(10) =

1 1000.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0(2) =

1 1000.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0(2) × 20 =

1.1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0(2) × 24


7. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): 4

Mantissa (not normalized):
1.1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

4 + 2(11-1) - 1 =

(4 + 1 023)(10) =

1 027(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 1 027 ÷ 2 = 513 + 1;
  • 513 ÷ 2 = 256 + 1;
  • 256 ÷ 2 = 128 + 0;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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) =

1027(10) =

100 0000 0011(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 52 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. 1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1 1000 =

1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

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

Exponent (11 bits) =
100 0000 0011

Mantissa (52 bits) =
1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001


Decimal number 24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 768 6 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0011 - 1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001

» Convert decimal 24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 764 to 64 bit double precision IEEE 754 binary floating point representation standard

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100