24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 804 1 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 804 1(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
24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 804 1(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: 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 804 1.

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) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 804 1 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 608 2;
  • 2) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 608 2 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 216 4;
  • 3) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 216 4 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 432 8;
  • 4) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 432 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 865 6;
  • 5) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 865 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 889 731 2;
  • 6) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 889 731 2 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 779 462 4;
  • 7) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 779 462 4 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 558 924 8;
  • 8) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 558 924 8 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 117 849 6;
  • 9) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 117 849 6 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 235 699 2;
  • 10) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 235 699 2 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 471 398 4;
  • 11) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 471 398 4 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 942 796 8;
  • 12) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 942 796 8 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 885 593 6;
  • 13) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 885 593 6 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 771 187 2;
  • 14) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 771 187 2 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 542 374 4;
  • 15) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 542 374 4 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 223 084 748 8;
  • 16) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 223 084 748 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 446 169 497 6;
  • 17) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 446 169 497 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 892 338 995 2;
  • 18) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 892 338 995 2 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 784 677 990 4;
  • 19) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 784 677 990 4 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 569 355 980 8;
  • 20) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 569 355 980 8 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 138 711 961 6;
  • 21) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 138 711 961 6 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 277 423 923 2;
  • 22) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 277 423 923 2 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 554 847 846 4;
  • 23) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 554 847 846 4 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 889 109 695 692 8;
  • 24) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 889 109 695 692 8 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 778 219 391 385 6;
  • 25) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 778 219 391 385 6 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 556 438 782 771 2;
  • 26) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 556 438 782 771 2 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 112 877 565 542 4;
  • 27) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 112 877 565 542 4 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 225 755 131 084 8;
  • 28) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 225 755 131 084 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 451 510 262 169 6;
  • 29) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 451 510 262 169 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 903 020 524 339 2;
  • 30) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 903 020 524 339 2 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 806 041 048 678 4;
  • 31) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 806 041 048 678 4 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 612 082 097 356 8;
  • 32) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 612 082 097 356 8 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 224 164 194 713 6;
  • 33) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 224 164 194 713 6 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 448 328 389 427 2;
  • 34) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 448 328 389 427 2 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 896 656 778 854 4;
  • 35) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 896 656 778 854 4 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 889 793 313 557 708 8;
  • 36) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 889 793 313 557 708 8 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 779 586 627 115 417 6;
  • 37) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 779 586 627 115 417 6 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 559 173 254 230 835 2;
  • 38) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 559 173 254 230 835 2 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 118 346 508 461 670 4;
  • 39) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 118 346 508 461 670 4 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 236 693 016 923 340 8;
  • 40) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 236 693 016 923 340 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 473 386 033 846 681 6;
  • 41) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 473 386 033 846 681 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 946 772 067 693 363 2;
  • 42) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 946 772 067 693 363 2 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 893 544 135 386 726 4;
  • 43) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 893 544 135 386 726 4 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 787 088 270 773 452 8;
  • 44) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 787 088 270 773 452 8 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 574 176 541 546 905 6;
  • 45) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 574 176 541 546 905 6 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 223 148 353 083 093 811 2;
  • 46) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 223 148 353 083 093 811 2 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 446 296 706 166 187 622 4;
  • 47) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 446 296 706 166 187 622 4 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 892 593 412 332 375 244 8;
  • 48) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 892 593 412 332 375 244 8 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 785 186 824 664 750 489 6;
  • 49) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 785 186 824 664 750 489 6 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 570 373 649 329 500 979 2;
  • 50) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 570 373 649 329 500 979 2 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 140 747 298 659 001 958 4;
  • 51) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 140 747 298 659 001 958 4 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 281 494 597 318 003 916 8;
  • 52) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 281 494 597 318 003 916 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 562 989 194 636 007 833 6;
  • 53) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 562 989 194 636 007 833 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 889 125 978 389 272 015 667 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:


0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 804 1(10) =

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

5. Positive number before normalization:

24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 804 1(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:


24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 804 1(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 24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 804 1 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 24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 800 1 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