0.000 000 000 000 000 000 008 485 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 485(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
0.000 000 000 000 000 000 008 485(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: 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: 0.000 000 000 000 000 000 008 485.

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) 0.000 000 000 000 000 000 008 485 × 2 = 0 + 0.000 000 000 000 000 000 016 97;
  • 2) 0.000 000 000 000 000 000 016 97 × 2 = 0 + 0.000 000 000 000 000 000 033 94;
  • 3) 0.000 000 000 000 000 000 033 94 × 2 = 0 + 0.000 000 000 000 000 000 067 88;
  • 4) 0.000 000 000 000 000 000 067 88 × 2 = 0 + 0.000 000 000 000 000 000 135 76;
  • 5) 0.000 000 000 000 000 000 135 76 × 2 = 0 + 0.000 000 000 000 000 000 271 52;
  • 6) 0.000 000 000 000 000 000 271 52 × 2 = 0 + 0.000 000 000 000 000 000 543 04;
  • 7) 0.000 000 000 000 000 000 543 04 × 2 = 0 + 0.000 000 000 000 000 001 086 08;
  • 8) 0.000 000 000 000 000 001 086 08 × 2 = 0 + 0.000 000 000 000 000 002 172 16;
  • 9) 0.000 000 000 000 000 002 172 16 × 2 = 0 + 0.000 000 000 000 000 004 344 32;
  • 10) 0.000 000 000 000 000 004 344 32 × 2 = 0 + 0.000 000 000 000 000 008 688 64;
  • 11) 0.000 000 000 000 000 008 688 64 × 2 = 0 + 0.000 000 000 000 000 017 377 28;
  • 12) 0.000 000 000 000 000 017 377 28 × 2 = 0 + 0.000 000 000 000 000 034 754 56;
  • 13) 0.000 000 000 000 000 034 754 56 × 2 = 0 + 0.000 000 000 000 000 069 509 12;
  • 14) 0.000 000 000 000 000 069 509 12 × 2 = 0 + 0.000 000 000 000 000 139 018 24;
  • 15) 0.000 000 000 000 000 139 018 24 × 2 = 0 + 0.000 000 000 000 000 278 036 48;
  • 16) 0.000 000 000 000 000 278 036 48 × 2 = 0 + 0.000 000 000 000 000 556 072 96;
  • 17) 0.000 000 000 000 000 556 072 96 × 2 = 0 + 0.000 000 000 000 001 112 145 92;
  • 18) 0.000 000 000 000 001 112 145 92 × 2 = 0 + 0.000 000 000 000 002 224 291 84;
  • 19) 0.000 000 000 000 002 224 291 84 × 2 = 0 + 0.000 000 000 000 004 448 583 68;
  • 20) 0.000 000 000 000 004 448 583 68 × 2 = 0 + 0.000 000 000 000 008 897 167 36;
  • 21) 0.000 000 000 000 008 897 167 36 × 2 = 0 + 0.000 000 000 000 017 794 334 72;
  • 22) 0.000 000 000 000 017 794 334 72 × 2 = 0 + 0.000 000 000 000 035 588 669 44;
  • 23) 0.000 000 000 000 035 588 669 44 × 2 = 0 + 0.000 000 000 000 071 177 338 88;
  • 24) 0.000 000 000 000 071 177 338 88 × 2 = 0 + 0.000 000 000 000 142 354 677 76;
  • 25) 0.000 000 000 000 142 354 677 76 × 2 = 0 + 0.000 000 000 000 284 709 355 52;
  • 26) 0.000 000 000 000 284 709 355 52 × 2 = 0 + 0.000 000 000 000 569 418 711 04;
  • 27) 0.000 000 000 000 569 418 711 04 × 2 = 0 + 0.000 000 000 001 138 837 422 08;
  • 28) 0.000 000 000 001 138 837 422 08 × 2 = 0 + 0.000 000 000 002 277 674 844 16;
  • 29) 0.000 000 000 002 277 674 844 16 × 2 = 0 + 0.000 000 000 004 555 349 688 32;
  • 30) 0.000 000 000 004 555 349 688 32 × 2 = 0 + 0.000 000 000 009 110 699 376 64;
  • 31) 0.000 000 000 009 110 699 376 64 × 2 = 0 + 0.000 000 000 018 221 398 753 28;
  • 32) 0.000 000 000 018 221 398 753 28 × 2 = 0 + 0.000 000 000 036 442 797 506 56;
  • 33) 0.000 000 000 036 442 797 506 56 × 2 = 0 + 0.000 000 000 072 885 595 013 12;
  • 34) 0.000 000 000 072 885 595 013 12 × 2 = 0 + 0.000 000 000 145 771 190 026 24;
  • 35) 0.000 000 000 145 771 190 026 24 × 2 = 0 + 0.000 000 000 291 542 380 052 48;
  • 36) 0.000 000 000 291 542 380 052 48 × 2 = 0 + 0.000 000 000 583 084 760 104 96;
  • 37) 0.000 000 000 583 084 760 104 96 × 2 = 0 + 0.000 000 001 166 169 520 209 92;
  • 38) 0.000 000 001 166 169 520 209 92 × 2 = 0 + 0.000 000 002 332 339 040 419 84;
  • 39) 0.000 000 002 332 339 040 419 84 × 2 = 0 + 0.000 000 004 664 678 080 839 68;
  • 40) 0.000 000 004 664 678 080 839 68 × 2 = 0 + 0.000 000 009 329 356 161 679 36;
  • 41) 0.000 000 009 329 356 161 679 36 × 2 = 0 + 0.000 000 018 658 712 323 358 72;
  • 42) 0.000 000 018 658 712 323 358 72 × 2 = 0 + 0.000 000 037 317 424 646 717 44;
  • 43) 0.000 000 037 317 424 646 717 44 × 2 = 0 + 0.000 000 074 634 849 293 434 88;
  • 44) 0.000 000 074 634 849 293 434 88 × 2 = 0 + 0.000 000 149 269 698 586 869 76;
  • 45) 0.000 000 149 269 698 586 869 76 × 2 = 0 + 0.000 000 298 539 397 173 739 52;
  • 46) 0.000 000 298 539 397 173 739 52 × 2 = 0 + 0.000 000 597 078 794 347 479 04;
  • 47) 0.000 000 597 078 794 347 479 04 × 2 = 0 + 0.000 001 194 157 588 694 958 08;
  • 48) 0.000 001 194 157 588 694 958 08 × 2 = 0 + 0.000 002 388 315 177 389 916 16;
  • 49) 0.000 002 388 315 177 389 916 16 × 2 = 0 + 0.000 004 776 630 354 779 832 32;
  • 50) 0.000 004 776 630 354 779 832 32 × 2 = 0 + 0.000 009 553 260 709 559 664 64;
  • 51) 0.000 009 553 260 709 559 664 64 × 2 = 0 + 0.000 019 106 521 419 119 329 28;
  • 52) 0.000 019 106 521 419 119 329 28 × 2 = 0 + 0.000 038 213 042 838 238 658 56;
  • 53) 0.000 038 213 042 838 238 658 56 × 2 = 0 + 0.000 076 426 085 676 477 317 12;
  • 54) 0.000 076 426 085 676 477 317 12 × 2 = 0 + 0.000 152 852 171 352 954 634 24;
  • 55) 0.000 152 852 171 352 954 634 24 × 2 = 0 + 0.000 305 704 342 705 909 268 48;
  • 56) 0.000 305 704 342 705 909 268 48 × 2 = 0 + 0.000 611 408 685 411 818 536 96;
  • 57) 0.000 611 408 685 411 818 536 96 × 2 = 0 + 0.001 222 817 370 823 637 073 92;
  • 58) 0.001 222 817 370 823 637 073 92 × 2 = 0 + 0.002 445 634 741 647 274 147 84;
  • 59) 0.002 445 634 741 647 274 147 84 × 2 = 0 + 0.004 891 269 483 294 548 295 68;
  • 60) 0.004 891 269 483 294 548 295 68 × 2 = 0 + 0.009 782 538 966 589 096 591 36;
  • 61) 0.009 782 538 966 589 096 591 36 × 2 = 0 + 0.019 565 077 933 178 193 182 72;
  • 62) 0.019 565 077 933 178 193 182 72 × 2 = 0 + 0.039 130 155 866 356 386 365 44;
  • 63) 0.039 130 155 866 356 386 365 44 × 2 = 0 + 0.078 260 311 732 712 772 730 88;
  • 64) 0.078 260 311 732 712 772 730 88 × 2 = 0 + 0.156 520 623 465 425 545 461 76;
  • 65) 0.156 520 623 465 425 545 461 76 × 2 = 0 + 0.313 041 246 930 851 090 923 52;
  • 66) 0.313 041 246 930 851 090 923 52 × 2 = 0 + 0.626 082 493 861 702 181 847 04;
  • 67) 0.626 082 493 861 702 181 847 04 × 2 = 1 + 0.252 164 987 723 404 363 694 08;
  • 68) 0.252 164 987 723 404 363 694 08 × 2 = 0 + 0.504 329 975 446 808 727 388 16;
  • 69) 0.504 329 975 446 808 727 388 16 × 2 = 1 + 0.008 659 950 893 617 454 776 32;
  • 70) 0.008 659 950 893 617 454 776 32 × 2 = 0 + 0.017 319 901 787 234 909 552 64;
  • 71) 0.017 319 901 787 234 909 552 64 × 2 = 0 + 0.034 639 803 574 469 819 105 28;
  • 72) 0.034 639 803 574 469 819 105 28 × 2 = 0 + 0.069 279 607 148 939 638 210 56;
  • 73) 0.069 279 607 148 939 638 210 56 × 2 = 0 + 0.138 559 214 297 879 276 421 12;
  • 74) 0.138 559 214 297 879 276 421 12 × 2 = 0 + 0.277 118 428 595 758 552 842 24;
  • 75) 0.277 118 428 595 758 552 842 24 × 2 = 0 + 0.554 236 857 191 517 105 684 48;
  • 76) 0.554 236 857 191 517 105 684 48 × 2 = 1 + 0.108 473 714 383 034 211 368 96;
  • 77) 0.108 473 714 383 034 211 368 96 × 2 = 0 + 0.216 947 428 766 068 422 737 92;
  • 78) 0.216 947 428 766 068 422 737 92 × 2 = 0 + 0.433 894 857 532 136 845 475 84;
  • 79) 0.433 894 857 532 136 845 475 84 × 2 = 0 + 0.867 789 715 064 273 690 951 68;
  • 80) 0.867 789 715 064 273 690 951 68 × 2 = 1 + 0.735 579 430 128 547 381 903 36;
  • 81) 0.735 579 430 128 547 381 903 36 × 2 = 1 + 0.471 158 860 257 094 763 806 72;
  • 82) 0.471 158 860 257 094 763 806 72 × 2 = 0 + 0.942 317 720 514 189 527 613 44;
  • 83) 0.942 317 720 514 189 527 613 44 × 2 = 1 + 0.884 635 441 028 379 055 226 88;
  • 84) 0.884 635 441 028 379 055 226 88 × 2 = 1 + 0.769 270 882 056 758 110 453 76;
  • 85) 0.769 270 882 056 758 110 453 76 × 2 = 1 + 0.538 541 764 113 516 220 907 52;
  • 86) 0.538 541 764 113 516 220 907 52 × 2 = 1 + 0.077 083 528 227 032 441 815 04;
  • 87) 0.077 083 528 227 032 441 815 04 × 2 = 0 + 0.154 167 056 454 064 883 630 08;
  • 88) 0.154 167 056 454 064 883 630 08 × 2 = 0 + 0.308 334 112 908 129 767 260 16;
  • 89) 0.308 334 112 908 129 767 260 16 × 2 = 0 + 0.616 668 225 816 259 534 520 32;
  • 90) 0.616 668 225 816 259 534 520 32 × 2 = 1 + 0.233 336 451 632 519 069 040 64;
  • 91) 0.233 336 451 632 519 069 040 64 × 2 = 0 + 0.466 672 903 265 038 138 081 28;
  • 92) 0.466 672 903 265 038 138 081 28 × 2 = 0 + 0.933 345 806 530 076 276 162 56;
  • 93) 0.933 345 806 530 076 276 162 56 × 2 = 1 + 0.866 691 613 060 152 552 325 12;
  • 94) 0.866 691 613 060 152 552 325 12 × 2 = 1 + 0.733 383 226 120 305 104 650 24;
  • 95) 0.733 383 226 120 305 104 650 24 × 2 = 1 + 0.466 766 452 240 610 209 300 48;
  • 96) 0.466 766 452 240 610 209 300 48 × 2 = 0 + 0.933 532 904 481 220 418 600 96;
  • 97) 0.933 532 904 481 220 418 600 96 × 2 = 1 + 0.867 065 808 962 440 837 201 92;
  • 98) 0.867 065 808 962 440 837 201 92 × 2 = 1 + 0.734 131 617 924 881 674 403 84;
  • 99) 0.734 131 617 924 881 674 403 84 × 2 = 1 + 0.468 263 235 849 763 348 807 68;
  • 100) 0.468 263 235 849 763 348 807 68 × 2 = 0 + 0.936 526 471 699 526 697 615 36;
  • 101) 0.936 526 471 699 526 697 615 36 × 2 = 1 + 0.873 052 943 399 053 395 230 72;
  • 102) 0.873 052 943 399 053 395 230 72 × 2 = 1 + 0.746 105 886 798 106 790 461 44;
  • 103) 0.746 105 886 798 106 790 461 44 × 2 = 1 + 0.492 211 773 596 213 580 922 88;
  • 104) 0.492 211 773 596 213 580 922 88 × 2 = 0 + 0.984 423 547 192 427 161 845 76;
  • 105) 0.984 423 547 192 427 161 845 76 × 2 = 1 + 0.968 847 094 384 854 323 691 52;
  • 106) 0.968 847 094 384 854 323 691 52 × 2 = 1 + 0.937 694 188 769 708 647 383 04;
  • 107) 0.937 694 188 769 708 647 383 04 × 2 = 1 + 0.875 388 377 539 417 294 766 08;
  • 108) 0.875 388 377 539 417 294 766 08 × 2 = 1 + 0.750 776 755 078 834 589 532 16;
  • 109) 0.750 776 755 078 834 589 532 16 × 2 = 1 + 0.501 553 510 157 669 179 064 32;
  • 110) 0.501 553 510 157 669 179 064 32 × 2 = 1 + 0.003 107 020 315 338 358 128 64;
  • 111) 0.003 107 020 315 338 358 128 64 × 2 = 0 + 0.006 214 040 630 676 716 257 28;
  • 112) 0.006 214 040 630 676 716 257 28 × 2 = 0 + 0.012 428 081 261 353 432 514 56;
  • 113) 0.012 428 081 261 353 432 514 56 × 2 = 0 + 0.024 856 162 522 706 865 029 12;
  • 114) 0.024 856 162 522 706 865 029 12 × 2 = 0 + 0.049 712 325 045 413 730 058 24;
  • 115) 0.049 712 325 045 413 730 058 24 × 2 = 0 + 0.099 424 650 090 827 460 116 48;
  • 116) 0.099 424 650 090 827 460 116 48 × 2 = 0 + 0.198 849 300 181 654 920 232 96;
  • 117) 0.198 849 300 181 654 920 232 96 × 2 = 0 + 0.397 698 600 363 309 840 465 92;
  • 118) 0.397 698 600 363 309 840 465 92 × 2 = 0 + 0.795 397 200 726 619 680 931 84;
  • 119) 0.795 397 200 726 619 680 931 84 × 2 = 1 + 0.590 794 401 453 239 361 863 68;

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:


0.000 000 000 000 000 000 008 485(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0001 0001 1011 1100 0100 1110 1110 1110 1111 1100 0000 001(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 485(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0001 0001 1011 1100 0100 1110 1110 1110 1111 1100 0000 001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 67 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 008 485(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0001 0001 1011 1100 0100 1110 1110 1110 1111 1100 0000 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0001 0001 1011 1100 0100 1110 1110 1110 1111 1100 0000 001(2) × 20 =

1.0100 0000 1000 1101 1110 0010 0111 0111 0111 0111 1110 0000 0001(2) × 2-67


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): -67

Mantissa (not normalized):
1.0100 0000 1000 1101 1110 0010 0111 0111 0111 0111 1110 0000 0001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-67 + 2(11-1) - 1 =

(-67 + 1 023)(10) =

956(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;
  • 956 ÷ 2 = 478 + 0;
  • 478 ÷ 2 = 239 + 0;
  • 239 ÷ 2 = 119 + 1;
  • 119 ÷ 2 = 59 + 1;
  • 59 ÷ 2 = 29 + 1;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 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) =

956(10) =

011 1011 1100(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, only if necessary (not the case here).


Mantissa (normalized) =

1. 0100 0000 1000 1101 1110 0010 0111 0111 0111 0111 1110 0000 0001 =

0100 0000 1000 1101 1110 0010 0111 0111 0111 0111 1110 0000 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) =
011 1011 1100

Mantissa (52 bits) =
0100 0000 1000 1101 1110 0010 0111 0111 0111 0111 1110 0000 0001


Decimal number 0.000 000 000 000 000 000 008 485 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0000 1000 1101 1110 0010 0111 0111 0111 0111 1110 0000 0001

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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