0.000 000 000 000 000 000 008 537 79 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 537 79(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
0.000 000 000 000 000 000 008 537 79(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: 0.000 000 000 000 000 000 008 537 79.

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) 0.000 000 000 000 000 000 008 537 79 × 2 = 0 + 0.000 000 000 000 000 000 017 075 58;
  • 2) 0.000 000 000 000 000 000 017 075 58 × 2 = 0 + 0.000 000 000 000 000 000 034 151 16;
  • 3) 0.000 000 000 000 000 000 034 151 16 × 2 = 0 + 0.000 000 000 000 000 000 068 302 32;
  • 4) 0.000 000 000 000 000 000 068 302 32 × 2 = 0 + 0.000 000 000 000 000 000 136 604 64;
  • 5) 0.000 000 000 000 000 000 136 604 64 × 2 = 0 + 0.000 000 000 000 000 000 273 209 28;
  • 6) 0.000 000 000 000 000 000 273 209 28 × 2 = 0 + 0.000 000 000 000 000 000 546 418 56;
  • 7) 0.000 000 000 000 000 000 546 418 56 × 2 = 0 + 0.000 000 000 000 000 001 092 837 12;
  • 8) 0.000 000 000 000 000 001 092 837 12 × 2 = 0 + 0.000 000 000 000 000 002 185 674 24;
  • 9) 0.000 000 000 000 000 002 185 674 24 × 2 = 0 + 0.000 000 000 000 000 004 371 348 48;
  • 10) 0.000 000 000 000 000 004 371 348 48 × 2 = 0 + 0.000 000 000 000 000 008 742 696 96;
  • 11) 0.000 000 000 000 000 008 742 696 96 × 2 = 0 + 0.000 000 000 000 000 017 485 393 92;
  • 12) 0.000 000 000 000 000 017 485 393 92 × 2 = 0 + 0.000 000 000 000 000 034 970 787 84;
  • 13) 0.000 000 000 000 000 034 970 787 84 × 2 = 0 + 0.000 000 000 000 000 069 941 575 68;
  • 14) 0.000 000 000 000 000 069 941 575 68 × 2 = 0 + 0.000 000 000 000 000 139 883 151 36;
  • 15) 0.000 000 000 000 000 139 883 151 36 × 2 = 0 + 0.000 000 000 000 000 279 766 302 72;
  • 16) 0.000 000 000 000 000 279 766 302 72 × 2 = 0 + 0.000 000 000 000 000 559 532 605 44;
  • 17) 0.000 000 000 000 000 559 532 605 44 × 2 = 0 + 0.000 000 000 000 001 119 065 210 88;
  • 18) 0.000 000 000 000 001 119 065 210 88 × 2 = 0 + 0.000 000 000 000 002 238 130 421 76;
  • 19) 0.000 000 000 000 002 238 130 421 76 × 2 = 0 + 0.000 000 000 000 004 476 260 843 52;
  • 20) 0.000 000 000 000 004 476 260 843 52 × 2 = 0 + 0.000 000 000 000 008 952 521 687 04;
  • 21) 0.000 000 000 000 008 952 521 687 04 × 2 = 0 + 0.000 000 000 000 017 905 043 374 08;
  • 22) 0.000 000 000 000 017 905 043 374 08 × 2 = 0 + 0.000 000 000 000 035 810 086 748 16;
  • 23) 0.000 000 000 000 035 810 086 748 16 × 2 = 0 + 0.000 000 000 000 071 620 173 496 32;
  • 24) 0.000 000 000 000 071 620 173 496 32 × 2 = 0 + 0.000 000 000 000 143 240 346 992 64;
  • 25) 0.000 000 000 000 143 240 346 992 64 × 2 = 0 + 0.000 000 000 000 286 480 693 985 28;
  • 26) 0.000 000 000 000 286 480 693 985 28 × 2 = 0 + 0.000 000 000 000 572 961 387 970 56;
  • 27) 0.000 000 000 000 572 961 387 970 56 × 2 = 0 + 0.000 000 000 001 145 922 775 941 12;
  • 28) 0.000 000 000 001 145 922 775 941 12 × 2 = 0 + 0.000 000 000 002 291 845 551 882 24;
  • 29) 0.000 000 000 002 291 845 551 882 24 × 2 = 0 + 0.000 000 000 004 583 691 103 764 48;
  • 30) 0.000 000 000 004 583 691 103 764 48 × 2 = 0 + 0.000 000 000 009 167 382 207 528 96;
  • 31) 0.000 000 000 009 167 382 207 528 96 × 2 = 0 + 0.000 000 000 018 334 764 415 057 92;
  • 32) 0.000 000 000 018 334 764 415 057 92 × 2 = 0 + 0.000 000 000 036 669 528 830 115 84;
  • 33) 0.000 000 000 036 669 528 830 115 84 × 2 = 0 + 0.000 000 000 073 339 057 660 231 68;
  • 34) 0.000 000 000 073 339 057 660 231 68 × 2 = 0 + 0.000 000 000 146 678 115 320 463 36;
  • 35) 0.000 000 000 146 678 115 320 463 36 × 2 = 0 + 0.000 000 000 293 356 230 640 926 72;
  • 36) 0.000 000 000 293 356 230 640 926 72 × 2 = 0 + 0.000 000 000 586 712 461 281 853 44;
  • 37) 0.000 000 000 586 712 461 281 853 44 × 2 = 0 + 0.000 000 001 173 424 922 563 706 88;
  • 38) 0.000 000 001 173 424 922 563 706 88 × 2 = 0 + 0.000 000 002 346 849 845 127 413 76;
  • 39) 0.000 000 002 346 849 845 127 413 76 × 2 = 0 + 0.000 000 004 693 699 690 254 827 52;
  • 40) 0.000 000 004 693 699 690 254 827 52 × 2 = 0 + 0.000 000 009 387 399 380 509 655 04;
  • 41) 0.000 000 009 387 399 380 509 655 04 × 2 = 0 + 0.000 000 018 774 798 761 019 310 08;
  • 42) 0.000 000 018 774 798 761 019 310 08 × 2 = 0 + 0.000 000 037 549 597 522 038 620 16;
  • 43) 0.000 000 037 549 597 522 038 620 16 × 2 = 0 + 0.000 000 075 099 195 044 077 240 32;
  • 44) 0.000 000 075 099 195 044 077 240 32 × 2 = 0 + 0.000 000 150 198 390 088 154 480 64;
  • 45) 0.000 000 150 198 390 088 154 480 64 × 2 = 0 + 0.000 000 300 396 780 176 308 961 28;
  • 46) 0.000 000 300 396 780 176 308 961 28 × 2 = 0 + 0.000 000 600 793 560 352 617 922 56;
  • 47) 0.000 000 600 793 560 352 617 922 56 × 2 = 0 + 0.000 001 201 587 120 705 235 845 12;
  • 48) 0.000 001 201 587 120 705 235 845 12 × 2 = 0 + 0.000 002 403 174 241 410 471 690 24;
  • 49) 0.000 002 403 174 241 410 471 690 24 × 2 = 0 + 0.000 004 806 348 482 820 943 380 48;
  • 50) 0.000 004 806 348 482 820 943 380 48 × 2 = 0 + 0.000 009 612 696 965 641 886 760 96;
  • 51) 0.000 009 612 696 965 641 886 760 96 × 2 = 0 + 0.000 019 225 393 931 283 773 521 92;
  • 52) 0.000 019 225 393 931 283 773 521 92 × 2 = 0 + 0.000 038 450 787 862 567 547 043 84;
  • 53) 0.000 038 450 787 862 567 547 043 84 × 2 = 0 + 0.000 076 901 575 725 135 094 087 68;
  • 54) 0.000 076 901 575 725 135 094 087 68 × 2 = 0 + 0.000 153 803 151 450 270 188 175 36;
  • 55) 0.000 153 803 151 450 270 188 175 36 × 2 = 0 + 0.000 307 606 302 900 540 376 350 72;
  • 56) 0.000 307 606 302 900 540 376 350 72 × 2 = 0 + 0.000 615 212 605 801 080 752 701 44;
  • 57) 0.000 615 212 605 801 080 752 701 44 × 2 = 0 + 0.001 230 425 211 602 161 505 402 88;
  • 58) 0.001 230 425 211 602 161 505 402 88 × 2 = 0 + 0.002 460 850 423 204 323 010 805 76;
  • 59) 0.002 460 850 423 204 323 010 805 76 × 2 = 0 + 0.004 921 700 846 408 646 021 611 52;
  • 60) 0.004 921 700 846 408 646 021 611 52 × 2 = 0 + 0.009 843 401 692 817 292 043 223 04;
  • 61) 0.009 843 401 692 817 292 043 223 04 × 2 = 0 + 0.019 686 803 385 634 584 086 446 08;
  • 62) 0.019 686 803 385 634 584 086 446 08 × 2 = 0 + 0.039 373 606 771 269 168 172 892 16;
  • 63) 0.039 373 606 771 269 168 172 892 16 × 2 = 0 + 0.078 747 213 542 538 336 345 784 32;
  • 64) 0.078 747 213 542 538 336 345 784 32 × 2 = 0 + 0.157 494 427 085 076 672 691 568 64;
  • 65) 0.157 494 427 085 076 672 691 568 64 × 2 = 0 + 0.314 988 854 170 153 345 383 137 28;
  • 66) 0.314 988 854 170 153 345 383 137 28 × 2 = 0 + 0.629 977 708 340 306 690 766 274 56;
  • 67) 0.629 977 708 340 306 690 766 274 56 × 2 = 1 + 0.259 955 416 680 613 381 532 549 12;
  • 68) 0.259 955 416 680 613 381 532 549 12 × 2 = 0 + 0.519 910 833 361 226 763 065 098 24;
  • 69) 0.519 910 833 361 226 763 065 098 24 × 2 = 1 + 0.039 821 666 722 453 526 130 196 48;
  • 70) 0.039 821 666 722 453 526 130 196 48 × 2 = 0 + 0.079 643 333 444 907 052 260 392 96;
  • 71) 0.079 643 333 444 907 052 260 392 96 × 2 = 0 + 0.159 286 666 889 814 104 520 785 92;
  • 72) 0.159 286 666 889 814 104 520 785 92 × 2 = 0 + 0.318 573 333 779 628 209 041 571 84;
  • 73) 0.318 573 333 779 628 209 041 571 84 × 2 = 0 + 0.637 146 667 559 256 418 083 143 68;
  • 74) 0.637 146 667 559 256 418 083 143 68 × 2 = 1 + 0.274 293 335 118 512 836 166 287 36;
  • 75) 0.274 293 335 118 512 836 166 287 36 × 2 = 0 + 0.548 586 670 237 025 672 332 574 72;
  • 76) 0.548 586 670 237 025 672 332 574 72 × 2 = 1 + 0.097 173 340 474 051 344 665 149 44;
  • 77) 0.097 173 340 474 051 344 665 149 44 × 2 = 0 + 0.194 346 680 948 102 689 330 298 88;
  • 78) 0.194 346 680 948 102 689 330 298 88 × 2 = 0 + 0.388 693 361 896 205 378 660 597 76;
  • 79) 0.388 693 361 896 205 378 660 597 76 × 2 = 0 + 0.777 386 723 792 410 757 321 195 52;
  • 80) 0.777 386 723 792 410 757 321 195 52 × 2 = 1 + 0.554 773 447 584 821 514 642 391 04;
  • 81) 0.554 773 447 584 821 514 642 391 04 × 2 = 1 + 0.109 546 895 169 643 029 284 782 08;
  • 82) 0.109 546 895 169 643 029 284 782 08 × 2 = 0 + 0.219 093 790 339 286 058 569 564 16;
  • 83) 0.219 093 790 339 286 058 569 564 16 × 2 = 0 + 0.438 187 580 678 572 117 139 128 32;
  • 84) 0.438 187 580 678 572 117 139 128 32 × 2 = 0 + 0.876 375 161 357 144 234 278 256 64;
  • 85) 0.876 375 161 357 144 234 278 256 64 × 2 = 1 + 0.752 750 322 714 288 468 556 513 28;
  • 86) 0.752 750 322 714 288 468 556 513 28 × 2 = 1 + 0.505 500 645 428 576 937 113 026 56;
  • 87) 0.505 500 645 428 576 937 113 026 56 × 2 = 1 + 0.011 001 290 857 153 874 226 053 12;
  • 88) 0.011 001 290 857 153 874 226 053 12 × 2 = 0 + 0.022 002 581 714 307 748 452 106 24;
  • 89) 0.022 002 581 714 307 748 452 106 24 × 2 = 0 + 0.044 005 163 428 615 496 904 212 48;
  • 90) 0.044 005 163 428 615 496 904 212 48 × 2 = 0 + 0.088 010 326 857 230 993 808 424 96;
  • 91) 0.088 010 326 857 230 993 808 424 96 × 2 = 0 + 0.176 020 653 714 461 987 616 849 92;
  • 92) 0.176 020 653 714 461 987 616 849 92 × 2 = 0 + 0.352 041 307 428 923 975 233 699 84;
  • 93) 0.352 041 307 428 923 975 233 699 84 × 2 = 0 + 0.704 082 614 857 847 950 467 399 68;
  • 94) 0.704 082 614 857 847 950 467 399 68 × 2 = 1 + 0.408 165 229 715 695 900 934 799 36;
  • 95) 0.408 165 229 715 695 900 934 799 36 × 2 = 0 + 0.816 330 459 431 391 801 869 598 72;
  • 96) 0.816 330 459 431 391 801 869 598 72 × 2 = 1 + 0.632 660 918 862 783 603 739 197 44;
  • 97) 0.632 660 918 862 783 603 739 197 44 × 2 = 1 + 0.265 321 837 725 567 207 478 394 88;
  • 98) 0.265 321 837 725 567 207 478 394 88 × 2 = 0 + 0.530 643 675 451 134 414 956 789 76;
  • 99) 0.530 643 675 451 134 414 956 789 76 × 2 = 1 + 0.061 287 350 902 268 829 913 579 52;
  • 100) 0.061 287 350 902 268 829 913 579 52 × 2 = 0 + 0.122 574 701 804 537 659 827 159 04;
  • 101) 0.122 574 701 804 537 659 827 159 04 × 2 = 0 + 0.245 149 403 609 075 319 654 318 08;
  • 102) 0.245 149 403 609 075 319 654 318 08 × 2 = 0 + 0.490 298 807 218 150 639 308 636 16;
  • 103) 0.490 298 807 218 150 639 308 636 16 × 2 = 0 + 0.980 597 614 436 301 278 617 272 32;
  • 104) 0.980 597 614 436 301 278 617 272 32 × 2 = 1 + 0.961 195 228 872 602 557 234 544 64;
  • 105) 0.961 195 228 872 602 557 234 544 64 × 2 = 1 + 0.922 390 457 745 205 114 469 089 28;
  • 106) 0.922 390 457 745 205 114 469 089 28 × 2 = 1 + 0.844 780 915 490 410 228 938 178 56;
  • 107) 0.844 780 915 490 410 228 938 178 56 × 2 = 1 + 0.689 561 830 980 820 457 876 357 12;
  • 108) 0.689 561 830 980 820 457 876 357 12 × 2 = 1 + 0.379 123 661 961 640 915 752 714 24;
  • 109) 0.379 123 661 961 640 915 752 714 24 × 2 = 0 + 0.758 247 323 923 281 831 505 428 48;
  • 110) 0.758 247 323 923 281 831 505 428 48 × 2 = 1 + 0.516 494 647 846 563 663 010 856 96;
  • 111) 0.516 494 647 846 563 663 010 856 96 × 2 = 1 + 0.032 989 295 693 127 326 021 713 92;
  • 112) 0.032 989 295 693 127 326 021 713 92 × 2 = 0 + 0.065 978 591 386 254 652 043 427 84;
  • 113) 0.065 978 591 386 254 652 043 427 84 × 2 = 0 + 0.131 957 182 772 509 304 086 855 68;
  • 114) 0.131 957 182 772 509 304 086 855 68 × 2 = 0 + 0.263 914 365 545 018 608 173 711 36;
  • 115) 0.263 914 365 545 018 608 173 711 36 × 2 = 0 + 0.527 828 731 090 037 216 347 422 72;
  • 116) 0.527 828 731 090 037 216 347 422 72 × 2 = 1 + 0.055 657 462 180 074 432 694 845 44;
  • 117) 0.055 657 462 180 074 432 694 845 44 × 2 = 0 + 0.111 314 924 360 148 865 389 690 88;
  • 118) 0.111 314 924 360 148 865 389 690 88 × 2 = 0 + 0.222 629 848 720 297 730 779 381 76;
  • 119) 0.222 629 848 720 297 730 779 381 76 × 2 = 0 + 0.445 259 697 440 595 461 558 763 52;

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:


0.000 000 000 000 000 000 008 537 79(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 1000 1110 0000 0101 1010 0001 1111 0110 0001 000(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 537 79(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 1000 1110 0000 0101 1010 0001 1111 0110 0001 000(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:


0.000 000 000 000 000 000 008 537 79(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 1000 1110 0000 0101 1010 0001 1111 0110 0001 000(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 1000 1110 0000 0101 1010 0001 1111 0110 0001 000(2) × 20 =

1.0100 0010 1000 1100 0111 0000 0010 1101 0000 1111 1011 0000 1000(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 0010 1000 1100 0111 0000 0010 1101 0000 1111 1011 0000 1000


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 0010 1000 1100 0111 0000 0010 1101 0000 1111 1011 0000 1000 =

0100 0010 1000 1100 0111 0000 0010 1101 0000 1111 1011 0000 1000


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 0010 1000 1100 0111 0000 0010 1101 0000 1111 1011 0000 1000


Decimal number 0.000 000 000 000 000 000 008 537 79 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1000 1100 0111 0000 0010 1101 0000 1111 1011 0000 1000

Share

» Convert decimal 0.000 000 000 000 000 000 008 536 91 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 06, 2026 [June]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: June - by our visitors

» Improve your social skills: The Hidden Requirement in Every Single Job Description. NotebookLM YouTube Video of 8.15 minutes

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