0.000 000 000 634 751 367 889 197 801 221 261 633 73 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 634 751 367 889 197 801 221 261 633 73(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
0.000 000 000 634 751 367 889 197 801 221 261 633 73(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: 0.000 000 000 634 751 367 889 197 801 221 261 633 73.

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) 0.000 000 000 634 751 367 889 197 801 221 261 633 73 × 2 = 0 + 0.000 000 001 269 502 735 778 395 602 442 523 267 46;
  • 2) 0.000 000 001 269 502 735 778 395 602 442 523 267 46 × 2 = 0 + 0.000 000 002 539 005 471 556 791 204 885 046 534 92;
  • 3) 0.000 000 002 539 005 471 556 791 204 885 046 534 92 × 2 = 0 + 0.000 000 005 078 010 943 113 582 409 770 093 069 84;
  • 4) 0.000 000 005 078 010 943 113 582 409 770 093 069 84 × 2 = 0 + 0.000 000 010 156 021 886 227 164 819 540 186 139 68;
  • 5) 0.000 000 010 156 021 886 227 164 819 540 186 139 68 × 2 = 0 + 0.000 000 020 312 043 772 454 329 639 080 372 279 36;
  • 6) 0.000 000 020 312 043 772 454 329 639 080 372 279 36 × 2 = 0 + 0.000 000 040 624 087 544 908 659 278 160 744 558 72;
  • 7) 0.000 000 040 624 087 544 908 659 278 160 744 558 72 × 2 = 0 + 0.000 000 081 248 175 089 817 318 556 321 489 117 44;
  • 8) 0.000 000 081 248 175 089 817 318 556 321 489 117 44 × 2 = 0 + 0.000 000 162 496 350 179 634 637 112 642 978 234 88;
  • 9) 0.000 000 162 496 350 179 634 637 112 642 978 234 88 × 2 = 0 + 0.000 000 324 992 700 359 269 274 225 285 956 469 76;
  • 10) 0.000 000 324 992 700 359 269 274 225 285 956 469 76 × 2 = 0 + 0.000 000 649 985 400 718 538 548 450 571 912 939 52;
  • 11) 0.000 000 649 985 400 718 538 548 450 571 912 939 52 × 2 = 0 + 0.000 001 299 970 801 437 077 096 901 143 825 879 04;
  • 12) 0.000 001 299 970 801 437 077 096 901 143 825 879 04 × 2 = 0 + 0.000 002 599 941 602 874 154 193 802 287 651 758 08;
  • 13) 0.000 002 599 941 602 874 154 193 802 287 651 758 08 × 2 = 0 + 0.000 005 199 883 205 748 308 387 604 575 303 516 16;
  • 14) 0.000 005 199 883 205 748 308 387 604 575 303 516 16 × 2 = 0 + 0.000 010 399 766 411 496 616 775 209 150 607 032 32;
  • 15) 0.000 010 399 766 411 496 616 775 209 150 607 032 32 × 2 = 0 + 0.000 020 799 532 822 993 233 550 418 301 214 064 64;
  • 16) 0.000 020 799 532 822 993 233 550 418 301 214 064 64 × 2 = 0 + 0.000 041 599 065 645 986 467 100 836 602 428 129 28;
  • 17) 0.000 041 599 065 645 986 467 100 836 602 428 129 28 × 2 = 0 + 0.000 083 198 131 291 972 934 201 673 204 856 258 56;
  • 18) 0.000 083 198 131 291 972 934 201 673 204 856 258 56 × 2 = 0 + 0.000 166 396 262 583 945 868 403 346 409 712 517 12;
  • 19) 0.000 166 396 262 583 945 868 403 346 409 712 517 12 × 2 = 0 + 0.000 332 792 525 167 891 736 806 692 819 425 034 24;
  • 20) 0.000 332 792 525 167 891 736 806 692 819 425 034 24 × 2 = 0 + 0.000 665 585 050 335 783 473 613 385 638 850 068 48;
  • 21) 0.000 665 585 050 335 783 473 613 385 638 850 068 48 × 2 = 0 + 0.001 331 170 100 671 566 947 226 771 277 700 136 96;
  • 22) 0.001 331 170 100 671 566 947 226 771 277 700 136 96 × 2 = 0 + 0.002 662 340 201 343 133 894 453 542 555 400 273 92;
  • 23) 0.002 662 340 201 343 133 894 453 542 555 400 273 92 × 2 = 0 + 0.005 324 680 402 686 267 788 907 085 110 800 547 84;
  • 24) 0.005 324 680 402 686 267 788 907 085 110 800 547 84 × 2 = 0 + 0.010 649 360 805 372 535 577 814 170 221 601 095 68;
  • 25) 0.010 649 360 805 372 535 577 814 170 221 601 095 68 × 2 = 0 + 0.021 298 721 610 745 071 155 628 340 443 202 191 36;
  • 26) 0.021 298 721 610 745 071 155 628 340 443 202 191 36 × 2 = 0 + 0.042 597 443 221 490 142 311 256 680 886 404 382 72;
  • 27) 0.042 597 443 221 490 142 311 256 680 886 404 382 72 × 2 = 0 + 0.085 194 886 442 980 284 622 513 361 772 808 765 44;
  • 28) 0.085 194 886 442 980 284 622 513 361 772 808 765 44 × 2 = 0 + 0.170 389 772 885 960 569 245 026 723 545 617 530 88;
  • 29) 0.170 389 772 885 960 569 245 026 723 545 617 530 88 × 2 = 0 + 0.340 779 545 771 921 138 490 053 447 091 235 061 76;
  • 30) 0.340 779 545 771 921 138 490 053 447 091 235 061 76 × 2 = 0 + 0.681 559 091 543 842 276 980 106 894 182 470 123 52;
  • 31) 0.681 559 091 543 842 276 980 106 894 182 470 123 52 × 2 = 1 + 0.363 118 183 087 684 553 960 213 788 364 940 247 04;
  • 32) 0.363 118 183 087 684 553 960 213 788 364 940 247 04 × 2 = 0 + 0.726 236 366 175 369 107 920 427 576 729 880 494 08;
  • 33) 0.726 236 366 175 369 107 920 427 576 729 880 494 08 × 2 = 1 + 0.452 472 732 350 738 215 840 855 153 459 760 988 16;
  • 34) 0.452 472 732 350 738 215 840 855 153 459 760 988 16 × 2 = 0 + 0.904 945 464 701 476 431 681 710 306 919 521 976 32;
  • 35) 0.904 945 464 701 476 431 681 710 306 919 521 976 32 × 2 = 1 + 0.809 890 929 402 952 863 363 420 613 839 043 952 64;
  • 36) 0.809 890 929 402 952 863 363 420 613 839 043 952 64 × 2 = 1 + 0.619 781 858 805 905 726 726 841 227 678 087 905 28;
  • 37) 0.619 781 858 805 905 726 726 841 227 678 087 905 28 × 2 = 1 + 0.239 563 717 611 811 453 453 682 455 356 175 810 56;
  • 38) 0.239 563 717 611 811 453 453 682 455 356 175 810 56 × 2 = 0 + 0.479 127 435 223 622 906 907 364 910 712 351 621 12;
  • 39) 0.479 127 435 223 622 906 907 364 910 712 351 621 12 × 2 = 0 + 0.958 254 870 447 245 813 814 729 821 424 703 242 24;
  • 40) 0.958 254 870 447 245 813 814 729 821 424 703 242 24 × 2 = 1 + 0.916 509 740 894 491 627 629 459 642 849 406 484 48;
  • 41) 0.916 509 740 894 491 627 629 459 642 849 406 484 48 × 2 = 1 + 0.833 019 481 788 983 255 258 919 285 698 812 968 96;
  • 42) 0.833 019 481 788 983 255 258 919 285 698 812 968 96 × 2 = 1 + 0.666 038 963 577 966 510 517 838 571 397 625 937 92;
  • 43) 0.666 038 963 577 966 510 517 838 571 397 625 937 92 × 2 = 1 + 0.332 077 927 155 933 021 035 677 142 795 251 875 84;
  • 44) 0.332 077 927 155 933 021 035 677 142 795 251 875 84 × 2 = 0 + 0.664 155 854 311 866 042 071 354 285 590 503 751 68;
  • 45) 0.664 155 854 311 866 042 071 354 285 590 503 751 68 × 2 = 1 + 0.328 311 708 623 732 084 142 708 571 181 007 503 36;
  • 46) 0.328 311 708 623 732 084 142 708 571 181 007 503 36 × 2 = 0 + 0.656 623 417 247 464 168 285 417 142 362 015 006 72;
  • 47) 0.656 623 417 247 464 168 285 417 142 362 015 006 72 × 2 = 1 + 0.313 246 834 494 928 336 570 834 284 724 030 013 44;
  • 48) 0.313 246 834 494 928 336 570 834 284 724 030 013 44 × 2 = 0 + 0.626 493 668 989 856 673 141 668 569 448 060 026 88;
  • 49) 0.626 493 668 989 856 673 141 668 569 448 060 026 88 × 2 = 1 + 0.252 987 337 979 713 346 283 337 138 896 120 053 76;
  • 50) 0.252 987 337 979 713 346 283 337 138 896 120 053 76 × 2 = 0 + 0.505 974 675 959 426 692 566 674 277 792 240 107 52;
  • 51) 0.505 974 675 959 426 692 566 674 277 792 240 107 52 × 2 = 1 + 0.011 949 351 918 853 385 133 348 555 584 480 215 04;
  • 52) 0.011 949 351 918 853 385 133 348 555 584 480 215 04 × 2 = 0 + 0.023 898 703 837 706 770 266 697 111 168 960 430 08;
  • 53) 0.023 898 703 837 706 770 266 697 111 168 960 430 08 × 2 = 0 + 0.047 797 407 675 413 540 533 394 222 337 920 860 16;
  • 54) 0.047 797 407 675 413 540 533 394 222 337 920 860 16 × 2 = 0 + 0.095 594 815 350 827 081 066 788 444 675 841 720 32;
  • 55) 0.095 594 815 350 827 081 066 788 444 675 841 720 32 × 2 = 0 + 0.191 189 630 701 654 162 133 576 889 351 683 440 64;
  • 56) 0.191 189 630 701 654 162 133 576 889 351 683 440 64 × 2 = 0 + 0.382 379 261 403 308 324 267 153 778 703 366 881 28;
  • 57) 0.382 379 261 403 308 324 267 153 778 703 366 881 28 × 2 = 0 + 0.764 758 522 806 616 648 534 307 557 406 733 762 56;
  • 58) 0.764 758 522 806 616 648 534 307 557 406 733 762 56 × 2 = 1 + 0.529 517 045 613 233 297 068 615 114 813 467 525 12;
  • 59) 0.529 517 045 613 233 297 068 615 114 813 467 525 12 × 2 = 1 + 0.059 034 091 226 466 594 137 230 229 626 935 050 24;
  • 60) 0.059 034 091 226 466 594 137 230 229 626 935 050 24 × 2 = 0 + 0.118 068 182 452 933 188 274 460 459 253 870 100 48;
  • 61) 0.118 068 182 452 933 188 274 460 459 253 870 100 48 × 2 = 0 + 0.236 136 364 905 866 376 548 920 918 507 740 200 96;
  • 62) 0.236 136 364 905 866 376 548 920 918 507 740 200 96 × 2 = 0 + 0.472 272 729 811 732 753 097 841 837 015 480 401 92;
  • 63) 0.472 272 729 811 732 753 097 841 837 015 480 401 92 × 2 = 0 + 0.944 545 459 623 465 506 195 683 674 030 960 803 84;
  • 64) 0.944 545 459 623 465 506 195 683 674 030 960 803 84 × 2 = 1 + 0.889 090 919 246 931 012 391 367 348 061 921 607 68;
  • 65) 0.889 090 919 246 931 012 391 367 348 061 921 607 68 × 2 = 1 + 0.778 181 838 493 862 024 782 734 696 123 843 215 36;
  • 66) 0.778 181 838 493 862 024 782 734 696 123 843 215 36 × 2 = 1 + 0.556 363 676 987 724 049 565 469 392 247 686 430 72;
  • 67) 0.556 363 676 987 724 049 565 469 392 247 686 430 72 × 2 = 1 + 0.112 727 353 975 448 099 130 938 784 495 372 861 44;
  • 68) 0.112 727 353 975 448 099 130 938 784 495 372 861 44 × 2 = 0 + 0.225 454 707 950 896 198 261 877 568 990 745 722 88;
  • 69) 0.225 454 707 950 896 198 261 877 568 990 745 722 88 × 2 = 0 + 0.450 909 415 901 792 396 523 755 137 981 491 445 76;
  • 70) 0.450 909 415 901 792 396 523 755 137 981 491 445 76 × 2 = 0 + 0.901 818 831 803 584 793 047 510 275 962 982 891 52;
  • 71) 0.901 818 831 803 584 793 047 510 275 962 982 891 52 × 2 = 1 + 0.803 637 663 607 169 586 095 020 551 925 965 783 04;
  • 72) 0.803 637 663 607 169 586 095 020 551 925 965 783 04 × 2 = 1 + 0.607 275 327 214 339 172 190 041 103 851 931 566 08;
  • 73) 0.607 275 327 214 339 172 190 041 103 851 931 566 08 × 2 = 1 + 0.214 550 654 428 678 344 380 082 207 703 863 132 16;
  • 74) 0.214 550 654 428 678 344 380 082 207 703 863 132 16 × 2 = 0 + 0.429 101 308 857 356 688 760 164 415 407 726 264 32;
  • 75) 0.429 101 308 857 356 688 760 164 415 407 726 264 32 × 2 = 0 + 0.858 202 617 714 713 377 520 328 830 815 452 528 64;
  • 76) 0.858 202 617 714 713 377 520 328 830 815 452 528 64 × 2 = 1 + 0.716 405 235 429 426 755 040 657 661 630 905 057 28;
  • 77) 0.716 405 235 429 426 755 040 657 661 630 905 057 28 × 2 = 1 + 0.432 810 470 858 853 510 081 315 323 261 810 114 56;
  • 78) 0.432 810 470 858 853 510 081 315 323 261 810 114 56 × 2 = 0 + 0.865 620 941 717 707 020 162 630 646 523 620 229 12;
  • 79) 0.865 620 941 717 707 020 162 630 646 523 620 229 12 × 2 = 1 + 0.731 241 883 435 414 040 325 261 293 047 240 458 24;
  • 80) 0.731 241 883 435 414 040 325 261 293 047 240 458 24 × 2 = 1 + 0.462 483 766 870 828 080 650 522 586 094 480 916 48;
  • 81) 0.462 483 766 870 828 080 650 522 586 094 480 916 48 × 2 = 0 + 0.924 967 533 741 656 161 301 045 172 188 961 832 96;
  • 82) 0.924 967 533 741 656 161 301 045 172 188 961 832 96 × 2 = 1 + 0.849 935 067 483 312 322 602 090 344 377 923 665 92;
  • 83) 0.849 935 067 483 312 322 602 090 344 377 923 665 92 × 2 = 1 + 0.699 870 134 966 624 645 204 180 688 755 847 331 84;

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:


0.000 000 000 634 751 367 889 197 801 221 261 633 73(10) =

0.0000 0000 0000 0000 0000 0000 0000 0010 1011 1001 1110 1010 1010 0000 0110 0001 1110 0011 1001 1011 011(2)

5. Positive number before normalization:

0.000 000 000 634 751 367 889 197 801 221 261 633 73(10) =

0.0000 0000 0000 0000 0000 0000 0000 0010 1011 1001 1110 1010 1010 0000 0110 0001 1110 0011 1001 1011 011(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 31 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 634 751 367 889 197 801 221 261 633 73(10) =

0.0000 0000 0000 0000 0000 0000 0000 0010 1011 1001 1110 1010 1010 0000 0110 0001 1110 0011 1001 1011 011(2) =

0.0000 0000 0000 0000 0000 0000 0000 0010 1011 1001 1110 1010 1010 0000 0110 0001 1110 0011 1001 1011 011(2) × 20 =

1.0101 1100 1111 0101 0101 0000 0011 0000 1111 0001 1100 1101 1011(2) × 2-31


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): -31

Mantissa (not normalized):
1.0101 1100 1111 0101 0101 0000 0011 0000 1111 0001 1100 1101 1011


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-31 + 2(11-1) - 1 =

(-31 + 1 023)(10) =

992(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;
  • 992 ÷ 2 = 496 + 0;
  • 496 ÷ 2 = 248 + 0;
  • 248 ÷ 2 = 124 + 0;
  • 124 ÷ 2 = 62 + 0;
  • 62 ÷ 2 = 31 + 0;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 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) =

992(10) =

011 1110 0000(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. 0101 1100 1111 0101 0101 0000 0011 0000 1111 0001 1100 1101 1011 =

0101 1100 1111 0101 0101 0000 0011 0000 1111 0001 1100 1101 1011


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 1110 0000

Mantissa (52 bits) =
0101 1100 1111 0101 0101 0000 0011 0000 1111 0001 1100 1101 1011


Decimal number 0.000 000 000 634 751 367 889 197 801 221 261 633 73 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1110 0000 - 0101 1100 1111 0101 0101 0000 0011 0000 1111 0001 1100 1101 1011

» Convert decimal 0.000 000 000 634 751 367 889 197 801 221 261 633 25 to 64 bit double precision IEEE 754 binary floating point representation standard

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» Month 10, 2026 [October]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: October - by our visitors

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