0.000 000 000 634 751 367 889 197 801 221 261 633 52 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 52(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 52(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 52.

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 52 × 2 = 0 + 0.000 000 001 269 502 735 778 395 602 442 523 267 04;
  • 2) 0.000 000 001 269 502 735 778 395 602 442 523 267 04 × 2 = 0 + 0.000 000 002 539 005 471 556 791 204 885 046 534 08;
  • 3) 0.000 000 002 539 005 471 556 791 204 885 046 534 08 × 2 = 0 + 0.000 000 005 078 010 943 113 582 409 770 093 068 16;
  • 4) 0.000 000 005 078 010 943 113 582 409 770 093 068 16 × 2 = 0 + 0.000 000 010 156 021 886 227 164 819 540 186 136 32;
  • 5) 0.000 000 010 156 021 886 227 164 819 540 186 136 32 × 2 = 0 + 0.000 000 020 312 043 772 454 329 639 080 372 272 64;
  • 6) 0.000 000 020 312 043 772 454 329 639 080 372 272 64 × 2 = 0 + 0.000 000 040 624 087 544 908 659 278 160 744 545 28;
  • 7) 0.000 000 040 624 087 544 908 659 278 160 744 545 28 × 2 = 0 + 0.000 000 081 248 175 089 817 318 556 321 489 090 56;
  • 8) 0.000 000 081 248 175 089 817 318 556 321 489 090 56 × 2 = 0 + 0.000 000 162 496 350 179 634 637 112 642 978 181 12;
  • 9) 0.000 000 162 496 350 179 634 637 112 642 978 181 12 × 2 = 0 + 0.000 000 324 992 700 359 269 274 225 285 956 362 24;
  • 10) 0.000 000 324 992 700 359 269 274 225 285 956 362 24 × 2 = 0 + 0.000 000 649 985 400 718 538 548 450 571 912 724 48;
  • 11) 0.000 000 649 985 400 718 538 548 450 571 912 724 48 × 2 = 0 + 0.000 001 299 970 801 437 077 096 901 143 825 448 96;
  • 12) 0.000 001 299 970 801 437 077 096 901 143 825 448 96 × 2 = 0 + 0.000 002 599 941 602 874 154 193 802 287 650 897 92;
  • 13) 0.000 002 599 941 602 874 154 193 802 287 650 897 92 × 2 = 0 + 0.000 005 199 883 205 748 308 387 604 575 301 795 84;
  • 14) 0.000 005 199 883 205 748 308 387 604 575 301 795 84 × 2 = 0 + 0.000 010 399 766 411 496 616 775 209 150 603 591 68;
  • 15) 0.000 010 399 766 411 496 616 775 209 150 603 591 68 × 2 = 0 + 0.000 020 799 532 822 993 233 550 418 301 207 183 36;
  • 16) 0.000 020 799 532 822 993 233 550 418 301 207 183 36 × 2 = 0 + 0.000 041 599 065 645 986 467 100 836 602 414 366 72;
  • 17) 0.000 041 599 065 645 986 467 100 836 602 414 366 72 × 2 = 0 + 0.000 083 198 131 291 972 934 201 673 204 828 733 44;
  • 18) 0.000 083 198 131 291 972 934 201 673 204 828 733 44 × 2 = 0 + 0.000 166 396 262 583 945 868 403 346 409 657 466 88;
  • 19) 0.000 166 396 262 583 945 868 403 346 409 657 466 88 × 2 = 0 + 0.000 332 792 525 167 891 736 806 692 819 314 933 76;
  • 20) 0.000 332 792 525 167 891 736 806 692 819 314 933 76 × 2 = 0 + 0.000 665 585 050 335 783 473 613 385 638 629 867 52;
  • 21) 0.000 665 585 050 335 783 473 613 385 638 629 867 52 × 2 = 0 + 0.001 331 170 100 671 566 947 226 771 277 259 735 04;
  • 22) 0.001 331 170 100 671 566 947 226 771 277 259 735 04 × 2 = 0 + 0.002 662 340 201 343 133 894 453 542 554 519 470 08;
  • 23) 0.002 662 340 201 343 133 894 453 542 554 519 470 08 × 2 = 0 + 0.005 324 680 402 686 267 788 907 085 109 038 940 16;
  • 24) 0.005 324 680 402 686 267 788 907 085 109 038 940 16 × 2 = 0 + 0.010 649 360 805 372 535 577 814 170 218 077 880 32;
  • 25) 0.010 649 360 805 372 535 577 814 170 218 077 880 32 × 2 = 0 + 0.021 298 721 610 745 071 155 628 340 436 155 760 64;
  • 26) 0.021 298 721 610 745 071 155 628 340 436 155 760 64 × 2 = 0 + 0.042 597 443 221 490 142 311 256 680 872 311 521 28;
  • 27) 0.042 597 443 221 490 142 311 256 680 872 311 521 28 × 2 = 0 + 0.085 194 886 442 980 284 622 513 361 744 623 042 56;
  • 28) 0.085 194 886 442 980 284 622 513 361 744 623 042 56 × 2 = 0 + 0.170 389 772 885 960 569 245 026 723 489 246 085 12;
  • 29) 0.170 389 772 885 960 569 245 026 723 489 246 085 12 × 2 = 0 + 0.340 779 545 771 921 138 490 053 446 978 492 170 24;
  • 30) 0.340 779 545 771 921 138 490 053 446 978 492 170 24 × 2 = 0 + 0.681 559 091 543 842 276 980 106 893 956 984 340 48;
  • 31) 0.681 559 091 543 842 276 980 106 893 956 984 340 48 × 2 = 1 + 0.363 118 183 087 684 553 960 213 787 913 968 680 96;
  • 32) 0.363 118 183 087 684 553 960 213 787 913 968 680 96 × 2 = 0 + 0.726 236 366 175 369 107 920 427 575 827 937 361 92;
  • 33) 0.726 236 366 175 369 107 920 427 575 827 937 361 92 × 2 = 1 + 0.452 472 732 350 738 215 840 855 151 655 874 723 84;
  • 34) 0.452 472 732 350 738 215 840 855 151 655 874 723 84 × 2 = 0 + 0.904 945 464 701 476 431 681 710 303 311 749 447 68;
  • 35) 0.904 945 464 701 476 431 681 710 303 311 749 447 68 × 2 = 1 + 0.809 890 929 402 952 863 363 420 606 623 498 895 36;
  • 36) 0.809 890 929 402 952 863 363 420 606 623 498 895 36 × 2 = 1 + 0.619 781 858 805 905 726 726 841 213 246 997 790 72;
  • 37) 0.619 781 858 805 905 726 726 841 213 246 997 790 72 × 2 = 1 + 0.239 563 717 611 811 453 453 682 426 493 995 581 44;
  • 38) 0.239 563 717 611 811 453 453 682 426 493 995 581 44 × 2 = 0 + 0.479 127 435 223 622 906 907 364 852 987 991 162 88;
  • 39) 0.479 127 435 223 622 906 907 364 852 987 991 162 88 × 2 = 0 + 0.958 254 870 447 245 813 814 729 705 975 982 325 76;
  • 40) 0.958 254 870 447 245 813 814 729 705 975 982 325 76 × 2 = 1 + 0.916 509 740 894 491 627 629 459 411 951 964 651 52;
  • 41) 0.916 509 740 894 491 627 629 459 411 951 964 651 52 × 2 = 1 + 0.833 019 481 788 983 255 258 918 823 903 929 303 04;
  • 42) 0.833 019 481 788 983 255 258 918 823 903 929 303 04 × 2 = 1 + 0.666 038 963 577 966 510 517 837 647 807 858 606 08;
  • 43) 0.666 038 963 577 966 510 517 837 647 807 858 606 08 × 2 = 1 + 0.332 077 927 155 933 021 035 675 295 615 717 212 16;
  • 44) 0.332 077 927 155 933 021 035 675 295 615 717 212 16 × 2 = 0 + 0.664 155 854 311 866 042 071 350 591 231 434 424 32;
  • 45) 0.664 155 854 311 866 042 071 350 591 231 434 424 32 × 2 = 1 + 0.328 311 708 623 732 084 142 701 182 462 868 848 64;
  • 46) 0.328 311 708 623 732 084 142 701 182 462 868 848 64 × 2 = 0 + 0.656 623 417 247 464 168 285 402 364 925 737 697 28;
  • 47) 0.656 623 417 247 464 168 285 402 364 925 737 697 28 × 2 = 1 + 0.313 246 834 494 928 336 570 804 729 851 475 394 56;
  • 48) 0.313 246 834 494 928 336 570 804 729 851 475 394 56 × 2 = 0 + 0.626 493 668 989 856 673 141 609 459 702 950 789 12;
  • 49) 0.626 493 668 989 856 673 141 609 459 702 950 789 12 × 2 = 1 + 0.252 987 337 979 713 346 283 218 919 405 901 578 24;
  • 50) 0.252 987 337 979 713 346 283 218 919 405 901 578 24 × 2 = 0 + 0.505 974 675 959 426 692 566 437 838 811 803 156 48;
  • 51) 0.505 974 675 959 426 692 566 437 838 811 803 156 48 × 2 = 1 + 0.011 949 351 918 853 385 132 875 677 623 606 312 96;
  • 52) 0.011 949 351 918 853 385 132 875 677 623 606 312 96 × 2 = 0 + 0.023 898 703 837 706 770 265 751 355 247 212 625 92;
  • 53) 0.023 898 703 837 706 770 265 751 355 247 212 625 92 × 2 = 0 + 0.047 797 407 675 413 540 531 502 710 494 425 251 84;
  • 54) 0.047 797 407 675 413 540 531 502 710 494 425 251 84 × 2 = 0 + 0.095 594 815 350 827 081 063 005 420 988 850 503 68;
  • 55) 0.095 594 815 350 827 081 063 005 420 988 850 503 68 × 2 = 0 + 0.191 189 630 701 654 162 126 010 841 977 701 007 36;
  • 56) 0.191 189 630 701 654 162 126 010 841 977 701 007 36 × 2 = 0 + 0.382 379 261 403 308 324 252 021 683 955 402 014 72;
  • 57) 0.382 379 261 403 308 324 252 021 683 955 402 014 72 × 2 = 0 + 0.764 758 522 806 616 648 504 043 367 910 804 029 44;
  • 58) 0.764 758 522 806 616 648 504 043 367 910 804 029 44 × 2 = 1 + 0.529 517 045 613 233 297 008 086 735 821 608 058 88;
  • 59) 0.529 517 045 613 233 297 008 086 735 821 608 058 88 × 2 = 1 + 0.059 034 091 226 466 594 016 173 471 643 216 117 76;
  • 60) 0.059 034 091 226 466 594 016 173 471 643 216 117 76 × 2 = 0 + 0.118 068 182 452 933 188 032 346 943 286 432 235 52;
  • 61) 0.118 068 182 452 933 188 032 346 943 286 432 235 52 × 2 = 0 + 0.236 136 364 905 866 376 064 693 886 572 864 471 04;
  • 62) 0.236 136 364 905 866 376 064 693 886 572 864 471 04 × 2 = 0 + 0.472 272 729 811 732 752 129 387 773 145 728 942 08;
  • 63) 0.472 272 729 811 732 752 129 387 773 145 728 942 08 × 2 = 0 + 0.944 545 459 623 465 504 258 775 546 291 457 884 16;
  • 64) 0.944 545 459 623 465 504 258 775 546 291 457 884 16 × 2 = 1 + 0.889 090 919 246 931 008 517 551 092 582 915 768 32;
  • 65) 0.889 090 919 246 931 008 517 551 092 582 915 768 32 × 2 = 1 + 0.778 181 838 493 862 017 035 102 185 165 831 536 64;
  • 66) 0.778 181 838 493 862 017 035 102 185 165 831 536 64 × 2 = 1 + 0.556 363 676 987 724 034 070 204 370 331 663 073 28;
  • 67) 0.556 363 676 987 724 034 070 204 370 331 663 073 28 × 2 = 1 + 0.112 727 353 975 448 068 140 408 740 663 326 146 56;
  • 68) 0.112 727 353 975 448 068 140 408 740 663 326 146 56 × 2 = 0 + 0.225 454 707 950 896 136 280 817 481 326 652 293 12;
  • 69) 0.225 454 707 950 896 136 280 817 481 326 652 293 12 × 2 = 0 + 0.450 909 415 901 792 272 561 634 962 653 304 586 24;
  • 70) 0.450 909 415 901 792 272 561 634 962 653 304 586 24 × 2 = 0 + 0.901 818 831 803 584 545 123 269 925 306 609 172 48;
  • 71) 0.901 818 831 803 584 545 123 269 925 306 609 172 48 × 2 = 1 + 0.803 637 663 607 169 090 246 539 850 613 218 344 96;
  • 72) 0.803 637 663 607 169 090 246 539 850 613 218 344 96 × 2 = 1 + 0.607 275 327 214 338 180 493 079 701 226 436 689 92;
  • 73) 0.607 275 327 214 338 180 493 079 701 226 436 689 92 × 2 = 1 + 0.214 550 654 428 676 360 986 159 402 452 873 379 84;
  • 74) 0.214 550 654 428 676 360 986 159 402 452 873 379 84 × 2 = 0 + 0.429 101 308 857 352 721 972 318 804 905 746 759 68;
  • 75) 0.429 101 308 857 352 721 972 318 804 905 746 759 68 × 2 = 0 + 0.858 202 617 714 705 443 944 637 609 811 493 519 36;
  • 76) 0.858 202 617 714 705 443 944 637 609 811 493 519 36 × 2 = 1 + 0.716 405 235 429 410 887 889 275 219 622 987 038 72;
  • 77) 0.716 405 235 429 410 887 889 275 219 622 987 038 72 × 2 = 1 + 0.432 810 470 858 821 775 778 550 439 245 974 077 44;
  • 78) 0.432 810 470 858 821 775 778 550 439 245 974 077 44 × 2 = 0 + 0.865 620 941 717 643 551 557 100 878 491 948 154 88;
  • 79) 0.865 620 941 717 643 551 557 100 878 491 948 154 88 × 2 = 1 + 0.731 241 883 435 287 103 114 201 756 983 896 309 76;
  • 80) 0.731 241 883 435 287 103 114 201 756 983 896 309 76 × 2 = 1 + 0.462 483 766 870 574 206 228 403 513 967 792 619 52;
  • 81) 0.462 483 766 870 574 206 228 403 513 967 792 619 52 × 2 = 0 + 0.924 967 533 741 148 412 456 807 027 935 585 239 04;
  • 82) 0.924 967 533 741 148 412 456 807 027 935 585 239 04 × 2 = 1 + 0.849 935 067 482 296 824 913 614 055 871 170 478 08;
  • 83) 0.849 935 067 482 296 824 913 614 055 871 170 478 08 × 2 = 1 + 0.699 870 134 964 593 649 827 228 111 742 340 956 16;

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 52(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 52(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 52(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 52 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 73 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