0.000 000 000 634 751 367 889 197 801 221 261 632 19 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 634 751 367 889 197 801 221 261 632 19(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
0.000 000 000 634 751 367 889 197 801 221 261 632 19(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: 0.000 000 000 634 751 367 889 197 801 221 261 632 19.

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) 0.000 000 000 634 751 367 889 197 801 221 261 632 19 × 2 = 0 + 0.000 000 001 269 502 735 778 395 602 442 523 264 38;
  • 2) 0.000 000 001 269 502 735 778 395 602 442 523 264 38 × 2 = 0 + 0.000 000 002 539 005 471 556 791 204 885 046 528 76;
  • 3) 0.000 000 002 539 005 471 556 791 204 885 046 528 76 × 2 = 0 + 0.000 000 005 078 010 943 113 582 409 770 093 057 52;
  • 4) 0.000 000 005 078 010 943 113 582 409 770 093 057 52 × 2 = 0 + 0.000 000 010 156 021 886 227 164 819 540 186 115 04;
  • 5) 0.000 000 010 156 021 886 227 164 819 540 186 115 04 × 2 = 0 + 0.000 000 020 312 043 772 454 329 639 080 372 230 08;
  • 6) 0.000 000 020 312 043 772 454 329 639 080 372 230 08 × 2 = 0 + 0.000 000 040 624 087 544 908 659 278 160 744 460 16;
  • 7) 0.000 000 040 624 087 544 908 659 278 160 744 460 16 × 2 = 0 + 0.000 000 081 248 175 089 817 318 556 321 488 920 32;
  • 8) 0.000 000 081 248 175 089 817 318 556 321 488 920 32 × 2 = 0 + 0.000 000 162 496 350 179 634 637 112 642 977 840 64;
  • 9) 0.000 000 162 496 350 179 634 637 112 642 977 840 64 × 2 = 0 + 0.000 000 324 992 700 359 269 274 225 285 955 681 28;
  • 10) 0.000 000 324 992 700 359 269 274 225 285 955 681 28 × 2 = 0 + 0.000 000 649 985 400 718 538 548 450 571 911 362 56;
  • 11) 0.000 000 649 985 400 718 538 548 450 571 911 362 56 × 2 = 0 + 0.000 001 299 970 801 437 077 096 901 143 822 725 12;
  • 12) 0.000 001 299 970 801 437 077 096 901 143 822 725 12 × 2 = 0 + 0.000 002 599 941 602 874 154 193 802 287 645 450 24;
  • 13) 0.000 002 599 941 602 874 154 193 802 287 645 450 24 × 2 = 0 + 0.000 005 199 883 205 748 308 387 604 575 290 900 48;
  • 14) 0.000 005 199 883 205 748 308 387 604 575 290 900 48 × 2 = 0 + 0.000 010 399 766 411 496 616 775 209 150 581 800 96;
  • 15) 0.000 010 399 766 411 496 616 775 209 150 581 800 96 × 2 = 0 + 0.000 020 799 532 822 993 233 550 418 301 163 601 92;
  • 16) 0.000 020 799 532 822 993 233 550 418 301 163 601 92 × 2 = 0 + 0.000 041 599 065 645 986 467 100 836 602 327 203 84;
  • 17) 0.000 041 599 065 645 986 467 100 836 602 327 203 84 × 2 = 0 + 0.000 083 198 131 291 972 934 201 673 204 654 407 68;
  • 18) 0.000 083 198 131 291 972 934 201 673 204 654 407 68 × 2 = 0 + 0.000 166 396 262 583 945 868 403 346 409 308 815 36;
  • 19) 0.000 166 396 262 583 945 868 403 346 409 308 815 36 × 2 = 0 + 0.000 332 792 525 167 891 736 806 692 818 617 630 72;
  • 20) 0.000 332 792 525 167 891 736 806 692 818 617 630 72 × 2 = 0 + 0.000 665 585 050 335 783 473 613 385 637 235 261 44;
  • 21) 0.000 665 585 050 335 783 473 613 385 637 235 261 44 × 2 = 0 + 0.001 331 170 100 671 566 947 226 771 274 470 522 88;
  • 22) 0.001 331 170 100 671 566 947 226 771 274 470 522 88 × 2 = 0 + 0.002 662 340 201 343 133 894 453 542 548 941 045 76;
  • 23) 0.002 662 340 201 343 133 894 453 542 548 941 045 76 × 2 = 0 + 0.005 324 680 402 686 267 788 907 085 097 882 091 52;
  • 24) 0.005 324 680 402 686 267 788 907 085 097 882 091 52 × 2 = 0 + 0.010 649 360 805 372 535 577 814 170 195 764 183 04;
  • 25) 0.010 649 360 805 372 535 577 814 170 195 764 183 04 × 2 = 0 + 0.021 298 721 610 745 071 155 628 340 391 528 366 08;
  • 26) 0.021 298 721 610 745 071 155 628 340 391 528 366 08 × 2 = 0 + 0.042 597 443 221 490 142 311 256 680 783 056 732 16;
  • 27) 0.042 597 443 221 490 142 311 256 680 783 056 732 16 × 2 = 0 + 0.085 194 886 442 980 284 622 513 361 566 113 464 32;
  • 28) 0.085 194 886 442 980 284 622 513 361 566 113 464 32 × 2 = 0 + 0.170 389 772 885 960 569 245 026 723 132 226 928 64;
  • 29) 0.170 389 772 885 960 569 245 026 723 132 226 928 64 × 2 = 0 + 0.340 779 545 771 921 138 490 053 446 264 453 857 28;
  • 30) 0.340 779 545 771 921 138 490 053 446 264 453 857 28 × 2 = 0 + 0.681 559 091 543 842 276 980 106 892 528 907 714 56;
  • 31) 0.681 559 091 543 842 276 980 106 892 528 907 714 56 × 2 = 1 + 0.363 118 183 087 684 553 960 213 785 057 815 429 12;
  • 32) 0.363 118 183 087 684 553 960 213 785 057 815 429 12 × 2 = 0 + 0.726 236 366 175 369 107 920 427 570 115 630 858 24;
  • 33) 0.726 236 366 175 369 107 920 427 570 115 630 858 24 × 2 = 1 + 0.452 472 732 350 738 215 840 855 140 231 261 716 48;
  • 34) 0.452 472 732 350 738 215 840 855 140 231 261 716 48 × 2 = 0 + 0.904 945 464 701 476 431 681 710 280 462 523 432 96;
  • 35) 0.904 945 464 701 476 431 681 710 280 462 523 432 96 × 2 = 1 + 0.809 890 929 402 952 863 363 420 560 925 046 865 92;
  • 36) 0.809 890 929 402 952 863 363 420 560 925 046 865 92 × 2 = 1 + 0.619 781 858 805 905 726 726 841 121 850 093 731 84;
  • 37) 0.619 781 858 805 905 726 726 841 121 850 093 731 84 × 2 = 1 + 0.239 563 717 611 811 453 453 682 243 700 187 463 68;
  • 38) 0.239 563 717 611 811 453 453 682 243 700 187 463 68 × 2 = 0 + 0.479 127 435 223 622 906 907 364 487 400 374 927 36;
  • 39) 0.479 127 435 223 622 906 907 364 487 400 374 927 36 × 2 = 0 + 0.958 254 870 447 245 813 814 728 974 800 749 854 72;
  • 40) 0.958 254 870 447 245 813 814 728 974 800 749 854 72 × 2 = 1 + 0.916 509 740 894 491 627 629 457 949 601 499 709 44;
  • 41) 0.916 509 740 894 491 627 629 457 949 601 499 709 44 × 2 = 1 + 0.833 019 481 788 983 255 258 915 899 202 999 418 88;
  • 42) 0.833 019 481 788 983 255 258 915 899 202 999 418 88 × 2 = 1 + 0.666 038 963 577 966 510 517 831 798 405 998 837 76;
  • 43) 0.666 038 963 577 966 510 517 831 798 405 998 837 76 × 2 = 1 + 0.332 077 927 155 933 021 035 663 596 811 997 675 52;
  • 44) 0.332 077 927 155 933 021 035 663 596 811 997 675 52 × 2 = 0 + 0.664 155 854 311 866 042 071 327 193 623 995 351 04;
  • 45) 0.664 155 854 311 866 042 071 327 193 623 995 351 04 × 2 = 1 + 0.328 311 708 623 732 084 142 654 387 247 990 702 08;
  • 46) 0.328 311 708 623 732 084 142 654 387 247 990 702 08 × 2 = 0 + 0.656 623 417 247 464 168 285 308 774 495 981 404 16;
  • 47) 0.656 623 417 247 464 168 285 308 774 495 981 404 16 × 2 = 1 + 0.313 246 834 494 928 336 570 617 548 991 962 808 32;
  • 48) 0.313 246 834 494 928 336 570 617 548 991 962 808 32 × 2 = 0 + 0.626 493 668 989 856 673 141 235 097 983 925 616 64;
  • 49) 0.626 493 668 989 856 673 141 235 097 983 925 616 64 × 2 = 1 + 0.252 987 337 979 713 346 282 470 195 967 851 233 28;
  • 50) 0.252 987 337 979 713 346 282 470 195 967 851 233 28 × 2 = 0 + 0.505 974 675 959 426 692 564 940 391 935 702 466 56;
  • 51) 0.505 974 675 959 426 692 564 940 391 935 702 466 56 × 2 = 1 + 0.011 949 351 918 853 385 129 880 783 871 404 933 12;
  • 52) 0.011 949 351 918 853 385 129 880 783 871 404 933 12 × 2 = 0 + 0.023 898 703 837 706 770 259 761 567 742 809 866 24;
  • 53) 0.023 898 703 837 706 770 259 761 567 742 809 866 24 × 2 = 0 + 0.047 797 407 675 413 540 519 523 135 485 619 732 48;
  • 54) 0.047 797 407 675 413 540 519 523 135 485 619 732 48 × 2 = 0 + 0.095 594 815 350 827 081 039 046 270 971 239 464 96;
  • 55) 0.095 594 815 350 827 081 039 046 270 971 239 464 96 × 2 = 0 + 0.191 189 630 701 654 162 078 092 541 942 478 929 92;
  • 56) 0.191 189 630 701 654 162 078 092 541 942 478 929 92 × 2 = 0 + 0.382 379 261 403 308 324 156 185 083 884 957 859 84;
  • 57) 0.382 379 261 403 308 324 156 185 083 884 957 859 84 × 2 = 0 + 0.764 758 522 806 616 648 312 370 167 769 915 719 68;
  • 58) 0.764 758 522 806 616 648 312 370 167 769 915 719 68 × 2 = 1 + 0.529 517 045 613 233 296 624 740 335 539 831 439 36;
  • 59) 0.529 517 045 613 233 296 624 740 335 539 831 439 36 × 2 = 1 + 0.059 034 091 226 466 593 249 480 671 079 662 878 72;
  • 60) 0.059 034 091 226 466 593 249 480 671 079 662 878 72 × 2 = 0 + 0.118 068 182 452 933 186 498 961 342 159 325 757 44;
  • 61) 0.118 068 182 452 933 186 498 961 342 159 325 757 44 × 2 = 0 + 0.236 136 364 905 866 372 997 922 684 318 651 514 88;
  • 62) 0.236 136 364 905 866 372 997 922 684 318 651 514 88 × 2 = 0 + 0.472 272 729 811 732 745 995 845 368 637 303 029 76;
  • 63) 0.472 272 729 811 732 745 995 845 368 637 303 029 76 × 2 = 0 + 0.944 545 459 623 465 491 991 690 737 274 606 059 52;
  • 64) 0.944 545 459 623 465 491 991 690 737 274 606 059 52 × 2 = 1 + 0.889 090 919 246 930 983 983 381 474 549 212 119 04;
  • 65) 0.889 090 919 246 930 983 983 381 474 549 212 119 04 × 2 = 1 + 0.778 181 838 493 861 967 966 762 949 098 424 238 08;
  • 66) 0.778 181 838 493 861 967 966 762 949 098 424 238 08 × 2 = 1 + 0.556 363 676 987 723 935 933 525 898 196 848 476 16;
  • 67) 0.556 363 676 987 723 935 933 525 898 196 848 476 16 × 2 = 1 + 0.112 727 353 975 447 871 867 051 796 393 696 952 32;
  • 68) 0.112 727 353 975 447 871 867 051 796 393 696 952 32 × 2 = 0 + 0.225 454 707 950 895 743 734 103 592 787 393 904 64;
  • 69) 0.225 454 707 950 895 743 734 103 592 787 393 904 64 × 2 = 0 + 0.450 909 415 901 791 487 468 207 185 574 787 809 28;
  • 70) 0.450 909 415 901 791 487 468 207 185 574 787 809 28 × 2 = 0 + 0.901 818 831 803 582 974 936 414 371 149 575 618 56;
  • 71) 0.901 818 831 803 582 974 936 414 371 149 575 618 56 × 2 = 1 + 0.803 637 663 607 165 949 872 828 742 299 151 237 12;
  • 72) 0.803 637 663 607 165 949 872 828 742 299 151 237 12 × 2 = 1 + 0.607 275 327 214 331 899 745 657 484 598 302 474 24;
  • 73) 0.607 275 327 214 331 899 745 657 484 598 302 474 24 × 2 = 1 + 0.214 550 654 428 663 799 491 314 969 196 604 948 48;
  • 74) 0.214 550 654 428 663 799 491 314 969 196 604 948 48 × 2 = 0 + 0.429 101 308 857 327 598 982 629 938 393 209 896 96;
  • 75) 0.429 101 308 857 327 598 982 629 938 393 209 896 96 × 2 = 0 + 0.858 202 617 714 655 197 965 259 876 786 419 793 92;
  • 76) 0.858 202 617 714 655 197 965 259 876 786 419 793 92 × 2 = 1 + 0.716 405 235 429 310 395 930 519 753 572 839 587 84;
  • 77) 0.716 405 235 429 310 395 930 519 753 572 839 587 84 × 2 = 1 + 0.432 810 470 858 620 791 861 039 507 145 679 175 68;
  • 78) 0.432 810 470 858 620 791 861 039 507 145 679 175 68 × 2 = 0 + 0.865 620 941 717 241 583 722 079 014 291 358 351 36;
  • 79) 0.865 620 941 717 241 583 722 079 014 291 358 351 36 × 2 = 1 + 0.731 241 883 434 483 167 444 158 028 582 716 702 72;
  • 80) 0.731 241 883 434 483 167 444 158 028 582 716 702 72 × 2 = 1 + 0.462 483 766 868 966 334 888 316 057 165 433 405 44;
  • 81) 0.462 483 766 868 966 334 888 316 057 165 433 405 44 × 2 = 0 + 0.924 967 533 737 932 669 776 632 114 330 866 810 88;
  • 82) 0.924 967 533 737 932 669 776 632 114 330 866 810 88 × 2 = 1 + 0.849 935 067 475 865 339 553 264 228 661 733 621 76;
  • 83) 0.849 935 067 475 865 339 553 264 228 661 733 621 76 × 2 = 1 + 0.699 870 134 951 730 679 106 528 457 323 467 243 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:


0.000 000 000 634 751 367 889 197 801 221 261 632 19(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:

0.000 000 000 634 751 367 889 197 801 221 261 632 19(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:


0.000 000 000 634 751 367 889 197 801 221 261 632 19(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 0.000 000 000 634 751 367 889 197 801 221 261 632 19 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 0.000 000 000 634 751 367 889 197 801 221 261 632 94 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