0.000 000 000 634 751 367 889 197 801 221 261 633 16 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 16(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 16(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 16.

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 16 × 2 = 0 + 0.000 000 001 269 502 735 778 395 602 442 523 266 32;
  • 2) 0.000 000 001 269 502 735 778 395 602 442 523 266 32 × 2 = 0 + 0.000 000 002 539 005 471 556 791 204 885 046 532 64;
  • 3) 0.000 000 002 539 005 471 556 791 204 885 046 532 64 × 2 = 0 + 0.000 000 005 078 010 943 113 582 409 770 093 065 28;
  • 4) 0.000 000 005 078 010 943 113 582 409 770 093 065 28 × 2 = 0 + 0.000 000 010 156 021 886 227 164 819 540 186 130 56;
  • 5) 0.000 000 010 156 021 886 227 164 819 540 186 130 56 × 2 = 0 + 0.000 000 020 312 043 772 454 329 639 080 372 261 12;
  • 6) 0.000 000 020 312 043 772 454 329 639 080 372 261 12 × 2 = 0 + 0.000 000 040 624 087 544 908 659 278 160 744 522 24;
  • 7) 0.000 000 040 624 087 544 908 659 278 160 744 522 24 × 2 = 0 + 0.000 000 081 248 175 089 817 318 556 321 489 044 48;
  • 8) 0.000 000 081 248 175 089 817 318 556 321 489 044 48 × 2 = 0 + 0.000 000 162 496 350 179 634 637 112 642 978 088 96;
  • 9) 0.000 000 162 496 350 179 634 637 112 642 978 088 96 × 2 = 0 + 0.000 000 324 992 700 359 269 274 225 285 956 177 92;
  • 10) 0.000 000 324 992 700 359 269 274 225 285 956 177 92 × 2 = 0 + 0.000 000 649 985 400 718 538 548 450 571 912 355 84;
  • 11) 0.000 000 649 985 400 718 538 548 450 571 912 355 84 × 2 = 0 + 0.000 001 299 970 801 437 077 096 901 143 824 711 68;
  • 12) 0.000 001 299 970 801 437 077 096 901 143 824 711 68 × 2 = 0 + 0.000 002 599 941 602 874 154 193 802 287 649 423 36;
  • 13) 0.000 002 599 941 602 874 154 193 802 287 649 423 36 × 2 = 0 + 0.000 005 199 883 205 748 308 387 604 575 298 846 72;
  • 14) 0.000 005 199 883 205 748 308 387 604 575 298 846 72 × 2 = 0 + 0.000 010 399 766 411 496 616 775 209 150 597 693 44;
  • 15) 0.000 010 399 766 411 496 616 775 209 150 597 693 44 × 2 = 0 + 0.000 020 799 532 822 993 233 550 418 301 195 386 88;
  • 16) 0.000 020 799 532 822 993 233 550 418 301 195 386 88 × 2 = 0 + 0.000 041 599 065 645 986 467 100 836 602 390 773 76;
  • 17) 0.000 041 599 065 645 986 467 100 836 602 390 773 76 × 2 = 0 + 0.000 083 198 131 291 972 934 201 673 204 781 547 52;
  • 18) 0.000 083 198 131 291 972 934 201 673 204 781 547 52 × 2 = 0 + 0.000 166 396 262 583 945 868 403 346 409 563 095 04;
  • 19) 0.000 166 396 262 583 945 868 403 346 409 563 095 04 × 2 = 0 + 0.000 332 792 525 167 891 736 806 692 819 126 190 08;
  • 20) 0.000 332 792 525 167 891 736 806 692 819 126 190 08 × 2 = 0 + 0.000 665 585 050 335 783 473 613 385 638 252 380 16;
  • 21) 0.000 665 585 050 335 783 473 613 385 638 252 380 16 × 2 = 0 + 0.001 331 170 100 671 566 947 226 771 276 504 760 32;
  • 22) 0.001 331 170 100 671 566 947 226 771 276 504 760 32 × 2 = 0 + 0.002 662 340 201 343 133 894 453 542 553 009 520 64;
  • 23) 0.002 662 340 201 343 133 894 453 542 553 009 520 64 × 2 = 0 + 0.005 324 680 402 686 267 788 907 085 106 019 041 28;
  • 24) 0.005 324 680 402 686 267 788 907 085 106 019 041 28 × 2 = 0 + 0.010 649 360 805 372 535 577 814 170 212 038 082 56;
  • 25) 0.010 649 360 805 372 535 577 814 170 212 038 082 56 × 2 = 0 + 0.021 298 721 610 745 071 155 628 340 424 076 165 12;
  • 26) 0.021 298 721 610 745 071 155 628 340 424 076 165 12 × 2 = 0 + 0.042 597 443 221 490 142 311 256 680 848 152 330 24;
  • 27) 0.042 597 443 221 490 142 311 256 680 848 152 330 24 × 2 = 0 + 0.085 194 886 442 980 284 622 513 361 696 304 660 48;
  • 28) 0.085 194 886 442 980 284 622 513 361 696 304 660 48 × 2 = 0 + 0.170 389 772 885 960 569 245 026 723 392 609 320 96;
  • 29) 0.170 389 772 885 960 569 245 026 723 392 609 320 96 × 2 = 0 + 0.340 779 545 771 921 138 490 053 446 785 218 641 92;
  • 30) 0.340 779 545 771 921 138 490 053 446 785 218 641 92 × 2 = 0 + 0.681 559 091 543 842 276 980 106 893 570 437 283 84;
  • 31) 0.681 559 091 543 842 276 980 106 893 570 437 283 84 × 2 = 1 + 0.363 118 183 087 684 553 960 213 787 140 874 567 68;
  • 32) 0.363 118 183 087 684 553 960 213 787 140 874 567 68 × 2 = 0 + 0.726 236 366 175 369 107 920 427 574 281 749 135 36;
  • 33) 0.726 236 366 175 369 107 920 427 574 281 749 135 36 × 2 = 1 + 0.452 472 732 350 738 215 840 855 148 563 498 270 72;
  • 34) 0.452 472 732 350 738 215 840 855 148 563 498 270 72 × 2 = 0 + 0.904 945 464 701 476 431 681 710 297 126 996 541 44;
  • 35) 0.904 945 464 701 476 431 681 710 297 126 996 541 44 × 2 = 1 + 0.809 890 929 402 952 863 363 420 594 253 993 082 88;
  • 36) 0.809 890 929 402 952 863 363 420 594 253 993 082 88 × 2 = 1 + 0.619 781 858 805 905 726 726 841 188 507 986 165 76;
  • 37) 0.619 781 858 805 905 726 726 841 188 507 986 165 76 × 2 = 1 + 0.239 563 717 611 811 453 453 682 377 015 972 331 52;
  • 38) 0.239 563 717 611 811 453 453 682 377 015 972 331 52 × 2 = 0 + 0.479 127 435 223 622 906 907 364 754 031 944 663 04;
  • 39) 0.479 127 435 223 622 906 907 364 754 031 944 663 04 × 2 = 0 + 0.958 254 870 447 245 813 814 729 508 063 889 326 08;
  • 40) 0.958 254 870 447 245 813 814 729 508 063 889 326 08 × 2 = 1 + 0.916 509 740 894 491 627 629 459 016 127 778 652 16;
  • 41) 0.916 509 740 894 491 627 629 459 016 127 778 652 16 × 2 = 1 + 0.833 019 481 788 983 255 258 918 032 255 557 304 32;
  • 42) 0.833 019 481 788 983 255 258 918 032 255 557 304 32 × 2 = 1 + 0.666 038 963 577 966 510 517 836 064 511 114 608 64;
  • 43) 0.666 038 963 577 966 510 517 836 064 511 114 608 64 × 2 = 1 + 0.332 077 927 155 933 021 035 672 129 022 229 217 28;
  • 44) 0.332 077 927 155 933 021 035 672 129 022 229 217 28 × 2 = 0 + 0.664 155 854 311 866 042 071 344 258 044 458 434 56;
  • 45) 0.664 155 854 311 866 042 071 344 258 044 458 434 56 × 2 = 1 + 0.328 311 708 623 732 084 142 688 516 088 916 869 12;
  • 46) 0.328 311 708 623 732 084 142 688 516 088 916 869 12 × 2 = 0 + 0.656 623 417 247 464 168 285 377 032 177 833 738 24;
  • 47) 0.656 623 417 247 464 168 285 377 032 177 833 738 24 × 2 = 1 + 0.313 246 834 494 928 336 570 754 064 355 667 476 48;
  • 48) 0.313 246 834 494 928 336 570 754 064 355 667 476 48 × 2 = 0 + 0.626 493 668 989 856 673 141 508 128 711 334 952 96;
  • 49) 0.626 493 668 989 856 673 141 508 128 711 334 952 96 × 2 = 1 + 0.252 987 337 979 713 346 283 016 257 422 669 905 92;
  • 50) 0.252 987 337 979 713 346 283 016 257 422 669 905 92 × 2 = 0 + 0.505 974 675 959 426 692 566 032 514 845 339 811 84;
  • 51) 0.505 974 675 959 426 692 566 032 514 845 339 811 84 × 2 = 1 + 0.011 949 351 918 853 385 132 065 029 690 679 623 68;
  • 52) 0.011 949 351 918 853 385 132 065 029 690 679 623 68 × 2 = 0 + 0.023 898 703 837 706 770 264 130 059 381 359 247 36;
  • 53) 0.023 898 703 837 706 770 264 130 059 381 359 247 36 × 2 = 0 + 0.047 797 407 675 413 540 528 260 118 762 718 494 72;
  • 54) 0.047 797 407 675 413 540 528 260 118 762 718 494 72 × 2 = 0 + 0.095 594 815 350 827 081 056 520 237 525 436 989 44;
  • 55) 0.095 594 815 350 827 081 056 520 237 525 436 989 44 × 2 = 0 + 0.191 189 630 701 654 162 113 040 475 050 873 978 88;
  • 56) 0.191 189 630 701 654 162 113 040 475 050 873 978 88 × 2 = 0 + 0.382 379 261 403 308 324 226 080 950 101 747 957 76;
  • 57) 0.382 379 261 403 308 324 226 080 950 101 747 957 76 × 2 = 0 + 0.764 758 522 806 616 648 452 161 900 203 495 915 52;
  • 58) 0.764 758 522 806 616 648 452 161 900 203 495 915 52 × 2 = 1 + 0.529 517 045 613 233 296 904 323 800 406 991 831 04;
  • 59) 0.529 517 045 613 233 296 904 323 800 406 991 831 04 × 2 = 1 + 0.059 034 091 226 466 593 808 647 600 813 983 662 08;
  • 60) 0.059 034 091 226 466 593 808 647 600 813 983 662 08 × 2 = 0 + 0.118 068 182 452 933 187 617 295 201 627 967 324 16;
  • 61) 0.118 068 182 452 933 187 617 295 201 627 967 324 16 × 2 = 0 + 0.236 136 364 905 866 375 234 590 403 255 934 648 32;
  • 62) 0.236 136 364 905 866 375 234 590 403 255 934 648 32 × 2 = 0 + 0.472 272 729 811 732 750 469 180 806 511 869 296 64;
  • 63) 0.472 272 729 811 732 750 469 180 806 511 869 296 64 × 2 = 0 + 0.944 545 459 623 465 500 938 361 613 023 738 593 28;
  • 64) 0.944 545 459 623 465 500 938 361 613 023 738 593 28 × 2 = 1 + 0.889 090 919 246 931 001 876 723 226 047 477 186 56;
  • 65) 0.889 090 919 246 931 001 876 723 226 047 477 186 56 × 2 = 1 + 0.778 181 838 493 862 003 753 446 452 094 954 373 12;
  • 66) 0.778 181 838 493 862 003 753 446 452 094 954 373 12 × 2 = 1 + 0.556 363 676 987 724 007 506 892 904 189 908 746 24;
  • 67) 0.556 363 676 987 724 007 506 892 904 189 908 746 24 × 2 = 1 + 0.112 727 353 975 448 015 013 785 808 379 817 492 48;
  • 68) 0.112 727 353 975 448 015 013 785 808 379 817 492 48 × 2 = 0 + 0.225 454 707 950 896 030 027 571 616 759 634 984 96;
  • 69) 0.225 454 707 950 896 030 027 571 616 759 634 984 96 × 2 = 0 + 0.450 909 415 901 792 060 055 143 233 519 269 969 92;
  • 70) 0.450 909 415 901 792 060 055 143 233 519 269 969 92 × 2 = 0 + 0.901 818 831 803 584 120 110 286 467 038 539 939 84;
  • 71) 0.901 818 831 803 584 120 110 286 467 038 539 939 84 × 2 = 1 + 0.803 637 663 607 168 240 220 572 934 077 079 879 68;
  • 72) 0.803 637 663 607 168 240 220 572 934 077 079 879 68 × 2 = 1 + 0.607 275 327 214 336 480 441 145 868 154 159 759 36;
  • 73) 0.607 275 327 214 336 480 441 145 868 154 159 759 36 × 2 = 1 + 0.214 550 654 428 672 960 882 291 736 308 319 518 72;
  • 74) 0.214 550 654 428 672 960 882 291 736 308 319 518 72 × 2 = 0 + 0.429 101 308 857 345 921 764 583 472 616 639 037 44;
  • 75) 0.429 101 308 857 345 921 764 583 472 616 639 037 44 × 2 = 0 + 0.858 202 617 714 691 843 529 166 945 233 278 074 88;
  • 76) 0.858 202 617 714 691 843 529 166 945 233 278 074 88 × 2 = 1 + 0.716 405 235 429 383 687 058 333 890 466 556 149 76;
  • 77) 0.716 405 235 429 383 687 058 333 890 466 556 149 76 × 2 = 1 + 0.432 810 470 858 767 374 116 667 780 933 112 299 52;
  • 78) 0.432 810 470 858 767 374 116 667 780 933 112 299 52 × 2 = 0 + 0.865 620 941 717 534 748 233 335 561 866 224 599 04;
  • 79) 0.865 620 941 717 534 748 233 335 561 866 224 599 04 × 2 = 1 + 0.731 241 883 435 069 496 466 671 123 732 449 198 08;
  • 80) 0.731 241 883 435 069 496 466 671 123 732 449 198 08 × 2 = 1 + 0.462 483 766 870 138 992 933 342 247 464 898 396 16;
  • 81) 0.462 483 766 870 138 992 933 342 247 464 898 396 16 × 2 = 0 + 0.924 967 533 740 277 985 866 684 494 929 796 792 32;
  • 82) 0.924 967 533 740 277 985 866 684 494 929 796 792 32 × 2 = 1 + 0.849 935 067 480 555 971 733 368 989 859 593 584 64;
  • 83) 0.849 935 067 480 555 971 733 368 989 859 593 584 64 × 2 = 1 + 0.699 870 134 961 111 943 466 737 979 719 187 169 28;

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 16(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 16(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 16(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 16 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 58 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