Decimal to 64 Bit IEEE 754 Binary: Convert Number 0.000 000 000 634 751 367 889 197 801 221 261 631 67 to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From Base Ten Decimal System

Number 0.000 000 000 634 751 367 889 197 801 221 261 631 67(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation standard (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: 0.000 000 000 634 751 367 889 197 801 221 261 631 67.

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) 0.000 000 000 634 751 367 889 197 801 221 261 631 67 × 2 = 0 + 0.000 000 001 269 502 735 778 395 602 442 523 263 34;
  • 2) 0.000 000 001 269 502 735 778 395 602 442 523 263 34 × 2 = 0 + 0.000 000 002 539 005 471 556 791 204 885 046 526 68;
  • 3) 0.000 000 002 539 005 471 556 791 204 885 046 526 68 × 2 = 0 + 0.000 000 005 078 010 943 113 582 409 770 093 053 36;
  • 4) 0.000 000 005 078 010 943 113 582 409 770 093 053 36 × 2 = 0 + 0.000 000 010 156 021 886 227 164 819 540 186 106 72;
  • 5) 0.000 000 010 156 021 886 227 164 819 540 186 106 72 × 2 = 0 + 0.000 000 020 312 043 772 454 329 639 080 372 213 44;
  • 6) 0.000 000 020 312 043 772 454 329 639 080 372 213 44 × 2 = 0 + 0.000 000 040 624 087 544 908 659 278 160 744 426 88;
  • 7) 0.000 000 040 624 087 544 908 659 278 160 744 426 88 × 2 = 0 + 0.000 000 081 248 175 089 817 318 556 321 488 853 76;
  • 8) 0.000 000 081 248 175 089 817 318 556 321 488 853 76 × 2 = 0 + 0.000 000 162 496 350 179 634 637 112 642 977 707 52;
  • 9) 0.000 000 162 496 350 179 634 637 112 642 977 707 52 × 2 = 0 + 0.000 000 324 992 700 359 269 274 225 285 955 415 04;
  • 10) 0.000 000 324 992 700 359 269 274 225 285 955 415 04 × 2 = 0 + 0.000 000 649 985 400 718 538 548 450 571 910 830 08;
  • 11) 0.000 000 649 985 400 718 538 548 450 571 910 830 08 × 2 = 0 + 0.000 001 299 970 801 437 077 096 901 143 821 660 16;
  • 12) 0.000 001 299 970 801 437 077 096 901 143 821 660 16 × 2 = 0 + 0.000 002 599 941 602 874 154 193 802 287 643 320 32;
  • 13) 0.000 002 599 941 602 874 154 193 802 287 643 320 32 × 2 = 0 + 0.000 005 199 883 205 748 308 387 604 575 286 640 64;
  • 14) 0.000 005 199 883 205 748 308 387 604 575 286 640 64 × 2 = 0 + 0.000 010 399 766 411 496 616 775 209 150 573 281 28;
  • 15) 0.000 010 399 766 411 496 616 775 209 150 573 281 28 × 2 = 0 + 0.000 020 799 532 822 993 233 550 418 301 146 562 56;
  • 16) 0.000 020 799 532 822 993 233 550 418 301 146 562 56 × 2 = 0 + 0.000 041 599 065 645 986 467 100 836 602 293 125 12;
  • 17) 0.000 041 599 065 645 986 467 100 836 602 293 125 12 × 2 = 0 + 0.000 083 198 131 291 972 934 201 673 204 586 250 24;
  • 18) 0.000 083 198 131 291 972 934 201 673 204 586 250 24 × 2 = 0 + 0.000 166 396 262 583 945 868 403 346 409 172 500 48;
  • 19) 0.000 166 396 262 583 945 868 403 346 409 172 500 48 × 2 = 0 + 0.000 332 792 525 167 891 736 806 692 818 345 000 96;
  • 20) 0.000 332 792 525 167 891 736 806 692 818 345 000 96 × 2 = 0 + 0.000 665 585 050 335 783 473 613 385 636 690 001 92;
  • 21) 0.000 665 585 050 335 783 473 613 385 636 690 001 92 × 2 = 0 + 0.001 331 170 100 671 566 947 226 771 273 380 003 84;
  • 22) 0.001 331 170 100 671 566 947 226 771 273 380 003 84 × 2 = 0 + 0.002 662 340 201 343 133 894 453 542 546 760 007 68;
  • 23) 0.002 662 340 201 343 133 894 453 542 546 760 007 68 × 2 = 0 + 0.005 324 680 402 686 267 788 907 085 093 520 015 36;
  • 24) 0.005 324 680 402 686 267 788 907 085 093 520 015 36 × 2 = 0 + 0.010 649 360 805 372 535 577 814 170 187 040 030 72;
  • 25) 0.010 649 360 805 372 535 577 814 170 187 040 030 72 × 2 = 0 + 0.021 298 721 610 745 071 155 628 340 374 080 061 44;
  • 26) 0.021 298 721 610 745 071 155 628 340 374 080 061 44 × 2 = 0 + 0.042 597 443 221 490 142 311 256 680 748 160 122 88;
  • 27) 0.042 597 443 221 490 142 311 256 680 748 160 122 88 × 2 = 0 + 0.085 194 886 442 980 284 622 513 361 496 320 245 76;
  • 28) 0.085 194 886 442 980 284 622 513 361 496 320 245 76 × 2 = 0 + 0.170 389 772 885 960 569 245 026 722 992 640 491 52;
  • 29) 0.170 389 772 885 960 569 245 026 722 992 640 491 52 × 2 = 0 + 0.340 779 545 771 921 138 490 053 445 985 280 983 04;
  • 30) 0.340 779 545 771 921 138 490 053 445 985 280 983 04 × 2 = 0 + 0.681 559 091 543 842 276 980 106 891 970 561 966 08;
  • 31) 0.681 559 091 543 842 276 980 106 891 970 561 966 08 × 2 = 1 + 0.363 118 183 087 684 553 960 213 783 941 123 932 16;
  • 32) 0.363 118 183 087 684 553 960 213 783 941 123 932 16 × 2 = 0 + 0.726 236 366 175 369 107 920 427 567 882 247 864 32;
  • 33) 0.726 236 366 175 369 107 920 427 567 882 247 864 32 × 2 = 1 + 0.452 472 732 350 738 215 840 855 135 764 495 728 64;
  • 34) 0.452 472 732 350 738 215 840 855 135 764 495 728 64 × 2 = 0 + 0.904 945 464 701 476 431 681 710 271 528 991 457 28;
  • 35) 0.904 945 464 701 476 431 681 710 271 528 991 457 28 × 2 = 1 + 0.809 890 929 402 952 863 363 420 543 057 982 914 56;
  • 36) 0.809 890 929 402 952 863 363 420 543 057 982 914 56 × 2 = 1 + 0.619 781 858 805 905 726 726 841 086 115 965 829 12;
  • 37) 0.619 781 858 805 905 726 726 841 086 115 965 829 12 × 2 = 1 + 0.239 563 717 611 811 453 453 682 172 231 931 658 24;
  • 38) 0.239 563 717 611 811 453 453 682 172 231 931 658 24 × 2 = 0 + 0.479 127 435 223 622 906 907 364 344 463 863 316 48;
  • 39) 0.479 127 435 223 622 906 907 364 344 463 863 316 48 × 2 = 0 + 0.958 254 870 447 245 813 814 728 688 927 726 632 96;
  • 40) 0.958 254 870 447 245 813 814 728 688 927 726 632 96 × 2 = 1 + 0.916 509 740 894 491 627 629 457 377 855 453 265 92;
  • 41) 0.916 509 740 894 491 627 629 457 377 855 453 265 92 × 2 = 1 + 0.833 019 481 788 983 255 258 914 755 710 906 531 84;
  • 42) 0.833 019 481 788 983 255 258 914 755 710 906 531 84 × 2 = 1 + 0.666 038 963 577 966 510 517 829 511 421 813 063 68;
  • 43) 0.666 038 963 577 966 510 517 829 511 421 813 063 68 × 2 = 1 + 0.332 077 927 155 933 021 035 659 022 843 626 127 36;
  • 44) 0.332 077 927 155 933 021 035 659 022 843 626 127 36 × 2 = 0 + 0.664 155 854 311 866 042 071 318 045 687 252 254 72;
  • 45) 0.664 155 854 311 866 042 071 318 045 687 252 254 72 × 2 = 1 + 0.328 311 708 623 732 084 142 636 091 374 504 509 44;
  • 46) 0.328 311 708 623 732 084 142 636 091 374 504 509 44 × 2 = 0 + 0.656 623 417 247 464 168 285 272 182 749 009 018 88;
  • 47) 0.656 623 417 247 464 168 285 272 182 749 009 018 88 × 2 = 1 + 0.313 246 834 494 928 336 570 544 365 498 018 037 76;
  • 48) 0.313 246 834 494 928 336 570 544 365 498 018 037 76 × 2 = 0 + 0.626 493 668 989 856 673 141 088 730 996 036 075 52;
  • 49) 0.626 493 668 989 856 673 141 088 730 996 036 075 52 × 2 = 1 + 0.252 987 337 979 713 346 282 177 461 992 072 151 04;
  • 50) 0.252 987 337 979 713 346 282 177 461 992 072 151 04 × 2 = 0 + 0.505 974 675 959 426 692 564 354 923 984 144 302 08;
  • 51) 0.505 974 675 959 426 692 564 354 923 984 144 302 08 × 2 = 1 + 0.011 949 351 918 853 385 128 709 847 968 288 604 16;
  • 52) 0.011 949 351 918 853 385 128 709 847 968 288 604 16 × 2 = 0 + 0.023 898 703 837 706 770 257 419 695 936 577 208 32;
  • 53) 0.023 898 703 837 706 770 257 419 695 936 577 208 32 × 2 = 0 + 0.047 797 407 675 413 540 514 839 391 873 154 416 64;
  • 54) 0.047 797 407 675 413 540 514 839 391 873 154 416 64 × 2 = 0 + 0.095 594 815 350 827 081 029 678 783 746 308 833 28;
  • 55) 0.095 594 815 350 827 081 029 678 783 746 308 833 28 × 2 = 0 + 0.191 189 630 701 654 162 059 357 567 492 617 666 56;
  • 56) 0.191 189 630 701 654 162 059 357 567 492 617 666 56 × 2 = 0 + 0.382 379 261 403 308 324 118 715 134 985 235 333 12;
  • 57) 0.382 379 261 403 308 324 118 715 134 985 235 333 12 × 2 = 0 + 0.764 758 522 806 616 648 237 430 269 970 470 666 24;
  • 58) 0.764 758 522 806 616 648 237 430 269 970 470 666 24 × 2 = 1 + 0.529 517 045 613 233 296 474 860 539 940 941 332 48;
  • 59) 0.529 517 045 613 233 296 474 860 539 940 941 332 48 × 2 = 1 + 0.059 034 091 226 466 592 949 721 079 881 882 664 96;
  • 60) 0.059 034 091 226 466 592 949 721 079 881 882 664 96 × 2 = 0 + 0.118 068 182 452 933 185 899 442 159 763 765 329 92;
  • 61) 0.118 068 182 452 933 185 899 442 159 763 765 329 92 × 2 = 0 + 0.236 136 364 905 866 371 798 884 319 527 530 659 84;
  • 62) 0.236 136 364 905 866 371 798 884 319 527 530 659 84 × 2 = 0 + 0.472 272 729 811 732 743 597 768 639 055 061 319 68;
  • 63) 0.472 272 729 811 732 743 597 768 639 055 061 319 68 × 2 = 0 + 0.944 545 459 623 465 487 195 537 278 110 122 639 36;
  • 64) 0.944 545 459 623 465 487 195 537 278 110 122 639 36 × 2 = 1 + 0.889 090 919 246 930 974 391 074 556 220 245 278 72;
  • 65) 0.889 090 919 246 930 974 391 074 556 220 245 278 72 × 2 = 1 + 0.778 181 838 493 861 948 782 149 112 440 490 557 44;
  • 66) 0.778 181 838 493 861 948 782 149 112 440 490 557 44 × 2 = 1 + 0.556 363 676 987 723 897 564 298 224 880 981 114 88;
  • 67) 0.556 363 676 987 723 897 564 298 224 880 981 114 88 × 2 = 1 + 0.112 727 353 975 447 795 128 596 449 761 962 229 76;
  • 68) 0.112 727 353 975 447 795 128 596 449 761 962 229 76 × 2 = 0 + 0.225 454 707 950 895 590 257 192 899 523 924 459 52;
  • 69) 0.225 454 707 950 895 590 257 192 899 523 924 459 52 × 2 = 0 + 0.450 909 415 901 791 180 514 385 799 047 848 919 04;
  • 70) 0.450 909 415 901 791 180 514 385 799 047 848 919 04 × 2 = 0 + 0.901 818 831 803 582 361 028 771 598 095 697 838 08;
  • 71) 0.901 818 831 803 582 361 028 771 598 095 697 838 08 × 2 = 1 + 0.803 637 663 607 164 722 057 543 196 191 395 676 16;
  • 72) 0.803 637 663 607 164 722 057 543 196 191 395 676 16 × 2 = 1 + 0.607 275 327 214 329 444 115 086 392 382 791 352 32;
  • 73) 0.607 275 327 214 329 444 115 086 392 382 791 352 32 × 2 = 1 + 0.214 550 654 428 658 888 230 172 784 765 582 704 64;
  • 74) 0.214 550 654 428 658 888 230 172 784 765 582 704 64 × 2 = 0 + 0.429 101 308 857 317 776 460 345 569 531 165 409 28;
  • 75) 0.429 101 308 857 317 776 460 345 569 531 165 409 28 × 2 = 0 + 0.858 202 617 714 635 552 920 691 139 062 330 818 56;
  • 76) 0.858 202 617 714 635 552 920 691 139 062 330 818 56 × 2 = 1 + 0.716 405 235 429 271 105 841 382 278 124 661 637 12;
  • 77) 0.716 405 235 429 271 105 841 382 278 124 661 637 12 × 2 = 1 + 0.432 810 470 858 542 211 682 764 556 249 323 274 24;
  • 78) 0.432 810 470 858 542 211 682 764 556 249 323 274 24 × 2 = 0 + 0.865 620 941 717 084 423 365 529 112 498 646 548 48;
  • 79) 0.865 620 941 717 084 423 365 529 112 498 646 548 48 × 2 = 1 + 0.731 241 883 434 168 846 731 058 224 997 293 096 96;
  • 80) 0.731 241 883 434 168 846 731 058 224 997 293 096 96 × 2 = 1 + 0.462 483 766 868 337 693 462 116 449 994 586 193 92;
  • 81) 0.462 483 766 868 337 693 462 116 449 994 586 193 92 × 2 = 0 + 0.924 967 533 736 675 386 924 232 899 989 172 387 84;
  • 82) 0.924 967 533 736 675 386 924 232 899 989 172 387 84 × 2 = 1 + 0.849 935 067 473 350 773 848 465 799 978 344 775 68;
  • 83) 0.849 935 067 473 350 773 848 465 799 978 344 775 68 × 2 = 1 + 0.699 870 134 946 701 547 696 931 599 956 689 551 36;

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:


0.000 000 000 634 751 367 889 197 801 221 261 631 67(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:

0.000 000 000 634 751 367 889 197 801 221 261 631 67(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:


0.000 000 000 634 751 367 889 197 801 221 261 631 67(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


The base ten decimal number 0.000 000 000 634 751 367 889 197 801 221 261 631 67 converted and written in 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

» Decimal number in base ten 0.000 000 000 634 751 367 889 197 801 221 261 631 29 converted and written as 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