0.000 000 000 634 751 367 889 197 801 221 261 632 58 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 58(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 58(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 58.

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 58 × 2 = 0 + 0.000 000 001 269 502 735 778 395 602 442 523 265 16;
  • 2) 0.000 000 001 269 502 735 778 395 602 442 523 265 16 × 2 = 0 + 0.000 000 002 539 005 471 556 791 204 885 046 530 32;
  • 3) 0.000 000 002 539 005 471 556 791 204 885 046 530 32 × 2 = 0 + 0.000 000 005 078 010 943 113 582 409 770 093 060 64;
  • 4) 0.000 000 005 078 010 943 113 582 409 770 093 060 64 × 2 = 0 + 0.000 000 010 156 021 886 227 164 819 540 186 121 28;
  • 5) 0.000 000 010 156 021 886 227 164 819 540 186 121 28 × 2 = 0 + 0.000 000 020 312 043 772 454 329 639 080 372 242 56;
  • 6) 0.000 000 020 312 043 772 454 329 639 080 372 242 56 × 2 = 0 + 0.000 000 040 624 087 544 908 659 278 160 744 485 12;
  • 7) 0.000 000 040 624 087 544 908 659 278 160 744 485 12 × 2 = 0 + 0.000 000 081 248 175 089 817 318 556 321 488 970 24;
  • 8) 0.000 000 081 248 175 089 817 318 556 321 488 970 24 × 2 = 0 + 0.000 000 162 496 350 179 634 637 112 642 977 940 48;
  • 9) 0.000 000 162 496 350 179 634 637 112 642 977 940 48 × 2 = 0 + 0.000 000 324 992 700 359 269 274 225 285 955 880 96;
  • 10) 0.000 000 324 992 700 359 269 274 225 285 955 880 96 × 2 = 0 + 0.000 000 649 985 400 718 538 548 450 571 911 761 92;
  • 11) 0.000 000 649 985 400 718 538 548 450 571 911 761 92 × 2 = 0 + 0.000 001 299 970 801 437 077 096 901 143 823 523 84;
  • 12) 0.000 001 299 970 801 437 077 096 901 143 823 523 84 × 2 = 0 + 0.000 002 599 941 602 874 154 193 802 287 647 047 68;
  • 13) 0.000 002 599 941 602 874 154 193 802 287 647 047 68 × 2 = 0 + 0.000 005 199 883 205 748 308 387 604 575 294 095 36;
  • 14) 0.000 005 199 883 205 748 308 387 604 575 294 095 36 × 2 = 0 + 0.000 010 399 766 411 496 616 775 209 150 588 190 72;
  • 15) 0.000 010 399 766 411 496 616 775 209 150 588 190 72 × 2 = 0 + 0.000 020 799 532 822 993 233 550 418 301 176 381 44;
  • 16) 0.000 020 799 532 822 993 233 550 418 301 176 381 44 × 2 = 0 + 0.000 041 599 065 645 986 467 100 836 602 352 762 88;
  • 17) 0.000 041 599 065 645 986 467 100 836 602 352 762 88 × 2 = 0 + 0.000 083 198 131 291 972 934 201 673 204 705 525 76;
  • 18) 0.000 083 198 131 291 972 934 201 673 204 705 525 76 × 2 = 0 + 0.000 166 396 262 583 945 868 403 346 409 411 051 52;
  • 19) 0.000 166 396 262 583 945 868 403 346 409 411 051 52 × 2 = 0 + 0.000 332 792 525 167 891 736 806 692 818 822 103 04;
  • 20) 0.000 332 792 525 167 891 736 806 692 818 822 103 04 × 2 = 0 + 0.000 665 585 050 335 783 473 613 385 637 644 206 08;
  • 21) 0.000 665 585 050 335 783 473 613 385 637 644 206 08 × 2 = 0 + 0.001 331 170 100 671 566 947 226 771 275 288 412 16;
  • 22) 0.001 331 170 100 671 566 947 226 771 275 288 412 16 × 2 = 0 + 0.002 662 340 201 343 133 894 453 542 550 576 824 32;
  • 23) 0.002 662 340 201 343 133 894 453 542 550 576 824 32 × 2 = 0 + 0.005 324 680 402 686 267 788 907 085 101 153 648 64;
  • 24) 0.005 324 680 402 686 267 788 907 085 101 153 648 64 × 2 = 0 + 0.010 649 360 805 372 535 577 814 170 202 307 297 28;
  • 25) 0.010 649 360 805 372 535 577 814 170 202 307 297 28 × 2 = 0 + 0.021 298 721 610 745 071 155 628 340 404 614 594 56;
  • 26) 0.021 298 721 610 745 071 155 628 340 404 614 594 56 × 2 = 0 + 0.042 597 443 221 490 142 311 256 680 809 229 189 12;
  • 27) 0.042 597 443 221 490 142 311 256 680 809 229 189 12 × 2 = 0 + 0.085 194 886 442 980 284 622 513 361 618 458 378 24;
  • 28) 0.085 194 886 442 980 284 622 513 361 618 458 378 24 × 2 = 0 + 0.170 389 772 885 960 569 245 026 723 236 916 756 48;
  • 29) 0.170 389 772 885 960 569 245 026 723 236 916 756 48 × 2 = 0 + 0.340 779 545 771 921 138 490 053 446 473 833 512 96;
  • 30) 0.340 779 545 771 921 138 490 053 446 473 833 512 96 × 2 = 0 + 0.681 559 091 543 842 276 980 106 892 947 667 025 92;
  • 31) 0.681 559 091 543 842 276 980 106 892 947 667 025 92 × 2 = 1 + 0.363 118 183 087 684 553 960 213 785 895 334 051 84;
  • 32) 0.363 118 183 087 684 553 960 213 785 895 334 051 84 × 2 = 0 + 0.726 236 366 175 369 107 920 427 571 790 668 103 68;
  • 33) 0.726 236 366 175 369 107 920 427 571 790 668 103 68 × 2 = 1 + 0.452 472 732 350 738 215 840 855 143 581 336 207 36;
  • 34) 0.452 472 732 350 738 215 840 855 143 581 336 207 36 × 2 = 0 + 0.904 945 464 701 476 431 681 710 287 162 672 414 72;
  • 35) 0.904 945 464 701 476 431 681 710 287 162 672 414 72 × 2 = 1 + 0.809 890 929 402 952 863 363 420 574 325 344 829 44;
  • 36) 0.809 890 929 402 952 863 363 420 574 325 344 829 44 × 2 = 1 + 0.619 781 858 805 905 726 726 841 148 650 689 658 88;
  • 37) 0.619 781 858 805 905 726 726 841 148 650 689 658 88 × 2 = 1 + 0.239 563 717 611 811 453 453 682 297 301 379 317 76;
  • 38) 0.239 563 717 611 811 453 453 682 297 301 379 317 76 × 2 = 0 + 0.479 127 435 223 622 906 907 364 594 602 758 635 52;
  • 39) 0.479 127 435 223 622 906 907 364 594 602 758 635 52 × 2 = 0 + 0.958 254 870 447 245 813 814 729 189 205 517 271 04;
  • 40) 0.958 254 870 447 245 813 814 729 189 205 517 271 04 × 2 = 1 + 0.916 509 740 894 491 627 629 458 378 411 034 542 08;
  • 41) 0.916 509 740 894 491 627 629 458 378 411 034 542 08 × 2 = 1 + 0.833 019 481 788 983 255 258 916 756 822 069 084 16;
  • 42) 0.833 019 481 788 983 255 258 916 756 822 069 084 16 × 2 = 1 + 0.666 038 963 577 966 510 517 833 513 644 138 168 32;
  • 43) 0.666 038 963 577 966 510 517 833 513 644 138 168 32 × 2 = 1 + 0.332 077 927 155 933 021 035 667 027 288 276 336 64;
  • 44) 0.332 077 927 155 933 021 035 667 027 288 276 336 64 × 2 = 0 + 0.664 155 854 311 866 042 071 334 054 576 552 673 28;
  • 45) 0.664 155 854 311 866 042 071 334 054 576 552 673 28 × 2 = 1 + 0.328 311 708 623 732 084 142 668 109 153 105 346 56;
  • 46) 0.328 311 708 623 732 084 142 668 109 153 105 346 56 × 2 = 0 + 0.656 623 417 247 464 168 285 336 218 306 210 693 12;
  • 47) 0.656 623 417 247 464 168 285 336 218 306 210 693 12 × 2 = 1 + 0.313 246 834 494 928 336 570 672 436 612 421 386 24;
  • 48) 0.313 246 834 494 928 336 570 672 436 612 421 386 24 × 2 = 0 + 0.626 493 668 989 856 673 141 344 873 224 842 772 48;
  • 49) 0.626 493 668 989 856 673 141 344 873 224 842 772 48 × 2 = 1 + 0.252 987 337 979 713 346 282 689 746 449 685 544 96;
  • 50) 0.252 987 337 979 713 346 282 689 746 449 685 544 96 × 2 = 0 + 0.505 974 675 959 426 692 565 379 492 899 371 089 92;
  • 51) 0.505 974 675 959 426 692 565 379 492 899 371 089 92 × 2 = 1 + 0.011 949 351 918 853 385 130 758 985 798 742 179 84;
  • 52) 0.011 949 351 918 853 385 130 758 985 798 742 179 84 × 2 = 0 + 0.023 898 703 837 706 770 261 517 971 597 484 359 68;
  • 53) 0.023 898 703 837 706 770 261 517 971 597 484 359 68 × 2 = 0 + 0.047 797 407 675 413 540 523 035 943 194 968 719 36;
  • 54) 0.047 797 407 675 413 540 523 035 943 194 968 719 36 × 2 = 0 + 0.095 594 815 350 827 081 046 071 886 389 937 438 72;
  • 55) 0.095 594 815 350 827 081 046 071 886 389 937 438 72 × 2 = 0 + 0.191 189 630 701 654 162 092 143 772 779 874 877 44;
  • 56) 0.191 189 630 701 654 162 092 143 772 779 874 877 44 × 2 = 0 + 0.382 379 261 403 308 324 184 287 545 559 749 754 88;
  • 57) 0.382 379 261 403 308 324 184 287 545 559 749 754 88 × 2 = 0 + 0.764 758 522 806 616 648 368 575 091 119 499 509 76;
  • 58) 0.764 758 522 806 616 648 368 575 091 119 499 509 76 × 2 = 1 + 0.529 517 045 613 233 296 737 150 182 238 999 019 52;
  • 59) 0.529 517 045 613 233 296 737 150 182 238 999 019 52 × 2 = 1 + 0.059 034 091 226 466 593 474 300 364 477 998 039 04;
  • 60) 0.059 034 091 226 466 593 474 300 364 477 998 039 04 × 2 = 0 + 0.118 068 182 452 933 186 948 600 728 955 996 078 08;
  • 61) 0.118 068 182 452 933 186 948 600 728 955 996 078 08 × 2 = 0 + 0.236 136 364 905 866 373 897 201 457 911 992 156 16;
  • 62) 0.236 136 364 905 866 373 897 201 457 911 992 156 16 × 2 = 0 + 0.472 272 729 811 732 747 794 402 915 823 984 312 32;
  • 63) 0.472 272 729 811 732 747 794 402 915 823 984 312 32 × 2 = 0 + 0.944 545 459 623 465 495 588 805 831 647 968 624 64;
  • 64) 0.944 545 459 623 465 495 588 805 831 647 968 624 64 × 2 = 1 + 0.889 090 919 246 930 991 177 611 663 295 937 249 28;
  • 65) 0.889 090 919 246 930 991 177 611 663 295 937 249 28 × 2 = 1 + 0.778 181 838 493 861 982 355 223 326 591 874 498 56;
  • 66) 0.778 181 838 493 861 982 355 223 326 591 874 498 56 × 2 = 1 + 0.556 363 676 987 723 964 710 446 653 183 748 997 12;
  • 67) 0.556 363 676 987 723 964 710 446 653 183 748 997 12 × 2 = 1 + 0.112 727 353 975 447 929 420 893 306 367 497 994 24;
  • 68) 0.112 727 353 975 447 929 420 893 306 367 497 994 24 × 2 = 0 + 0.225 454 707 950 895 858 841 786 612 734 995 988 48;
  • 69) 0.225 454 707 950 895 858 841 786 612 734 995 988 48 × 2 = 0 + 0.450 909 415 901 791 717 683 573 225 469 991 976 96;
  • 70) 0.450 909 415 901 791 717 683 573 225 469 991 976 96 × 2 = 0 + 0.901 818 831 803 583 435 367 146 450 939 983 953 92;
  • 71) 0.901 818 831 803 583 435 367 146 450 939 983 953 92 × 2 = 1 + 0.803 637 663 607 166 870 734 292 901 879 967 907 84;
  • 72) 0.803 637 663 607 166 870 734 292 901 879 967 907 84 × 2 = 1 + 0.607 275 327 214 333 741 468 585 803 759 935 815 68;
  • 73) 0.607 275 327 214 333 741 468 585 803 759 935 815 68 × 2 = 1 + 0.214 550 654 428 667 482 937 171 607 519 871 631 36;
  • 74) 0.214 550 654 428 667 482 937 171 607 519 871 631 36 × 2 = 0 + 0.429 101 308 857 334 965 874 343 215 039 743 262 72;
  • 75) 0.429 101 308 857 334 965 874 343 215 039 743 262 72 × 2 = 0 + 0.858 202 617 714 669 931 748 686 430 079 486 525 44;
  • 76) 0.858 202 617 714 669 931 748 686 430 079 486 525 44 × 2 = 1 + 0.716 405 235 429 339 863 497 372 860 158 973 050 88;
  • 77) 0.716 405 235 429 339 863 497 372 860 158 973 050 88 × 2 = 1 + 0.432 810 470 858 679 726 994 745 720 317 946 101 76;
  • 78) 0.432 810 470 858 679 726 994 745 720 317 946 101 76 × 2 = 0 + 0.865 620 941 717 359 453 989 491 440 635 892 203 52;
  • 79) 0.865 620 941 717 359 453 989 491 440 635 892 203 52 × 2 = 1 + 0.731 241 883 434 718 907 978 982 881 271 784 407 04;
  • 80) 0.731 241 883 434 718 907 978 982 881 271 784 407 04 × 2 = 1 + 0.462 483 766 869 437 815 957 965 762 543 568 814 08;
  • 81) 0.462 483 766 869 437 815 957 965 762 543 568 814 08 × 2 = 0 + 0.924 967 533 738 875 631 915 931 525 087 137 628 16;
  • 82) 0.924 967 533 738 875 631 915 931 525 087 137 628 16 × 2 = 1 + 0.849 935 067 477 751 263 831 863 050 174 275 256 32;
  • 83) 0.849 935 067 477 751 263 831 863 050 174 275 256 32 × 2 = 1 + 0.699 870 134 955 502 527 663 726 100 348 550 512 64;

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 58(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 58(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 58(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 58 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 28 to 64 bit double precision IEEE 754 binary floating point representation standard

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

How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100