0.000 000 000 000 000 013 248 735 989 328 012 498 115 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 013 248 735 989 328 012 498 115 9(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
0.000 000 000 000 000 013 248 735 989 328 012 498 115 9(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: 0.000 000 000 000 000 013 248 735 989 328 012 498 115 9.

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) 0.000 000 000 000 000 013 248 735 989 328 012 498 115 9 × 2 = 0 + 0.000 000 000 000 000 026 497 471 978 656 024 996 231 8;
  • 2) 0.000 000 000 000 000 026 497 471 978 656 024 996 231 8 × 2 = 0 + 0.000 000 000 000 000 052 994 943 957 312 049 992 463 6;
  • 3) 0.000 000 000 000 000 052 994 943 957 312 049 992 463 6 × 2 = 0 + 0.000 000 000 000 000 105 989 887 914 624 099 984 927 2;
  • 4) 0.000 000 000 000 000 105 989 887 914 624 099 984 927 2 × 2 = 0 + 0.000 000 000 000 000 211 979 775 829 248 199 969 854 4;
  • 5) 0.000 000 000 000 000 211 979 775 829 248 199 969 854 4 × 2 = 0 + 0.000 000 000 000 000 423 959 551 658 496 399 939 708 8;
  • 6) 0.000 000 000 000 000 423 959 551 658 496 399 939 708 8 × 2 = 0 + 0.000 000 000 000 000 847 919 103 316 992 799 879 417 6;
  • 7) 0.000 000 000 000 000 847 919 103 316 992 799 879 417 6 × 2 = 0 + 0.000 000 000 000 001 695 838 206 633 985 599 758 835 2;
  • 8) 0.000 000 000 000 001 695 838 206 633 985 599 758 835 2 × 2 = 0 + 0.000 000 000 000 003 391 676 413 267 971 199 517 670 4;
  • 9) 0.000 000 000 000 003 391 676 413 267 971 199 517 670 4 × 2 = 0 + 0.000 000 000 000 006 783 352 826 535 942 399 035 340 8;
  • 10) 0.000 000 000 000 006 783 352 826 535 942 399 035 340 8 × 2 = 0 + 0.000 000 000 000 013 566 705 653 071 884 798 070 681 6;
  • 11) 0.000 000 000 000 013 566 705 653 071 884 798 070 681 6 × 2 = 0 + 0.000 000 000 000 027 133 411 306 143 769 596 141 363 2;
  • 12) 0.000 000 000 000 027 133 411 306 143 769 596 141 363 2 × 2 = 0 + 0.000 000 000 000 054 266 822 612 287 539 192 282 726 4;
  • 13) 0.000 000 000 000 054 266 822 612 287 539 192 282 726 4 × 2 = 0 + 0.000 000 000 000 108 533 645 224 575 078 384 565 452 8;
  • 14) 0.000 000 000 000 108 533 645 224 575 078 384 565 452 8 × 2 = 0 + 0.000 000 000 000 217 067 290 449 150 156 769 130 905 6;
  • 15) 0.000 000 000 000 217 067 290 449 150 156 769 130 905 6 × 2 = 0 + 0.000 000 000 000 434 134 580 898 300 313 538 261 811 2;
  • 16) 0.000 000 000 000 434 134 580 898 300 313 538 261 811 2 × 2 = 0 + 0.000 000 000 000 868 269 161 796 600 627 076 523 622 4;
  • 17) 0.000 000 000 000 868 269 161 796 600 627 076 523 622 4 × 2 = 0 + 0.000 000 000 001 736 538 323 593 201 254 153 047 244 8;
  • 18) 0.000 000 000 001 736 538 323 593 201 254 153 047 244 8 × 2 = 0 + 0.000 000 000 003 473 076 647 186 402 508 306 094 489 6;
  • 19) 0.000 000 000 003 473 076 647 186 402 508 306 094 489 6 × 2 = 0 + 0.000 000 000 006 946 153 294 372 805 016 612 188 979 2;
  • 20) 0.000 000 000 006 946 153 294 372 805 016 612 188 979 2 × 2 = 0 + 0.000 000 000 013 892 306 588 745 610 033 224 377 958 4;
  • 21) 0.000 000 000 013 892 306 588 745 610 033 224 377 958 4 × 2 = 0 + 0.000 000 000 027 784 613 177 491 220 066 448 755 916 8;
  • 22) 0.000 000 000 027 784 613 177 491 220 066 448 755 916 8 × 2 = 0 + 0.000 000 000 055 569 226 354 982 440 132 897 511 833 6;
  • 23) 0.000 000 000 055 569 226 354 982 440 132 897 511 833 6 × 2 = 0 + 0.000 000 000 111 138 452 709 964 880 265 795 023 667 2;
  • 24) 0.000 000 000 111 138 452 709 964 880 265 795 023 667 2 × 2 = 0 + 0.000 000 000 222 276 905 419 929 760 531 590 047 334 4;
  • 25) 0.000 000 000 222 276 905 419 929 760 531 590 047 334 4 × 2 = 0 + 0.000 000 000 444 553 810 839 859 521 063 180 094 668 8;
  • 26) 0.000 000 000 444 553 810 839 859 521 063 180 094 668 8 × 2 = 0 + 0.000 000 000 889 107 621 679 719 042 126 360 189 337 6;
  • 27) 0.000 000 000 889 107 621 679 719 042 126 360 189 337 6 × 2 = 0 + 0.000 000 001 778 215 243 359 438 084 252 720 378 675 2;
  • 28) 0.000 000 001 778 215 243 359 438 084 252 720 378 675 2 × 2 = 0 + 0.000 000 003 556 430 486 718 876 168 505 440 757 350 4;
  • 29) 0.000 000 003 556 430 486 718 876 168 505 440 757 350 4 × 2 = 0 + 0.000 000 007 112 860 973 437 752 337 010 881 514 700 8;
  • 30) 0.000 000 007 112 860 973 437 752 337 010 881 514 700 8 × 2 = 0 + 0.000 000 014 225 721 946 875 504 674 021 763 029 401 6;
  • 31) 0.000 000 014 225 721 946 875 504 674 021 763 029 401 6 × 2 = 0 + 0.000 000 028 451 443 893 751 009 348 043 526 058 803 2;
  • 32) 0.000 000 028 451 443 893 751 009 348 043 526 058 803 2 × 2 = 0 + 0.000 000 056 902 887 787 502 018 696 087 052 117 606 4;
  • 33) 0.000 000 056 902 887 787 502 018 696 087 052 117 606 4 × 2 = 0 + 0.000 000 113 805 775 575 004 037 392 174 104 235 212 8;
  • 34) 0.000 000 113 805 775 575 004 037 392 174 104 235 212 8 × 2 = 0 + 0.000 000 227 611 551 150 008 074 784 348 208 470 425 6;
  • 35) 0.000 000 227 611 551 150 008 074 784 348 208 470 425 6 × 2 = 0 + 0.000 000 455 223 102 300 016 149 568 696 416 940 851 2;
  • 36) 0.000 000 455 223 102 300 016 149 568 696 416 940 851 2 × 2 = 0 + 0.000 000 910 446 204 600 032 299 137 392 833 881 702 4;
  • 37) 0.000 000 910 446 204 600 032 299 137 392 833 881 702 4 × 2 = 0 + 0.000 001 820 892 409 200 064 598 274 785 667 763 404 8;
  • 38) 0.000 001 820 892 409 200 064 598 274 785 667 763 404 8 × 2 = 0 + 0.000 003 641 784 818 400 129 196 549 571 335 526 809 6;
  • 39) 0.000 003 641 784 818 400 129 196 549 571 335 526 809 6 × 2 = 0 + 0.000 007 283 569 636 800 258 393 099 142 671 053 619 2;
  • 40) 0.000 007 283 569 636 800 258 393 099 142 671 053 619 2 × 2 = 0 + 0.000 014 567 139 273 600 516 786 198 285 342 107 238 4;
  • 41) 0.000 014 567 139 273 600 516 786 198 285 342 107 238 4 × 2 = 0 + 0.000 029 134 278 547 201 033 572 396 570 684 214 476 8;
  • 42) 0.000 029 134 278 547 201 033 572 396 570 684 214 476 8 × 2 = 0 + 0.000 058 268 557 094 402 067 144 793 141 368 428 953 6;
  • 43) 0.000 058 268 557 094 402 067 144 793 141 368 428 953 6 × 2 = 0 + 0.000 116 537 114 188 804 134 289 586 282 736 857 907 2;
  • 44) 0.000 116 537 114 188 804 134 289 586 282 736 857 907 2 × 2 = 0 + 0.000 233 074 228 377 608 268 579 172 565 473 715 814 4;
  • 45) 0.000 233 074 228 377 608 268 579 172 565 473 715 814 4 × 2 = 0 + 0.000 466 148 456 755 216 537 158 345 130 947 431 628 8;
  • 46) 0.000 466 148 456 755 216 537 158 345 130 947 431 628 8 × 2 = 0 + 0.000 932 296 913 510 433 074 316 690 261 894 863 257 6;
  • 47) 0.000 932 296 913 510 433 074 316 690 261 894 863 257 6 × 2 = 0 + 0.001 864 593 827 020 866 148 633 380 523 789 726 515 2;
  • 48) 0.001 864 593 827 020 866 148 633 380 523 789 726 515 2 × 2 = 0 + 0.003 729 187 654 041 732 297 266 761 047 579 453 030 4;
  • 49) 0.003 729 187 654 041 732 297 266 761 047 579 453 030 4 × 2 = 0 + 0.007 458 375 308 083 464 594 533 522 095 158 906 060 8;
  • 50) 0.007 458 375 308 083 464 594 533 522 095 158 906 060 8 × 2 = 0 + 0.014 916 750 616 166 929 189 067 044 190 317 812 121 6;
  • 51) 0.014 916 750 616 166 929 189 067 044 190 317 812 121 6 × 2 = 0 + 0.029 833 501 232 333 858 378 134 088 380 635 624 243 2;
  • 52) 0.029 833 501 232 333 858 378 134 088 380 635 624 243 2 × 2 = 0 + 0.059 667 002 464 667 716 756 268 176 761 271 248 486 4;
  • 53) 0.059 667 002 464 667 716 756 268 176 761 271 248 486 4 × 2 = 0 + 0.119 334 004 929 335 433 512 536 353 522 542 496 972 8;
  • 54) 0.119 334 004 929 335 433 512 536 353 522 542 496 972 8 × 2 = 0 + 0.238 668 009 858 670 867 025 072 707 045 084 993 945 6;
  • 55) 0.238 668 009 858 670 867 025 072 707 045 084 993 945 6 × 2 = 0 + 0.477 336 019 717 341 734 050 145 414 090 169 987 891 2;
  • 56) 0.477 336 019 717 341 734 050 145 414 090 169 987 891 2 × 2 = 0 + 0.954 672 039 434 683 468 100 290 828 180 339 975 782 4;
  • 57) 0.954 672 039 434 683 468 100 290 828 180 339 975 782 4 × 2 = 1 + 0.909 344 078 869 366 936 200 581 656 360 679 951 564 8;
  • 58) 0.909 344 078 869 366 936 200 581 656 360 679 951 564 8 × 2 = 1 + 0.818 688 157 738 733 872 401 163 312 721 359 903 129 6;
  • 59) 0.818 688 157 738 733 872 401 163 312 721 359 903 129 6 × 2 = 1 + 0.637 376 315 477 467 744 802 326 625 442 719 806 259 2;
  • 60) 0.637 376 315 477 467 744 802 326 625 442 719 806 259 2 × 2 = 1 + 0.274 752 630 954 935 489 604 653 250 885 439 612 518 4;
  • 61) 0.274 752 630 954 935 489 604 653 250 885 439 612 518 4 × 2 = 0 + 0.549 505 261 909 870 979 209 306 501 770 879 225 036 8;
  • 62) 0.549 505 261 909 870 979 209 306 501 770 879 225 036 8 × 2 = 1 + 0.099 010 523 819 741 958 418 613 003 541 758 450 073 6;
  • 63) 0.099 010 523 819 741 958 418 613 003 541 758 450 073 6 × 2 = 0 + 0.198 021 047 639 483 916 837 226 007 083 516 900 147 2;
  • 64) 0.198 021 047 639 483 916 837 226 007 083 516 900 147 2 × 2 = 0 + 0.396 042 095 278 967 833 674 452 014 167 033 800 294 4;
  • 65) 0.396 042 095 278 967 833 674 452 014 167 033 800 294 4 × 2 = 0 + 0.792 084 190 557 935 667 348 904 028 334 067 600 588 8;
  • 66) 0.792 084 190 557 935 667 348 904 028 334 067 600 588 8 × 2 = 1 + 0.584 168 381 115 871 334 697 808 056 668 135 201 177 6;
  • 67) 0.584 168 381 115 871 334 697 808 056 668 135 201 177 6 × 2 = 1 + 0.168 336 762 231 742 669 395 616 113 336 270 402 355 2;
  • 68) 0.168 336 762 231 742 669 395 616 113 336 270 402 355 2 × 2 = 0 + 0.336 673 524 463 485 338 791 232 226 672 540 804 710 4;
  • 69) 0.336 673 524 463 485 338 791 232 226 672 540 804 710 4 × 2 = 0 + 0.673 347 048 926 970 677 582 464 453 345 081 609 420 8;
  • 70) 0.673 347 048 926 970 677 582 464 453 345 081 609 420 8 × 2 = 1 + 0.346 694 097 853 941 355 164 928 906 690 163 218 841 6;
  • 71) 0.346 694 097 853 941 355 164 928 906 690 163 218 841 6 × 2 = 0 + 0.693 388 195 707 882 710 329 857 813 380 326 437 683 2;
  • 72) 0.693 388 195 707 882 710 329 857 813 380 326 437 683 2 × 2 = 1 + 0.386 776 391 415 765 420 659 715 626 760 652 875 366 4;
  • 73) 0.386 776 391 415 765 420 659 715 626 760 652 875 366 4 × 2 = 0 + 0.773 552 782 831 530 841 319 431 253 521 305 750 732 8;
  • 74) 0.773 552 782 831 530 841 319 431 253 521 305 750 732 8 × 2 = 1 + 0.547 105 565 663 061 682 638 862 507 042 611 501 465 6;
  • 75) 0.547 105 565 663 061 682 638 862 507 042 611 501 465 6 × 2 = 1 + 0.094 211 131 326 123 365 277 725 014 085 223 002 931 2;
  • 76) 0.094 211 131 326 123 365 277 725 014 085 223 002 931 2 × 2 = 0 + 0.188 422 262 652 246 730 555 450 028 170 446 005 862 4;
  • 77) 0.188 422 262 652 246 730 555 450 028 170 446 005 862 4 × 2 = 0 + 0.376 844 525 304 493 461 110 900 056 340 892 011 724 8;
  • 78) 0.376 844 525 304 493 461 110 900 056 340 892 011 724 8 × 2 = 0 + 0.753 689 050 608 986 922 221 800 112 681 784 023 449 6;
  • 79) 0.753 689 050 608 986 922 221 800 112 681 784 023 449 6 × 2 = 1 + 0.507 378 101 217 973 844 443 600 225 363 568 046 899 2;
  • 80) 0.507 378 101 217 973 844 443 600 225 363 568 046 899 2 × 2 = 1 + 0.014 756 202 435 947 688 887 200 450 727 136 093 798 4;
  • 81) 0.014 756 202 435 947 688 887 200 450 727 136 093 798 4 × 2 = 0 + 0.029 512 404 871 895 377 774 400 901 454 272 187 596 8;
  • 82) 0.029 512 404 871 895 377 774 400 901 454 272 187 596 8 × 2 = 0 + 0.059 024 809 743 790 755 548 801 802 908 544 375 193 6;
  • 83) 0.059 024 809 743 790 755 548 801 802 908 544 375 193 6 × 2 = 0 + 0.118 049 619 487 581 511 097 603 605 817 088 750 387 2;
  • 84) 0.118 049 619 487 581 511 097 603 605 817 088 750 387 2 × 2 = 0 + 0.236 099 238 975 163 022 195 207 211 634 177 500 774 4;
  • 85) 0.236 099 238 975 163 022 195 207 211 634 177 500 774 4 × 2 = 0 + 0.472 198 477 950 326 044 390 414 423 268 355 001 548 8;
  • 86) 0.472 198 477 950 326 044 390 414 423 268 355 001 548 8 × 2 = 0 + 0.944 396 955 900 652 088 780 828 846 536 710 003 097 6;
  • 87) 0.944 396 955 900 652 088 780 828 846 536 710 003 097 6 × 2 = 1 + 0.888 793 911 801 304 177 561 657 693 073 420 006 195 2;
  • 88) 0.888 793 911 801 304 177 561 657 693 073 420 006 195 2 × 2 = 1 + 0.777 587 823 602 608 355 123 315 386 146 840 012 390 4;
  • 89) 0.777 587 823 602 608 355 123 315 386 146 840 012 390 4 × 2 = 1 + 0.555 175 647 205 216 710 246 630 772 293 680 024 780 8;
  • 90) 0.555 175 647 205 216 710 246 630 772 293 680 024 780 8 × 2 = 1 + 0.110 351 294 410 433 420 493 261 544 587 360 049 561 6;
  • 91) 0.110 351 294 410 433 420 493 261 544 587 360 049 561 6 × 2 = 0 + 0.220 702 588 820 866 840 986 523 089 174 720 099 123 2;
  • 92) 0.220 702 588 820 866 840 986 523 089 174 720 099 123 2 × 2 = 0 + 0.441 405 177 641 733 681 973 046 178 349 440 198 246 4;
  • 93) 0.441 405 177 641 733 681 973 046 178 349 440 198 246 4 × 2 = 0 + 0.882 810 355 283 467 363 946 092 356 698 880 396 492 8;
  • 94) 0.882 810 355 283 467 363 946 092 356 698 880 396 492 8 × 2 = 1 + 0.765 620 710 566 934 727 892 184 713 397 760 792 985 6;
  • 95) 0.765 620 710 566 934 727 892 184 713 397 760 792 985 6 × 2 = 1 + 0.531 241 421 133 869 455 784 369 426 795 521 585 971 2;
  • 96) 0.531 241 421 133 869 455 784 369 426 795 521 585 971 2 × 2 = 1 + 0.062 482 842 267 738 911 568 738 853 591 043 171 942 4;
  • 97) 0.062 482 842 267 738 911 568 738 853 591 043 171 942 4 × 2 = 0 + 0.124 965 684 535 477 823 137 477 707 182 086 343 884 8;
  • 98) 0.124 965 684 535 477 823 137 477 707 182 086 343 884 8 × 2 = 0 + 0.249 931 369 070 955 646 274 955 414 364 172 687 769 6;
  • 99) 0.249 931 369 070 955 646 274 955 414 364 172 687 769 6 × 2 = 0 + 0.499 862 738 141 911 292 549 910 828 728 345 375 539 2;
  • 100) 0.499 862 738 141 911 292 549 910 828 728 345 375 539 2 × 2 = 0 + 0.999 725 476 283 822 585 099 821 657 456 690 751 078 4;
  • 101) 0.999 725 476 283 822 585 099 821 657 456 690 751 078 4 × 2 = 1 + 0.999 450 952 567 645 170 199 643 314 913 381 502 156 8;
  • 102) 0.999 450 952 567 645 170 199 643 314 913 381 502 156 8 × 2 = 1 + 0.998 901 905 135 290 340 399 286 629 826 763 004 313 6;
  • 103) 0.998 901 905 135 290 340 399 286 629 826 763 004 313 6 × 2 = 1 + 0.997 803 810 270 580 680 798 573 259 653 526 008 627 2;
  • 104) 0.997 803 810 270 580 680 798 573 259 653 526 008 627 2 × 2 = 1 + 0.995 607 620 541 161 361 597 146 519 307 052 017 254 4;
  • 105) 0.995 607 620 541 161 361 597 146 519 307 052 017 254 4 × 2 = 1 + 0.991 215 241 082 322 723 194 293 038 614 104 034 508 8;
  • 106) 0.991 215 241 082 322 723 194 293 038 614 104 034 508 8 × 2 = 1 + 0.982 430 482 164 645 446 388 586 077 228 208 069 017 6;
  • 107) 0.982 430 482 164 645 446 388 586 077 228 208 069 017 6 × 2 = 1 + 0.964 860 964 329 290 892 777 172 154 456 416 138 035 2;
  • 108) 0.964 860 964 329 290 892 777 172 154 456 416 138 035 2 × 2 = 1 + 0.929 721 928 658 581 785 554 344 308 912 832 276 070 4;
  • 109) 0.929 721 928 658 581 785 554 344 308 912 832 276 070 4 × 2 = 1 + 0.859 443 857 317 163 571 108 688 617 825 664 552 140 8;

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:


0.000 000 000 000 000 013 248 735 989 328 012 498 115 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0110 0011 0000 0011 1100 0111 0000 1111 1111 1(2)

5. Positive number before normalization:

0.000 000 000 000 000 013 248 735 989 328 012 498 115 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0110 0011 0000 0011 1100 0111 0000 1111 1111 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 57 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 013 248 735 989 328 012 498 115 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0110 0011 0000 0011 1100 0111 0000 1111 1111 1(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0110 0011 0000 0011 1100 0111 0000 1111 1111 1(2) × 20 =

1.1110 1000 1100 1010 1100 0110 0000 0111 1000 1110 0001 1111 1111(2) × 2-57


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): -57

Mantissa (not normalized):
1.1110 1000 1100 1010 1100 0110 0000 0111 1000 1110 0001 1111 1111


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-57 + 2(11-1) - 1 =

(-57 + 1 023)(10) =

966(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;
  • 966 ÷ 2 = 483 + 0;
  • 483 ÷ 2 = 241 + 1;
  • 241 ÷ 2 = 120 + 1;
  • 120 ÷ 2 = 60 + 0;
  • 60 ÷ 2 = 30 + 0;
  • 30 ÷ 2 = 15 + 0;
  • 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) =

966(10) =

011 1100 0110(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. 1110 1000 1100 1010 1100 0110 0000 0111 1000 1110 0001 1111 1111 =

1110 1000 1100 1010 1100 0110 0000 0111 1000 1110 0001 1111 1111


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 1100 0110

Mantissa (52 bits) =
1110 1000 1100 1010 1100 0110 0000 0111 1000 1110 0001 1111 1111


Decimal number 0.000 000 000 000 000 013 248 735 989 328 012 498 115 9 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1100 0110 - 1110 1000 1100 1010 1100 0110 0000 0111 1000 1110 0001 1111 1111

» Convert decimal 0.000 000 000 000 000 013 248 735 989 328 012 498 109 2 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