0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 116 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 116(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 116(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 116.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 116 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 232;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 232 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 464;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 464 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 224 928;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 224 928 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 449 856;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 449 856 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 899 712;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 899 712 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 799 424;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 799 424 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 598 848;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 598 848 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 197 696;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 197 696 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 395 392;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 395 392 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 790 784;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 790 784 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 581 568;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 581 568 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 787 163 136;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 787 163 136 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 574 326 272;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 574 326 272 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 148 652 544;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 148 652 544 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 297 305 088;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 297 305 088 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 594 610 176;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 594 610 176 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 189 220 352;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 189 220 352 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 378 440 704;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 378 440 704 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 756 881 408;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 756 881 408 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 513 762 816;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 513 762 816 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 923 027 525 632;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 923 027 525 632 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 846 055 051 264;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 846 055 051 264 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 692 110 102 528;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 692 110 102 528 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 384 220 205 056;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 384 220 205 056 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 768 440 410 112;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 768 440 410 112 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 536 880 820 224;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 536 880 820 224 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 073 761 640 448;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 073 761 640 448 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 147 523 280 896;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 147 523 280 896 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 295 046 561 792;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 295 046 561 792 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 590 093 123 584;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 590 093 123 584 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 180 186 247 168;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 180 186 247 168 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 098 360 372 494 336;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 098 360 372 494 336 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 196 720 744 988 672;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 196 720 744 988 672 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 393 441 489 977 344;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 393 441 489 977 344 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 786 882 979 954 688;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 786 882 979 954 688 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 573 765 959 909 376;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 573 765 959 909 376 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 147 531 919 818 752;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 147 531 919 818 752 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 295 063 839 637 504;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 295 063 839 637 504 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 590 127 679 275 008;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 590 127 679 275 008 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 180 255 358 550 016;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 180 255 358 550 016 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 360 510 717 100 032;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 360 510 717 100 032 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 721 021 434 200 064;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 721 021 434 200 064 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 425 442 042 868 400 128;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 425 442 042 868 400 128 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 850 884 085 736 800 256;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 850 884 085 736 800 256 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 701 768 171 473 600 512;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 701 768 171 473 600 512 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 403 536 342 947 201 024;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 403 536 342 947 201 024 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 807 072 685 894 402 048;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 807 072 685 894 402 048 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 614 145 371 788 804 096;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 614 145 371 788 804 096 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 228 290 743 577 608 192;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 228 290 743 577 608 192 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 456 581 487 155 216 384;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 456 581 487 155 216 384 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 913 162 974 310 432 768;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 913 162 974 310 432 768 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 826 325 948 620 865 536;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 826 325 948 620 865 536 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 563 652 651 897 241 731 072;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 563 652 651 897 241 731 072 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 127 305 303 794 483 462 144;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 127 305 303 794 483 462 144 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 254 610 607 588 966 924 288;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 254 610 607 588 966 924 288 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 509 221 215 177 933 848 576;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 509 221 215 177 933 848 576 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 018 442 430 355 867 697 152;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 018 442 430 355 867 697 152 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 036 884 860 711 735 394 304;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 036 884 860 711 735 394 304 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 073 769 721 423 470 788 608;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 073 769 721 423 470 788 608 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 147 539 442 846 941 577 216;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 147 539 442 846 941 577 216 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 295 078 885 693 883 154 432;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 295 078 885 693 883 154 432 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 590 157 771 387 766 308 864;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 590 157 771 387 766 308 864 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 001 180 315 542 775 532 617 728;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 001 180 315 542 775 532 617 728 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 002 360 631 085 551 065 235 456;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 002 360 631 085 551 065 235 456 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 004 721 262 171 102 130 470 912;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 004 721 262 171 102 130 470 912 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 009 442 524 342 204 260 941 824;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 009 442 524 342 204 260 941 824 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 018 885 048 684 408 521 883 648;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 018 885 048 684 408 521 883 648 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 037 770 097 368 817 043 767 296;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 037 770 097 368 817 043 767 296 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 075 540 194 737 634 087 534 592;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 075 540 194 737 634 087 534 592 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 151 080 389 475 268 175 069 184;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 151 080 389 475 268 175 069 184 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 302 160 778 950 536 350 138 368;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 302 160 778 950 536 350 138 368 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 604 321 557 901 072 700 276 736;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 604 321 557 901 072 700 276 736 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 001 208 643 115 802 145 400 553 472;
  • 74) 0.462 318 420 410 156 250 000 000 000 001 208 643 115 802 145 400 553 472 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 002 417 286 231 604 290 801 106 944;
  • 75) 0.924 636 840 820 312 500 000 000 000 002 417 286 231 604 290 801 106 944 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 004 834 572 463 208 581 602 213 888;
  • 76) 0.849 273 681 640 625 000 000 000 000 004 834 572 463 208 581 602 213 888 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 009 669 144 926 417 163 204 427 776;
  • 77) 0.698 547 363 281 250 000 000 000 000 009 669 144 926 417 163 204 427 776 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 019 338 289 852 834 326 408 855 552;
  • 78) 0.397 094 726 562 500 000 000 000 000 019 338 289 852 834 326 408 855 552 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 038 676 579 705 668 652 817 711 104;
  • 79) 0.794 189 453 125 000 000 000 000 000 038 676 579 705 668 652 817 711 104 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 077 353 159 411 337 305 635 422 208;
  • 80) 0.588 378 906 250 000 000 000 000 000 077 353 159 411 337 305 635 422 208 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 154 706 318 822 674 611 270 844 416;
  • 81) 0.176 757 812 500 000 000 000 000 000 154 706 318 822 674 611 270 844 416 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 309 412 637 645 349 222 541 688 832;
  • 82) 0.353 515 625 000 000 000 000 000 000 309 412 637 645 349 222 541 688 832 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 618 825 275 290 698 445 083 377 664;
  • 83) 0.707 031 250 000 000 000 000 000 000 618 825 275 290 698 445 083 377 664 × 2 = 1 + 0.414 062 500 000 000 000 000 000 001 237 650 550 581 396 890 166 755 328;
  • 84) 0.414 062 500 000 000 000 000 000 001 237 650 550 581 396 890 166 755 328 × 2 = 0 + 0.828 125 000 000 000 000 000 000 002 475 301 101 162 793 780 333 510 656;
  • 85) 0.828 125 000 000 000 000 000 000 002 475 301 101 162 793 780 333 510 656 × 2 = 1 + 0.656 250 000 000 000 000 000 000 004 950 602 202 325 587 560 667 021 312;
  • 86) 0.656 250 000 000 000 000 000 000 004 950 602 202 325 587 560 667 021 312 × 2 = 1 + 0.312 500 000 000 000 000 000 000 009 901 204 404 651 175 121 334 042 624;
  • 87) 0.312 500 000 000 000 000 000 000 009 901 204 404 651 175 121 334 042 624 × 2 = 0 + 0.625 000 000 000 000 000 000 000 019 802 408 809 302 350 242 668 085 248;
  • 88) 0.625 000 000 000 000 000 000 000 019 802 408 809 302 350 242 668 085 248 × 2 = 1 + 0.250 000 000 000 000 000 000 000 039 604 817 618 604 700 485 336 170 496;
  • 89) 0.250 000 000 000 000 000 000 000 039 604 817 618 604 700 485 336 170 496 × 2 = 0 + 0.500 000 000 000 000 000 000 000 079 209 635 237 209 400 970 672 340 992;
  • 90) 0.500 000 000 000 000 000 000 000 079 209 635 237 209 400 970 672 340 992 × 2 = 1 + 0.000 000 000 000 000 000 000 000 158 419 270 474 418 801 941 344 681 984;
  • 91) 0.000 000 000 000 000 000 000 000 158 419 270 474 418 801 941 344 681 984 × 2 = 0 + 0.000 000 000 000 000 000 000 000 316 838 540 948 837 603 882 689 363 968;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 116(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 116(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 39 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 116(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010(2) × 2-39


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): -39

Mantissa (not normalized):
1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-39 + 2(11-1) - 1 =

(-39 + 1 023)(10) =

984(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;
  • 984 ÷ 2 = 492 + 0;
  • 492 ÷ 2 = 246 + 0;
  • 246 ÷ 2 = 123 + 0;
  • 123 ÷ 2 = 61 + 1;
  • 61 ÷ 2 = 30 + 1;
  • 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) =

984(10) =

011 1101 1000(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. 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010 =

1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010


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 1101 1000

Mantissa (52 bits) =
1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 116 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1101 1000 - 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010

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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