0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 061 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 061(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 061(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 061.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 061 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 122;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 122 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 244;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 244 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 488;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 488 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 976;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 976 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 952;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 952 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 235 904;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 235 904 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 471 808;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 471 808 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 943 616;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 943 616 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 887 232;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 887 232 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 774 464;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 774 464 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 548 928;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 548 928 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 639 097 856;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 639 097 856 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 278 195 712;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 278 195 712 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 556 391 424;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 556 391 424 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 112 782 848;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 112 782 848 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 225 565 696;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 225 565 696 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 451 131 392;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 451 131 392 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 902 262 784;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 902 262 784 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 804 525 568;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 804 525 568 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 609 051 136;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 609 051 136 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 218 102 272;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 218 102 272 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 436 204 544;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 436 204 544 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 872 409 088;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 872 409 088 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 744 818 176;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 744 818 176 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 147 489 636 352;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 147 489 636 352 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 294 979 272 704;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 294 979 272 704 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 589 958 545 408;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 589 958 545 408 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 179 917 090 816;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 179 917 090 816 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 359 834 181 632;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 359 834 181 632 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 719 668 363 264;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 719 668 363 264 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 439 336 726 528;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 439 336 726 528 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 878 673 453 056;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 878 673 453 056 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 757 346 906 112;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 757 346 906 112 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 514 693 812 224;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 514 693 812 224 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 487 029 387 624 448;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 487 029 387 624 448 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 974 058 775 248 896;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 974 058 775 248 896 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 948 117 550 497 792;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 948 117 550 497 792 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 896 235 100 995 584;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 896 235 100 995 584 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 792 470 201 991 168;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 792 470 201 991 168 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 584 940 403 982 336;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 584 940 403 982 336 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 169 880 807 964 672;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 169 880 807 964 672 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 339 761 615 929 344;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 339 761 615 929 344 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 679 523 231 858 688;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 679 523 231 858 688 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 633 359 046 463 717 376;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 633 359 046 463 717 376 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 266 718 092 927 434 752;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 266 718 092 927 434 752 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 533 436 185 854 869 504;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 533 436 185 854 869 504 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 066 872 371 709 739 008;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 066 872 371 709 739 008 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 133 744 743 419 478 016;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 133 744 743 419 478 016 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 267 489 486 838 956 032;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 267 489 486 838 956 032 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 534 978 973 677 912 064;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 534 978 973 677 912 064 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 069 957 947 355 824 128;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 069 957 947 355 824 128 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 139 915 894 711 648 256;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 139 915 894 711 648 256 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 279 831 789 423 296 512;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 279 831 789 423 296 512 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 559 663 578 846 593 024;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 559 663 578 846 593 024 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 001 119 327 157 693 186 048;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 001 119 327 157 693 186 048 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 002 238 654 315 386 372 096;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 002 238 654 315 386 372 096 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 004 477 308 630 772 744 192;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 004 477 308 630 772 744 192 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 008 954 617 261 545 488 384;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 008 954 617 261 545 488 384 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 017 909 234 523 090 976 768;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 017 909 234 523 090 976 768 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 035 818 469 046 181 953 536;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 035 818 469 046 181 953 536 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 071 636 938 092 363 907 072;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 071 636 938 092 363 907 072 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 143 273 876 184 727 814 144;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 143 273 876 184 727 814 144 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 286 547 752 369 455 628 288;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 286 547 752 369 455 628 288 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 573 095 504 738 911 256 576;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 573 095 504 738 911 256 576 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 001 146 191 009 477 822 513 152;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 001 146 191 009 477 822 513 152 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 002 292 382 018 955 645 026 304;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 002 292 382 018 955 645 026 304 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 004 584 764 037 911 290 052 608;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 004 584 764 037 911 290 052 608 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 009 169 528 075 822 580 105 216;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 009 169 528 075 822 580 105 216 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 018 339 056 151 645 160 210 432;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 018 339 056 151 645 160 210 432 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 036 678 112 303 290 320 420 864;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 036 678 112 303 290 320 420 864 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 073 356 224 606 580 640 841 728;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 073 356 224 606 580 640 841 728 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 146 712 449 213 161 281 683 456;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 146 712 449 213 161 281 683 456 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 293 424 898 426 322 563 366 912;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 293 424 898 426 322 563 366 912 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 586 849 796 852 645 126 733 824;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 586 849 796 852 645 126 733 824 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 001 173 699 593 705 290 253 467 648;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 001 173 699 593 705 290 253 467 648 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 002 347 399 187 410 580 506 935 296;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 002 347 399 187 410 580 506 935 296 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 004 694 798 374 821 161 013 870 592;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 004 694 798 374 821 161 013 870 592 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 009 389 596 749 642 322 027 741 184;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 009 389 596 749 642 322 027 741 184 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 018 779 193 499 284 644 055 482 368;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 018 779 193 499 284 644 055 482 368 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 037 558 386 998 569 288 110 964 736;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 037 558 386 998 569 288 110 964 736 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 075 116 773 997 138 576 221 929 472;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 075 116 773 997 138 576 221 929 472 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 150 233 547 994 277 152 443 858 944;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 150 233 547 994 277 152 443 858 944 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 300 467 095 988 554 304 887 717 888;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 300 467 095 988 554 304 887 717 888 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 600 934 191 977 108 609 775 435 776;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 600 934 191 977 108 609 775 435 776 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 001 201 868 383 954 217 219 550 871 552;
  • 86) 0.656 250 000 000 000 000 000 000 000 001 201 868 383 954 217 219 550 871 552 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 002 403 736 767 908 434 439 101 743 104;
  • 87) 0.312 500 000 000 000 000 000 000 000 002 403 736 767 908 434 439 101 743 104 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 004 807 473 535 816 868 878 203 486 208;
  • 88) 0.625 000 000 000 000 000 000 000 000 004 807 473 535 816 868 878 203 486 208 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 009 614 947 071 633 737 756 406 972 416;
  • 89) 0.250 000 000 000 000 000 000 000 000 009 614 947 071 633 737 756 406 972 416 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 019 229 894 143 267 475 512 813 944 832;
  • 90) 0.500 000 000 000 000 000 000 000 000 019 229 894 143 267 475 512 813 944 832 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 038 459 788 286 534 951 025 627 889 664;
  • 91) 0.000 000 000 000 000 000 000 000 000 038 459 788 286 534 951 025 627 889 664 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 076 919 576 573 069 902 051 255 779 328;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 061(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 061(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 061(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 061 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