0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 782 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 782(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 782(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 782.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 782 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 564;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 564 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 128;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 128 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 678 256;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 678 256 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 356 512;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 356 512 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 713 024;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 713 024 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 426 048;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 426 048 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 852 096;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 852 096 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 704 192;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 704 192 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 408 384;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 408 384 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 614 816 768;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 614 816 768 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 229 633 536;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 229 633 536 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 459 267 072;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 459 267 072 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 918 534 144;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 918 534 144 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 837 068 288;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 837 068 288 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 674 136 576;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 674 136 576 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 348 273 152;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 348 273 152 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 696 546 304;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 696 546 304 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 393 092 608;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 393 092 608 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 786 185 216;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 786 185 216 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 572 370 432;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 572 370 432 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 144 740 864;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 144 740 864 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 342 289 481 728;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 342 289 481 728 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 684 578 963 456;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 684 578 963 456 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 369 157 926 912;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 369 157 926 912 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 738 315 853 824;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 738 315 853 824 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 476 631 707 648;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 476 631 707 648 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 953 263 415 296;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 953 263 415 296 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 906 526 830 592;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 906 526 830 592 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 813 053 661 184;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 813 053 661 184 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 271 626 107 322 368;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 271 626 107 322 368 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 543 252 214 644 736;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 543 252 214 644 736 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 086 504 429 289 472;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 086 504 429 289 472 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 173 008 858 578 944;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 173 008 858 578 944 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 346 017 717 157 888;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 346 017 717 157 888 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 692 035 434 315 776;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 692 035 434 315 776 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 384 070 868 631 552;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 384 070 868 631 552 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 768 141 737 263 104;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 768 141 737 263 104 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 536 283 474 526 208;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 536 283 474 526 208 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 072 566 949 052 416;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 072 566 949 052 416 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 145 133 898 104 832;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 145 133 898 104 832 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 292 290 267 796 209 664;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 292 290 267 796 209 664 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 584 580 535 592 419 328;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 584 580 535 592 419 328 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 169 161 071 184 838 656;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 169 161 071 184 838 656 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 338 322 142 369 677 312;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 338 322 142 369 677 312 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 676 644 284 739 354 624;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 676 644 284 739 354 624 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 353 288 569 478 709 248;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 353 288 569 478 709 248 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 706 577 138 957 418 496;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 706 577 138 957 418 496 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 413 154 277 914 836 992;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 413 154 277 914 836 992 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 810 826 308 555 829 673 984;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 810 826 308 555 829 673 984 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 621 652 617 111 659 347 968;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 621 652 617 111 659 347 968 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 243 305 234 223 318 695 936;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 243 305 234 223 318 695 936 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 486 610 468 446 637 391 872;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 486 610 468 446 637 391 872 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 973 220 936 893 274 783 744;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 973 220 936 893 274 783 744 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 946 441 873 786 549 567 488;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 946 441 873 786 549 567 488 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 892 883 747 573 099 134 976;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 892 883 747 573 099 134 976 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 785 767 495 146 198 269 952;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 785 767 495 146 198 269 952 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 571 534 990 292 396 539 904;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 571 534 990 292 396 539 904 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 143 069 980 584 793 079 808;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 143 069 980 584 793 079 808 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 286 139 961 169 586 159 616;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 286 139 961 169 586 159 616 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 572 279 922 339 172 319 232;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 572 279 922 339 172 319 232 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 144 559 844 678 344 638 464;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 144 559 844 678 344 638 464 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 498 289 119 689 356 689 276 928;
  • 63) 0.676 983 553 916 215 896 606 445 312 498 289 119 689 356 689 276 928 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 996 578 239 378 713 378 553 856;
  • 64) 0.353 967 107 832 431 793 212 890 624 996 578 239 378 713 378 553 856 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 993 156 478 757 426 757 107 712;
  • 65) 0.707 934 215 664 863 586 425 781 249 993 156 478 757 426 757 107 712 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 986 312 957 514 853 514 215 424;
  • 66) 0.415 868 431 329 727 172 851 562 499 986 312 957 514 853 514 215 424 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 972 625 915 029 707 028 430 848;
  • 67) 0.831 736 862 659 454 345 703 124 999 972 625 915 029 707 028 430 848 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 945 251 830 059 414 056 861 696;
  • 68) 0.663 473 725 318 908 691 406 249 999 945 251 830 059 414 056 861 696 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 890 503 660 118 828 113 723 392;
  • 69) 0.326 947 450 637 817 382 812 499 999 890 503 660 118 828 113 723 392 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 781 007 320 237 656 227 446 784;
  • 70) 0.653 894 901 275 634 765 624 999 999 781 007 320 237 656 227 446 784 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 562 014 640 475 312 454 893 568;
  • 71) 0.307 789 802 551 269 531 249 999 999 562 014 640 475 312 454 893 568 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 124 029 280 950 624 909 787 136;
  • 72) 0.615 579 605 102 539 062 499 999 999 124 029 280 950 624 909 787 136 × 2 = 1 + 0.231 159 210 205 078 124 999 999 998 248 058 561 901 249 819 574 272;
  • 73) 0.231 159 210 205 078 124 999 999 998 248 058 561 901 249 819 574 272 × 2 = 0 + 0.462 318 420 410 156 249 999 999 996 496 117 123 802 499 639 148 544;
  • 74) 0.462 318 420 410 156 249 999 999 996 496 117 123 802 499 639 148 544 × 2 = 0 + 0.924 636 840 820 312 499 999 999 992 992 234 247 604 999 278 297 088;
  • 75) 0.924 636 840 820 312 499 999 999 992 992 234 247 604 999 278 297 088 × 2 = 1 + 0.849 273 681 640 624 999 999 999 985 984 468 495 209 998 556 594 176;
  • 76) 0.849 273 681 640 624 999 999 999 985 984 468 495 209 998 556 594 176 × 2 = 1 + 0.698 547 363 281 249 999 999 999 971 968 936 990 419 997 113 188 352;
  • 77) 0.698 547 363 281 249 999 999 999 971 968 936 990 419 997 113 188 352 × 2 = 1 + 0.397 094 726 562 499 999 999 999 943 937 873 980 839 994 226 376 704;
  • 78) 0.397 094 726 562 499 999 999 999 943 937 873 980 839 994 226 376 704 × 2 = 0 + 0.794 189 453 124 999 999 999 999 887 875 747 961 679 988 452 753 408;
  • 79) 0.794 189 453 124 999 999 999 999 887 875 747 961 679 988 452 753 408 × 2 = 1 + 0.588 378 906 249 999 999 999 999 775 751 495 923 359 976 905 506 816;
  • 80) 0.588 378 906 249 999 999 999 999 775 751 495 923 359 976 905 506 816 × 2 = 1 + 0.176 757 812 499 999 999 999 999 551 502 991 846 719 953 811 013 632;
  • 81) 0.176 757 812 499 999 999 999 999 551 502 991 846 719 953 811 013 632 × 2 = 0 + 0.353 515 624 999 999 999 999 999 103 005 983 693 439 907 622 027 264;
  • 82) 0.353 515 624 999 999 999 999 999 103 005 983 693 439 907 622 027 264 × 2 = 0 + 0.707 031 249 999 999 999 999 998 206 011 967 386 879 815 244 054 528;
  • 83) 0.707 031 249 999 999 999 999 998 206 011 967 386 879 815 244 054 528 × 2 = 1 + 0.414 062 499 999 999 999 999 996 412 023 934 773 759 630 488 109 056;
  • 84) 0.414 062 499 999 999 999 999 996 412 023 934 773 759 630 488 109 056 × 2 = 0 + 0.828 124 999 999 999 999 999 992 824 047 869 547 519 260 976 218 112;
  • 85) 0.828 124 999 999 999 999 999 992 824 047 869 547 519 260 976 218 112 × 2 = 1 + 0.656 249 999 999 999 999 999 985 648 095 739 095 038 521 952 436 224;
  • 86) 0.656 249 999 999 999 999 999 985 648 095 739 095 038 521 952 436 224 × 2 = 1 + 0.312 499 999 999 999 999 999 971 296 191 478 190 077 043 904 872 448;
  • 87) 0.312 499 999 999 999 999 999 971 296 191 478 190 077 043 904 872 448 × 2 = 0 + 0.624 999 999 999 999 999 999 942 592 382 956 380 154 087 809 744 896;
  • 88) 0.624 999 999 999 999 999 999 942 592 382 956 380 154 087 809 744 896 × 2 = 1 + 0.249 999 999 999 999 999 999 885 184 765 912 760 308 175 619 489 792;
  • 89) 0.249 999 999 999 999 999 999 885 184 765 912 760 308 175 619 489 792 × 2 = 0 + 0.499 999 999 999 999 999 999 770 369 531 825 520 616 351 238 979 584;
  • 90) 0.499 999 999 999 999 999 999 770 369 531 825 520 616 351 238 979 584 × 2 = 0 + 0.999 999 999 999 999 999 999 540 739 063 651 041 232 702 477 959 168;
  • 91) 0.999 999 999 999 999 999 999 540 739 063 651 041 232 702 477 959 168 × 2 = 1 + 0.999 999 999 999 999 999 999 081 478 127 302 082 465 404 955 918 336;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 782(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 782(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 782(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001(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 1001


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 1001 =

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


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 1001


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 782 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 1001

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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