0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 141 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 141(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 141(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 141.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 141 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 282;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 282 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 564;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 564 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 128;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 128 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 256;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 256 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 512;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 512 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 024;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 024 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 898 048;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 898 048 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 796 096;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 796 096 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 592 192;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 592 192 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 184 384;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 184 384 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 368 768;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 368 768 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 737 536;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 737 536 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 475 072;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 475 072 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 842 950 144;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 842 950 144 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 685 900 288;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 685 900 288 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 371 800 576;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 371 800 576 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 743 601 152;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 743 601 152 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 487 202 304;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 487 202 304 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 974 404 608;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 974 404 608 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 948 809 216;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 948 809 216 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 897 618 432;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 897 618 432 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 795 236 864;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 795 236 864 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 590 473 728;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 590 473 728 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 180 947 456;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 180 947 456 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 361 894 912;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 361 894 912 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 500 723 789 824;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 500 723 789 824 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 001 447 579 648;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 001 447 579 648 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 002 895 159 296;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 002 895 159 296 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 005 790 318 592;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 005 790 318 592 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 011 580 637 184;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 011 580 637 184 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 023 161 274 368;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 023 161 274 368 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 046 322 548 736;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 046 322 548 736 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 092 645 097 472;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 092 645 097 472 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 185 290 194 944;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 185 290 194 944 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 370 580 389 888;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 370 580 389 888 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 408 741 160 779 776;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 408 741 160 779 776 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 817 482 321 559 552;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 817 482 321 559 552 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 634 964 643 119 104;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 634 964 643 119 104 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 269 929 286 238 208;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 269 929 286 238 208 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 539 858 572 476 416;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 539 858 572 476 416 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 079 717 144 952 832;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 079 717 144 952 832 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 159 434 289 905 664;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 159 434 289 905 664 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 318 868 579 811 328;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 318 868 579 811 328 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 637 737 159 622 656;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 637 737 159 622 656 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 275 474 319 245 312;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 275 474 319 245 312 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 378 550 948 638 490 624;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 378 550 948 638 490 624 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 757 101 897 276 981 248;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 757 101 897 276 981 248 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 514 203 794 553 962 496;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 514 203 794 553 962 496 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 028 407 589 107 924 992;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 028 407 589 107 924 992 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 056 815 178 215 849 984;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 056 815 178 215 849 984 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 113 630 356 431 699 968;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 113 630 356 431 699 968 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 227 260 712 863 399 936;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 227 260 712 863 399 936 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 454 521 425 726 799 872;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 454 521 425 726 799 872 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 952 909 042 851 453 599 744;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 952 909 042 851 453 599 744 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 905 818 085 702 907 199 488;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 905 818 085 702 907 199 488 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 811 636 171 405 814 398 976;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 811 636 171 405 814 398 976 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 623 272 342 811 628 797 952;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 623 272 342 811 628 797 952 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 246 544 685 623 257 595 904;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 246 544 685 623 257 595 904 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 493 089 371 246 515 191 808;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 493 089 371 246 515 191 808 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 986 178 742 493 030 383 616;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 986 178 742 493 030 383 616 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 972 357 484 986 060 767 232;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 972 357 484 986 060 767 232 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 944 714 969 972 121 534 464;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 944 714 969 972 121 534 464 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 889 429 939 944 243 068 928;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 889 429 939 944 243 068 928 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 778 859 879 888 486 137 856;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 778 859 879 888 486 137 856 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 557 719 759 776 972 275 712;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 557 719 759 776 972 275 712 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 115 439 519 553 944 551 424;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 115 439 519 553 944 551 424 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 998 230 879 039 107 889 102 848;
  • 68) 0.663 473 725 318 908 691 406 249 999 998 230 879 039 107 889 102 848 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 996 461 758 078 215 778 205 696;
  • 69) 0.326 947 450 637 817 382 812 499 999 996 461 758 078 215 778 205 696 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 992 923 516 156 431 556 411 392;
  • 70) 0.653 894 901 275 634 765 624 999 999 992 923 516 156 431 556 411 392 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 985 847 032 312 863 112 822 784;
  • 71) 0.307 789 802 551 269 531 249 999 999 985 847 032 312 863 112 822 784 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 971 694 064 625 726 225 645 568;
  • 72) 0.615 579 605 102 539 062 499 999 999 971 694 064 625 726 225 645 568 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 943 388 129 251 452 451 291 136;
  • 73) 0.231 159 210 205 078 124 999 999 999 943 388 129 251 452 451 291 136 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 886 776 258 502 904 902 582 272;
  • 74) 0.462 318 420 410 156 249 999 999 999 886 776 258 502 904 902 582 272 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 773 552 517 005 809 805 164 544;
  • 75) 0.924 636 840 820 312 499 999 999 999 773 552 517 005 809 805 164 544 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 547 105 034 011 619 610 329 088;
  • 76) 0.849 273 681 640 624 999 999 999 999 547 105 034 011 619 610 329 088 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 094 210 068 023 239 220 658 176;
  • 77) 0.698 547 363 281 249 999 999 999 999 094 210 068 023 239 220 658 176 × 2 = 1 + 0.397 094 726 562 499 999 999 999 998 188 420 136 046 478 441 316 352;
  • 78) 0.397 094 726 562 499 999 999 999 998 188 420 136 046 478 441 316 352 × 2 = 0 + 0.794 189 453 124 999 999 999 999 996 376 840 272 092 956 882 632 704;
  • 79) 0.794 189 453 124 999 999 999 999 996 376 840 272 092 956 882 632 704 × 2 = 1 + 0.588 378 906 249 999 999 999 999 992 753 680 544 185 913 765 265 408;
  • 80) 0.588 378 906 249 999 999 999 999 992 753 680 544 185 913 765 265 408 × 2 = 1 + 0.176 757 812 499 999 999 999 999 985 507 361 088 371 827 530 530 816;
  • 81) 0.176 757 812 499 999 999 999 999 985 507 361 088 371 827 530 530 816 × 2 = 0 + 0.353 515 624 999 999 999 999 999 971 014 722 176 743 655 061 061 632;
  • 82) 0.353 515 624 999 999 999 999 999 971 014 722 176 743 655 061 061 632 × 2 = 0 + 0.707 031 249 999 999 999 999 999 942 029 444 353 487 310 122 123 264;
  • 83) 0.707 031 249 999 999 999 999 999 942 029 444 353 487 310 122 123 264 × 2 = 1 + 0.414 062 499 999 999 999 999 999 884 058 888 706 974 620 244 246 528;
  • 84) 0.414 062 499 999 999 999 999 999 884 058 888 706 974 620 244 246 528 × 2 = 0 + 0.828 124 999 999 999 999 999 999 768 117 777 413 949 240 488 493 056;
  • 85) 0.828 124 999 999 999 999 999 999 768 117 777 413 949 240 488 493 056 × 2 = 1 + 0.656 249 999 999 999 999 999 999 536 235 554 827 898 480 976 986 112;
  • 86) 0.656 249 999 999 999 999 999 999 536 235 554 827 898 480 976 986 112 × 2 = 1 + 0.312 499 999 999 999 999 999 999 072 471 109 655 796 961 953 972 224;
  • 87) 0.312 499 999 999 999 999 999 999 072 471 109 655 796 961 953 972 224 × 2 = 0 + 0.624 999 999 999 999 999 999 998 144 942 219 311 593 923 907 944 448;
  • 88) 0.624 999 999 999 999 999 999 998 144 942 219 311 593 923 907 944 448 × 2 = 1 + 0.249 999 999 999 999 999 999 996 289 884 438 623 187 847 815 888 896;
  • 89) 0.249 999 999 999 999 999 999 996 289 884 438 623 187 847 815 888 896 × 2 = 0 + 0.499 999 999 999 999 999 999 992 579 768 877 246 375 695 631 777 792;
  • 90) 0.499 999 999 999 999 999 999 992 579 768 877 246 375 695 631 777 792 × 2 = 0 + 0.999 999 999 999 999 999 999 985 159 537 754 492 751 391 263 555 584;
  • 91) 0.999 999 999 999 999 999 999 985 159 537 754 492 751 391 263 555 584 × 2 = 1 + 0.999 999 999 999 999 999 999 970 319 075 508 985 502 782 527 111 168;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 141(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 141(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 141(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 141 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