0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 688 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 688(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 688(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 688.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 688 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 975 376;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 975 376 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 950 752;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 950 752 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 901 504;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 901 504 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 803 008;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 803 008 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 606 016;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 606 016 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 212 032;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 212 032 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 424 064;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 424 064 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 848 128;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 848 128 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 696 256;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 696 256 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 392 512;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 392 512 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 318 785 024;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 318 785 024 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 637 570 048;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 637 570 048 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 275 140 096;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 275 140 096 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 550 280 192;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 550 280 192 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 100 560 384;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 100 560 384 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 201 120 768;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 201 120 768 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 402 241 536;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 402 241 536 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 804 483 072;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 804 483 072 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 608 966 144;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 608 966 144 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 217 932 288;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 217 932 288 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 758 435 864 576;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 758 435 864 576 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 516 871 729 152;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 516 871 729 152 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 033 743 458 304;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 033 743 458 304 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 067 486 916 608;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 067 486 916 608 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 134 973 833 216;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 134 973 833 216 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 269 947 666 432;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 269 947 666 432 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 539 895 332 864;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 539 895 332 864 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 079 790 665 728;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 079 790 665 728 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 159 581 331 456;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 159 581 331 456 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 319 162 662 912;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 319 162 662 912 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 904 638 325 325 824;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 904 638 325 325 824 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 809 276 650 651 648;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 809 276 650 651 648 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 618 553 301 303 296;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 618 553 301 303 296 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 237 106 602 606 592;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 237 106 602 606 592 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 474 213 205 213 184;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 474 213 205 213 184 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 948 426 410 426 368;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 948 426 410 426 368 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 896 852 820 852 736;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 896 852 820 852 736 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 793 705 641 705 472;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 793 705 641 705 472 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 587 411 283 410 944;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 587 411 283 410 944 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 174 822 566 821 888;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 174 822 566 821 888 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 078 349 645 133 643 776;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 078 349 645 133 643 776 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 156 699 290 267 287 552;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 156 699 290 267 287 552 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 313 398 580 534 575 104;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 313 398 580 534 575 104 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 626 797 161 069 150 208;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 626 797 161 069 150 208 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 253 594 322 138 300 416;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 253 594 322 138 300 416 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 507 188 644 276 600 832;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 507 188 644 276 600 832 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 014 377 288 553 201 664;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 014 377 288 553 201 664 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 028 754 577 106 403 328;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 028 754 577 106 403 328 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 057 509 154 212 806 656;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 057 509 154 212 806 656 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 115 018 308 425 613 312;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 115 018 308 425 613 312 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 624 230 036 616 851 226 624;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 624 230 036 616 851 226 624 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 248 460 073 233 702 453 248;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 248 460 073 233 702 453 248 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 496 920 146 467 404 906 496;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 496 920 146 467 404 906 496 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 993 840 292 934 809 812 992;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 993 840 292 934 809 812 992 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 987 680 585 869 619 625 984;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 987 680 585 869 619 625 984 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 499 975 361 171 739 239 251 968;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 499 975 361 171 739 239 251 968 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 999 950 722 343 478 478 503 936;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 999 950 722 343 478 478 503 936 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 999 901 444 686 956 957 007 872;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 999 901 444 686 956 957 007 872 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 999 802 889 373 913 914 015 744;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 999 802 889 373 913 914 015 744 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 999 605 778 747 827 828 031 488;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 999 605 778 747 827 828 031 488 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 999 211 557 495 655 656 062 976;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 999 211 557 495 655 656 062 976 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 998 423 114 991 311 312 125 952;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 998 423 114 991 311 312 125 952 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 996 846 229 982 622 624 251 904;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 996 846 229 982 622 624 251 904 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 993 692 459 965 245 248 503 808;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 993 692 459 965 245 248 503 808 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 987 384 919 930 490 497 007 616;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 987 384 919 930 490 497 007 616 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 999 974 769 839 860 980 994 015 232;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 999 974 769 839 860 980 994 015 232 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 999 949 539 679 721 961 988 030 464;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 999 949 539 679 721 961 988 030 464 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 999 899 079 359 443 923 976 060 928;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 999 899 079 359 443 923 976 060 928 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 999 798 158 718 887 847 952 121 856;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 999 798 158 718 887 847 952 121 856 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 999 596 317 437 775 695 904 243 712;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 999 596 317 437 775 695 904 243 712 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 999 192 634 875 551 391 808 487 424;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 999 192 634 875 551 391 808 487 424 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 998 385 269 751 102 783 616 974 848;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 998 385 269 751 102 783 616 974 848 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 996 770 539 502 205 567 233 949 696;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 996 770 539 502 205 567 233 949 696 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 993 541 079 004 411 134 467 899 392;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 993 541 079 004 411 134 467 899 392 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 987 082 158 008 822 268 935 798 784;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 987 082 158 008 822 268 935 798 784 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 999 974 164 316 017 644 537 871 597 568;
  • 77) 0.698 547 363 281 249 999 999 999 999 999 974 164 316 017 644 537 871 597 568 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 999 948 328 632 035 289 075 743 195 136;
  • 78) 0.397 094 726 562 499 999 999 999 999 999 948 328 632 035 289 075 743 195 136 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 999 896 657 264 070 578 151 486 390 272;
  • 79) 0.794 189 453 124 999 999 999 999 999 999 896 657 264 070 578 151 486 390 272 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 999 793 314 528 141 156 302 972 780 544;
  • 80) 0.588 378 906 249 999 999 999 999 999 999 793 314 528 141 156 302 972 780 544 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 999 586 629 056 282 312 605 945 561 088;
  • 81) 0.176 757 812 499 999 999 999 999 999 999 586 629 056 282 312 605 945 561 088 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 999 173 258 112 564 625 211 891 122 176;
  • 82) 0.353 515 624 999 999 999 999 999 999 999 173 258 112 564 625 211 891 122 176 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 998 346 516 225 129 250 423 782 244 352;
  • 83) 0.707 031 249 999 999 999 999 999 999 998 346 516 225 129 250 423 782 244 352 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 996 693 032 450 258 500 847 564 488 704;
  • 84) 0.414 062 499 999 999 999 999 999 999 996 693 032 450 258 500 847 564 488 704 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 993 386 064 900 517 001 695 128 977 408;
  • 85) 0.828 124 999 999 999 999 999 999 999 993 386 064 900 517 001 695 128 977 408 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 986 772 129 801 034 003 390 257 954 816;
  • 86) 0.656 249 999 999 999 999 999 999 999 986 772 129 801 034 003 390 257 954 816 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 973 544 259 602 068 006 780 515 909 632;
  • 87) 0.312 499 999 999 999 999 999 999 999 973 544 259 602 068 006 780 515 909 632 × 2 = 0 + 0.624 999 999 999 999 999 999 999 999 947 088 519 204 136 013 561 031 819 264;
  • 88) 0.624 999 999 999 999 999 999 999 999 947 088 519 204 136 013 561 031 819 264 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 894 177 038 408 272 027 122 063 638 528;
  • 89) 0.249 999 999 999 999 999 999 999 999 894 177 038 408 272 027 122 063 638 528 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 788 354 076 816 544 054 244 127 277 056;
  • 90) 0.499 999 999 999 999 999 999 999 999 788 354 076 816 544 054 244 127 277 056 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 576 708 153 633 088 108 488 254 554 112;
  • 91) 0.999 999 999 999 999 999 999 999 999 576 708 153 633 088 108 488 254 554 112 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 153 416 307 266 176 216 976 509 108 224;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 688(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 688(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 688(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 688 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