0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 386 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 386(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 386(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 386.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 386 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 772;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 772 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 341 544;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 341 544 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 683 088;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 683 088 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 366 176;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 366 176 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 732 352;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 732 352 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 464 704;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 464 704 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 929 408;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 929 408 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 858 816;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 858 816 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 717 632;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 717 632 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 435 264;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 435 264 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 870 528;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 870 528 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 461 741 056;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 461 741 056 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 923 482 112;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 923 482 112 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 846 964 224;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 846 964 224 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 693 928 448;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 693 928 448 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 387 856 896;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 387 856 896 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 775 713 792;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 775 713 792 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 551 427 584;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 551 427 584 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 102 855 168;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 102 855 168 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 086 205 710 336;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 086 205 710 336 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 172 411 420 672;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 172 411 420 672 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 344 822 841 344;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 344 822 841 344 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 689 645 682 688;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 689 645 682 688 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 379 291 365 376;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 379 291 365 376 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 758 582 730 752;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 758 582 730 752 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 517 165 461 504;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 517 165 461 504 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 034 330 923 008;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 034 330 923 008 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 068 661 846 016;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 068 661 846 016 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 137 323 692 032;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 137 323 692 032 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 274 647 384 064;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 274 647 384 064 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 549 294 768 128;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 549 294 768 128 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 089 098 589 536 256;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 089 098 589 536 256 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 178 197 179 072 512;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 178 197 179 072 512 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 356 394 358 145 024;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 356 394 358 145 024 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 712 788 716 290 048;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 712 788 716 290 048 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 425 577 432 580 096;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 425 577 432 580 096 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 851 154 865 160 192;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 851 154 865 160 192 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 702 309 730 320 384;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 702 309 730 320 384 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 404 619 460 640 768;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 404 619 460 640 768 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 809 238 921 281 536;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 809 238 921 281 536 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 618 477 842 563 072;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 618 477 842 563 072 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 587 236 955 685 126 144;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 587 236 955 685 126 144 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 174 473 911 370 252 288;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 174 473 911 370 252 288 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 348 947 822 740 504 576;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 348 947 822 740 504 576 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 697 895 645 481 009 152;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 697 895 645 481 009 152 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 395 791 290 962 018 304;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 395 791 290 962 018 304 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 791 582 581 924 036 608;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 791 582 581 924 036 608 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 583 165 163 848 073 216;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 583 165 163 848 073 216 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 166 330 327 696 146 432;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 166 330 327 696 146 432 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 332 660 655 392 292 864;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 332 660 655 392 292 864 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 665 321 310 784 585 728;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 665 321 310 784 585 728 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 489 330 642 621 569 171 456;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 489 330 642 621 569 171 456 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 978 661 285 243 138 342 912;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 978 661 285 243 138 342 912 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 957 322 570 486 276 685 824;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 957 322 570 486 276 685 824 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 914 645 140 972 553 371 648;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 914 645 140 972 553 371 648 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 829 290 281 945 106 743 296;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 829 290 281 945 106 743 296 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 658 580 563 890 213 486 592;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 658 580 563 890 213 486 592 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 317 161 127 780 426 973 184;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 317 161 127 780 426 973 184 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 634 322 255 560 853 946 368;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 634 322 255 560 853 946 368 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 268 644 511 121 707 892 736;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 268 644 511 121 707 892 736 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 537 289 022 243 415 785 472;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 537 289 022 243 415 785 472 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 501 074 578 044 486 831 570 944;
  • 63) 0.676 983 553 916 215 896 606 445 312 501 074 578 044 486 831 570 944 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 002 149 156 088 973 663 141 888;
  • 64) 0.353 967 107 832 431 793 212 890 625 002 149 156 088 973 663 141 888 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 004 298 312 177 947 326 283 776;
  • 65) 0.707 934 215 664 863 586 425 781 250 004 298 312 177 947 326 283 776 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 008 596 624 355 894 652 567 552;
  • 66) 0.415 868 431 329 727 172 851 562 500 008 596 624 355 894 652 567 552 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 017 193 248 711 789 305 135 104;
  • 67) 0.831 736 862 659 454 345 703 125 000 017 193 248 711 789 305 135 104 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 034 386 497 423 578 610 270 208;
  • 68) 0.663 473 725 318 908 691 406 250 000 034 386 497 423 578 610 270 208 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 068 772 994 847 157 220 540 416;
  • 69) 0.326 947 450 637 817 382 812 500 000 068 772 994 847 157 220 540 416 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 137 545 989 694 314 441 080 832;
  • 70) 0.653 894 901 275 634 765 625 000 000 137 545 989 694 314 441 080 832 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 275 091 979 388 628 882 161 664;
  • 71) 0.307 789 802 551 269 531 250 000 000 275 091 979 388 628 882 161 664 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 550 183 958 777 257 764 323 328;
  • 72) 0.615 579 605 102 539 062 500 000 000 550 183 958 777 257 764 323 328 × 2 = 1 + 0.231 159 210 205 078 125 000 000 001 100 367 917 554 515 528 646 656;
  • 73) 0.231 159 210 205 078 125 000 000 001 100 367 917 554 515 528 646 656 × 2 = 0 + 0.462 318 420 410 156 250 000 000 002 200 735 835 109 031 057 293 312;
  • 74) 0.462 318 420 410 156 250 000 000 002 200 735 835 109 031 057 293 312 × 2 = 0 + 0.924 636 840 820 312 500 000 000 004 401 471 670 218 062 114 586 624;
  • 75) 0.924 636 840 820 312 500 000 000 004 401 471 670 218 062 114 586 624 × 2 = 1 + 0.849 273 681 640 625 000 000 000 008 802 943 340 436 124 229 173 248;
  • 76) 0.849 273 681 640 625 000 000 000 008 802 943 340 436 124 229 173 248 × 2 = 1 + 0.698 547 363 281 250 000 000 000 017 605 886 680 872 248 458 346 496;
  • 77) 0.698 547 363 281 250 000 000 000 017 605 886 680 872 248 458 346 496 × 2 = 1 + 0.397 094 726 562 500 000 000 000 035 211 773 361 744 496 916 692 992;
  • 78) 0.397 094 726 562 500 000 000 000 035 211 773 361 744 496 916 692 992 × 2 = 0 + 0.794 189 453 125 000 000 000 000 070 423 546 723 488 993 833 385 984;
  • 79) 0.794 189 453 125 000 000 000 000 070 423 546 723 488 993 833 385 984 × 2 = 1 + 0.588 378 906 250 000 000 000 000 140 847 093 446 977 987 666 771 968;
  • 80) 0.588 378 906 250 000 000 000 000 140 847 093 446 977 987 666 771 968 × 2 = 1 + 0.176 757 812 500 000 000 000 000 281 694 186 893 955 975 333 543 936;
  • 81) 0.176 757 812 500 000 000 000 000 281 694 186 893 955 975 333 543 936 × 2 = 0 + 0.353 515 625 000 000 000 000 000 563 388 373 787 911 950 667 087 872;
  • 82) 0.353 515 625 000 000 000 000 000 563 388 373 787 911 950 667 087 872 × 2 = 0 + 0.707 031 250 000 000 000 000 001 126 776 747 575 823 901 334 175 744;
  • 83) 0.707 031 250 000 000 000 000 001 126 776 747 575 823 901 334 175 744 × 2 = 1 + 0.414 062 500 000 000 000 000 002 253 553 495 151 647 802 668 351 488;
  • 84) 0.414 062 500 000 000 000 000 002 253 553 495 151 647 802 668 351 488 × 2 = 0 + 0.828 125 000 000 000 000 000 004 507 106 990 303 295 605 336 702 976;
  • 85) 0.828 125 000 000 000 000 000 004 507 106 990 303 295 605 336 702 976 × 2 = 1 + 0.656 250 000 000 000 000 000 009 014 213 980 606 591 210 673 405 952;
  • 86) 0.656 250 000 000 000 000 000 009 014 213 980 606 591 210 673 405 952 × 2 = 1 + 0.312 500 000 000 000 000 000 018 028 427 961 213 182 421 346 811 904;
  • 87) 0.312 500 000 000 000 000 000 018 028 427 961 213 182 421 346 811 904 × 2 = 0 + 0.625 000 000 000 000 000 000 036 056 855 922 426 364 842 693 623 808;
  • 88) 0.625 000 000 000 000 000 000 036 056 855 922 426 364 842 693 623 808 × 2 = 1 + 0.250 000 000 000 000 000 000 072 113 711 844 852 729 685 387 247 616;
  • 89) 0.250 000 000 000 000 000 000 072 113 711 844 852 729 685 387 247 616 × 2 = 0 + 0.500 000 000 000 000 000 000 144 227 423 689 705 459 370 774 495 232;
  • 90) 0.500 000 000 000 000 000 000 144 227 423 689 705 459 370 774 495 232 × 2 = 1 + 0.000 000 000 000 000 000 000 288 454 847 379 410 918 741 548 990 464;
  • 91) 0.000 000 000 000 000 000 000 288 454 847 379 410 918 741 548 990 464 × 2 = 0 + 0.000 000 000 000 000 000 000 576 909 694 758 821 837 483 097 980 928;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 386(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 386(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 386(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 386 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