0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 037 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 037(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 037(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 037.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 037 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 074;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 074 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 148;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 148 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 224 296;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 224 296 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 448 592;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 448 592 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 897 184;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 897 184 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 794 368;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 794 368 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 588 736;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 588 736 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 177 472;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 177 472 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 354 944;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 354 944 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 709 888;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 709 888 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 419 776;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 419 776 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 839 552;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 839 552 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 679 104;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 679 104 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 147 358 208;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 147 358 208 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 294 716 416;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 294 716 416 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 589 432 832;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 589 432 832 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 178 865 664;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 178 865 664 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 357 731 328;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 357 731 328 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 715 462 656;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 715 462 656 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 430 925 312;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 430 925 312 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 861 850 624;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 861 850 624 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 723 701 248;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 723 701 248 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 447 402 496;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 447 402 496 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 894 804 992;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 894 804 992 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 765 789 609 984;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 765 789 609 984 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 531 579 219 968;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 531 579 219 968 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 063 158 439 936;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 063 158 439 936 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 126 316 879 872;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 126 316 879 872 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 252 633 759 744;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 252 633 759 744 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 505 267 519 488;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 505 267 519 488 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 010 535 038 976;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 010 535 038 976 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 098 021 070 077 952;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 098 021 070 077 952 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 196 042 140 155 904;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 196 042 140 155 904 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 392 084 280 311 808;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 392 084 280 311 808 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 784 168 560 623 616;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 784 168 560 623 616 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 568 337 121 247 232;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 568 337 121 247 232 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 136 674 242 494 464;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 136 674 242 494 464 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 273 348 484 988 928;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 273 348 484 988 928 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 546 696 969 977 856;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 546 696 969 977 856 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 093 393 939 955 712;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 093 393 939 955 712 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 186 787 879 911 424;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 186 787 879 911 424 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 373 575 759 822 848;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 373 575 759 822 848 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 747 151 519 645 696;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 747 151 519 645 696 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 849 494 303 039 291 392;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 849 494 303 039 291 392 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 698 988 606 078 582 784;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 698 988 606 078 582 784 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 397 977 212 157 165 568;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 397 977 212 157 165 568 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 795 954 424 314 331 136;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 795 954 424 314 331 136 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 591 908 848 628 662 272;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 591 908 848 628 662 272 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 183 817 697 257 324 544;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 183 817 697 257 324 544 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 367 635 394 514 649 088;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 367 635 394 514 649 088 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 735 270 789 029 298 176;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 735 270 789 029 298 176 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 470 541 578 058 596 352;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 470 541 578 058 596 352 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 941 083 156 117 192 704;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 941 083 156 117 192 704 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 882 166 312 234 385 408;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 882 166 312 234 385 408 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 251 764 332 624 468 770 816;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 251 764 332 624 468 770 816 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 503 528 665 248 937 541 632;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 503 528 665 248 937 541 632 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 007 057 330 497 875 083 264;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 007 057 330 497 875 083 264 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 014 114 660 995 750 166 528;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 014 114 660 995 750 166 528 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 028 229 321 991 500 333 056;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 028 229 321 991 500 333 056 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 056 458 643 983 000 666 112;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 056 458 643 983 000 666 112 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 112 917 287 966 001 332 224;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 112 917 287 966 001 332 224 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 225 834 575 932 002 664 448;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 225 834 575 932 002 664 448 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 451 669 151 864 005 328 896;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 451 669 151 864 005 328 896 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 903 338 303 728 010 657 792;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 903 338 303 728 010 657 792 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 001 806 676 607 456 021 315 584;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 001 806 676 607 456 021 315 584 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 003 613 353 214 912 042 631 168;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 003 613 353 214 912 042 631 168 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 007 226 706 429 824 085 262 336;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 007 226 706 429 824 085 262 336 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 014 453 412 859 648 170 524 672;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 014 453 412 859 648 170 524 672 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 028 906 825 719 296 341 049 344;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 028 906 825 719 296 341 049 344 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 057 813 651 438 592 682 098 688;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 057 813 651 438 592 682 098 688 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 115 627 302 877 185 364 197 376;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 115 627 302 877 185 364 197 376 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 231 254 605 754 370 728 394 752;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 231 254 605 754 370 728 394 752 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 462 509 211 508 741 456 789 504;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 462 509 211 508 741 456 789 504 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 925 018 423 017 482 913 579 008;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 925 018 423 017 482 913 579 008 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 001 850 036 846 034 965 827 158 016;
  • 76) 0.849 273 681 640 625 000 000 000 000 001 850 036 846 034 965 827 158 016 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 003 700 073 692 069 931 654 316 032;
  • 77) 0.698 547 363 281 250 000 000 000 000 003 700 073 692 069 931 654 316 032 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 007 400 147 384 139 863 308 632 064;
  • 78) 0.397 094 726 562 500 000 000 000 000 007 400 147 384 139 863 308 632 064 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 014 800 294 768 279 726 617 264 128;
  • 79) 0.794 189 453 125 000 000 000 000 000 014 800 294 768 279 726 617 264 128 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 029 600 589 536 559 453 234 528 256;
  • 80) 0.588 378 906 250 000 000 000 000 000 029 600 589 536 559 453 234 528 256 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 059 201 179 073 118 906 469 056 512;
  • 81) 0.176 757 812 500 000 000 000 000 000 059 201 179 073 118 906 469 056 512 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 118 402 358 146 237 812 938 113 024;
  • 82) 0.353 515 625 000 000 000 000 000 000 118 402 358 146 237 812 938 113 024 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 236 804 716 292 475 625 876 226 048;
  • 83) 0.707 031 250 000 000 000 000 000 000 236 804 716 292 475 625 876 226 048 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 473 609 432 584 951 251 752 452 096;
  • 84) 0.414 062 500 000 000 000 000 000 000 473 609 432 584 951 251 752 452 096 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 947 218 865 169 902 503 504 904 192;
  • 85) 0.828 125 000 000 000 000 000 000 000 947 218 865 169 902 503 504 904 192 × 2 = 1 + 0.656 250 000 000 000 000 000 000 001 894 437 730 339 805 007 009 808 384;
  • 86) 0.656 250 000 000 000 000 000 000 001 894 437 730 339 805 007 009 808 384 × 2 = 1 + 0.312 500 000 000 000 000 000 000 003 788 875 460 679 610 014 019 616 768;
  • 87) 0.312 500 000 000 000 000 000 000 003 788 875 460 679 610 014 019 616 768 × 2 = 0 + 0.625 000 000 000 000 000 000 000 007 577 750 921 359 220 028 039 233 536;
  • 88) 0.625 000 000 000 000 000 000 000 007 577 750 921 359 220 028 039 233 536 × 2 = 1 + 0.250 000 000 000 000 000 000 000 015 155 501 842 718 440 056 078 467 072;
  • 89) 0.250 000 000 000 000 000 000 000 015 155 501 842 718 440 056 078 467 072 × 2 = 0 + 0.500 000 000 000 000 000 000 000 030 311 003 685 436 880 112 156 934 144;
  • 90) 0.500 000 000 000 000 000 000 000 030 311 003 685 436 880 112 156 934 144 × 2 = 1 + 0.000 000 000 000 000 000 000 000 060 622 007 370 873 760 224 313 868 288;
  • 91) 0.000 000 000 000 000 000 000 000 060 622 007 370 873 760 224 313 868 288 × 2 = 0 + 0.000 000 000 000 000 000 000 000 121 244 014 741 747 520 448 627 736 576;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 037(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 037(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 037(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 037 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

Share

» Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 082 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 07, 2026 [July]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: July - by our visitors

» Improve your social skills: Are You Using Communication by Design, or by Default? Gemini Notebook YouTube Video of 6.33 minutes

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