Convert 11 000 345 657 654 343 567 545 456 545 454 345 643 564 301 111.099 945 643 564 356 543 458 82 to 64 Bit Double Precision IEEE 754 Binary Floating Point Standard, From a Number in Base 10 Decimal System

11 000 345 657 654 343 567 545 456 545 454 345 643 564 301 111.099 945 643 564 356 543 458 82(10) to 64 bit double precision IEEE 754 binary floating point (1 bit for sign, 11 bits for exponent, 52 bits for mantissa) = ?

1. First, convert to the binary (base 2) the integer part: 11 000 345 657 654 343 567 545 456 545 454 345 643 564 301 111.
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;
  • 11 000 345 657 654 343 567 545 456 545 454 345 643 564 301 111 ÷ 2 = 5 500 172 828 827 171 783 772 728 272 727 172 821 782 150 555 + 1;
  • 5 500 172 828 827 171 783 772 728 272 727 172 821 782 150 555 ÷ 2 = 2 750 086 414 413 585 891 886 364 136 363 586 410 891 075 277 + 1;
  • 2 750 086 414 413 585 891 886 364 136 363 586 410 891 075 277 ÷ 2 = 1 375 043 207 206 792 945 943 182 068 181 793 205 445 537 638 + 1;
  • 1 375 043 207 206 792 945 943 182 068 181 793 205 445 537 638 ÷ 2 = 687 521 603 603 396 472 971 591 034 090 896 602 722 768 819 + 0;
  • 687 521 603 603 396 472 971 591 034 090 896 602 722 768 819 ÷ 2 = 343 760 801 801 698 236 485 795 517 045 448 301 361 384 409 + 1;
  • 343 760 801 801 698 236 485 795 517 045 448 301 361 384 409 ÷ 2 = 171 880 400 900 849 118 242 897 758 522 724 150 680 692 204 + 1;
  • 171 880 400 900 849 118 242 897 758 522 724 150 680 692 204 ÷ 2 = 85 940 200 450 424 559 121 448 879 261 362 075 340 346 102 + 0;
  • 85 940 200 450 424 559 121 448 879 261 362 075 340 346 102 ÷ 2 = 42 970 100 225 212 279 560 724 439 630 681 037 670 173 051 + 0;
  • 42 970 100 225 212 279 560 724 439 630 681 037 670 173 051 ÷ 2 = 21 485 050 112 606 139 780 362 219 815 340 518 835 086 525 + 1;
  • 21 485 050 112 606 139 780 362 219 815 340 518 835 086 525 ÷ 2 = 10 742 525 056 303 069 890 181 109 907 670 259 417 543 262 + 1;
  • 10 742 525 056 303 069 890 181 109 907 670 259 417 543 262 ÷ 2 = 5 371 262 528 151 534 945 090 554 953 835 129 708 771 631 + 0;
  • 5 371 262 528 151 534 945 090 554 953 835 129 708 771 631 ÷ 2 = 2 685 631 264 075 767 472 545 277 476 917 564 854 385 815 + 1;
  • 2 685 631 264 075 767 472 545 277 476 917 564 854 385 815 ÷ 2 = 1 342 815 632 037 883 736 272 638 738 458 782 427 192 907 + 1;
  • 1 342 815 632 037 883 736 272 638 738 458 782 427 192 907 ÷ 2 = 671 407 816 018 941 868 136 319 369 229 391 213 596 453 + 1;
  • 671 407 816 018 941 868 136 319 369 229 391 213 596 453 ÷ 2 = 335 703 908 009 470 934 068 159 684 614 695 606 798 226 + 1;
  • 335 703 908 009 470 934 068 159 684 614 695 606 798 226 ÷ 2 = 167 851 954 004 735 467 034 079 842 307 347 803 399 113 + 0;
  • 167 851 954 004 735 467 034 079 842 307 347 803 399 113 ÷ 2 = 83 925 977 002 367 733 517 039 921 153 673 901 699 556 + 1;
  • 83 925 977 002 367 733 517 039 921 153 673 901 699 556 ÷ 2 = 41 962 988 501 183 866 758 519 960 576 836 950 849 778 + 0;
  • 41 962 988 501 183 866 758 519 960 576 836 950 849 778 ÷ 2 = 20 981 494 250 591 933 379 259 980 288 418 475 424 889 + 0;
  • 20 981 494 250 591 933 379 259 980 288 418 475 424 889 ÷ 2 = 10 490 747 125 295 966 689 629 990 144 209 237 712 444 + 1;
  • 10 490 747 125 295 966 689 629 990 144 209 237 712 444 ÷ 2 = 5 245 373 562 647 983 344 814 995 072 104 618 856 222 + 0;
  • 5 245 373 562 647 983 344 814 995 072 104 618 856 222 ÷ 2 = 2 622 686 781 323 991 672 407 497 536 052 309 428 111 + 0;
  • 2 622 686 781 323 991 672 407 497 536 052 309 428 111 ÷ 2 = 1 311 343 390 661 995 836 203 748 768 026 154 714 055 + 1;
  • 1 311 343 390 661 995 836 203 748 768 026 154 714 055 ÷ 2 = 655 671 695 330 997 918 101 874 384 013 077 357 027 + 1;
  • 655 671 695 330 997 918 101 874 384 013 077 357 027 ÷ 2 = 327 835 847 665 498 959 050 937 192 006 538 678 513 + 1;
  • 327 835 847 665 498 959 050 937 192 006 538 678 513 ÷ 2 = 163 917 923 832 749 479 525 468 596 003 269 339 256 + 1;
  • 163 917 923 832 749 479 525 468 596 003 269 339 256 ÷ 2 = 81 958 961 916 374 739 762 734 298 001 634 669 628 + 0;
  • 81 958 961 916 374 739 762 734 298 001 634 669 628 ÷ 2 = 40 979 480 958 187 369 881 367 149 000 817 334 814 + 0;
  • 40 979 480 958 187 369 881 367 149 000 817 334 814 ÷ 2 = 20 489 740 479 093 684 940 683 574 500 408 667 407 + 0;
  • 20 489 740 479 093 684 940 683 574 500 408 667 407 ÷ 2 = 10 244 870 239 546 842 470 341 787 250 204 333 703 + 1;
  • 10 244 870 239 546 842 470 341 787 250 204 333 703 ÷ 2 = 5 122 435 119 773 421 235 170 893 625 102 166 851 + 1;
  • 5 122 435 119 773 421 235 170 893 625 102 166 851 ÷ 2 = 2 561 217 559 886 710 617 585 446 812 551 083 425 + 1;
  • 2 561 217 559 886 710 617 585 446 812 551 083 425 ÷ 2 = 1 280 608 779 943 355 308 792 723 406 275 541 712 + 1;
  • 1 280 608 779 943 355 308 792 723 406 275 541 712 ÷ 2 = 640 304 389 971 677 654 396 361 703 137 770 856 + 0;
  • 640 304 389 971 677 654 396 361 703 137 770 856 ÷ 2 = 320 152 194 985 838 827 198 180 851 568 885 428 + 0;
  • 320 152 194 985 838 827 198 180 851 568 885 428 ÷ 2 = 160 076 097 492 919 413 599 090 425 784 442 714 + 0;
  • 160 076 097 492 919 413 599 090 425 784 442 714 ÷ 2 = 80 038 048 746 459 706 799 545 212 892 221 357 + 0;
  • 80 038 048 746 459 706 799 545 212 892 221 357 ÷ 2 = 40 019 024 373 229 853 399 772 606 446 110 678 + 1;
  • 40 019 024 373 229 853 399 772 606 446 110 678 ÷ 2 = 20 009 512 186 614 926 699 886 303 223 055 339 + 0;
  • 20 009 512 186 614 926 699 886 303 223 055 339 ÷ 2 = 10 004 756 093 307 463 349 943 151 611 527 669 + 1;
  • 10 004 756 093 307 463 349 943 151 611 527 669 ÷ 2 = 5 002 378 046 653 731 674 971 575 805 763 834 + 1;
  • 5 002 378 046 653 731 674 971 575 805 763 834 ÷ 2 = 2 501 189 023 326 865 837 485 787 902 881 917 + 0;
  • 2 501 189 023 326 865 837 485 787 902 881 917 ÷ 2 = 1 250 594 511 663 432 918 742 893 951 440 958 + 1;
  • 1 250 594 511 663 432 918 742 893 951 440 958 ÷ 2 = 625 297 255 831 716 459 371 446 975 720 479 + 0;
  • 625 297 255 831 716 459 371 446 975 720 479 ÷ 2 = 312 648 627 915 858 229 685 723 487 860 239 + 1;
  • 312 648 627 915 858 229 685 723 487 860 239 ÷ 2 = 156 324 313 957 929 114 842 861 743 930 119 + 1;
  • 156 324 313 957 929 114 842 861 743 930 119 ÷ 2 = 78 162 156 978 964 557 421 430 871 965 059 + 1;
  • 78 162 156 978 964 557 421 430 871 965 059 ÷ 2 = 39 081 078 489 482 278 710 715 435 982 529 + 1;
  • 39 081 078 489 482 278 710 715 435 982 529 ÷ 2 = 19 540 539 244 741 139 355 357 717 991 264 + 1;
  • 19 540 539 244 741 139 355 357 717 991 264 ÷ 2 = 9 770 269 622 370 569 677 678 858 995 632 + 0;
  • 9 770 269 622 370 569 677 678 858 995 632 ÷ 2 = 4 885 134 811 185 284 838 839 429 497 816 + 0;
  • 4 885 134 811 185 284 838 839 429 497 816 ÷ 2 = 2 442 567 405 592 642 419 419 714 748 908 + 0;
  • 2 442 567 405 592 642 419 419 714 748 908 ÷ 2 = 1 221 283 702 796 321 209 709 857 374 454 + 0;
  • 1 221 283 702 796 321 209 709 857 374 454 ÷ 2 = 610 641 851 398 160 604 854 928 687 227 + 0;
  • 610 641 851 398 160 604 854 928 687 227 ÷ 2 = 305 320 925 699 080 302 427 464 343 613 + 1;
  • 305 320 925 699 080 302 427 464 343 613 ÷ 2 = 152 660 462 849 540 151 213 732 171 806 + 1;
  • 152 660 462 849 540 151 213 732 171 806 ÷ 2 = 76 330 231 424 770 075 606 866 085 903 + 0;
  • 76 330 231 424 770 075 606 866 085 903 ÷ 2 = 38 165 115 712 385 037 803 433 042 951 + 1;
  • 38 165 115 712 385 037 803 433 042 951 ÷ 2 = 19 082 557 856 192 518 901 716 521 475 + 1;
  • 19 082 557 856 192 518 901 716 521 475 ÷ 2 = 9 541 278 928 096 259 450 858 260 737 + 1;
  • 9 541 278 928 096 259 450 858 260 737 ÷ 2 = 4 770 639 464 048 129 725 429 130 368 + 1;
  • 4 770 639 464 048 129 725 429 130 368 ÷ 2 = 2 385 319 732 024 064 862 714 565 184 + 0;
  • 2 385 319 732 024 064 862 714 565 184 ÷ 2 = 1 192 659 866 012 032 431 357 282 592 + 0;
  • 1 192 659 866 012 032 431 357 282 592 ÷ 2 = 596 329 933 006 016 215 678 641 296 + 0;
  • 596 329 933 006 016 215 678 641 296 ÷ 2 = 298 164 966 503 008 107 839 320 648 + 0;
  • 298 164 966 503 008 107 839 320 648 ÷ 2 = 149 082 483 251 504 053 919 660 324 + 0;
  • 149 082 483 251 504 053 919 660 324 ÷ 2 = 74 541 241 625 752 026 959 830 162 + 0;
  • 74 541 241 625 752 026 959 830 162 ÷ 2 = 37 270 620 812 876 013 479 915 081 + 0;
  • 37 270 620 812 876 013 479 915 081 ÷ 2 = 18 635 310 406 438 006 739 957 540 + 1;
  • 18 635 310 406 438 006 739 957 540 ÷ 2 = 9 317 655 203 219 003 369 978 770 + 0;
  • 9 317 655 203 219 003 369 978 770 ÷ 2 = 4 658 827 601 609 501 684 989 385 + 0;
  • 4 658 827 601 609 501 684 989 385 ÷ 2 = 2 329 413 800 804 750 842 494 692 + 1;
  • 2 329 413 800 804 750 842 494 692 ÷ 2 = 1 164 706 900 402 375 421 247 346 + 0;
  • 1 164 706 900 402 375 421 247 346 ÷ 2 = 582 353 450 201 187 710 623 673 + 0;
  • 582 353 450 201 187 710 623 673 ÷ 2 = 291 176 725 100 593 855 311 836 + 1;
  • 291 176 725 100 593 855 311 836 ÷ 2 = 145 588 362 550 296 927 655 918 + 0;
  • 145 588 362 550 296 927 655 918 ÷ 2 = 72 794 181 275 148 463 827 959 + 0;
  • 72 794 181 275 148 463 827 959 ÷ 2 = 36 397 090 637 574 231 913 979 + 1;
  • 36 397 090 637 574 231 913 979 ÷ 2 = 18 198 545 318 787 115 956 989 + 1;
  • 18 198 545 318 787 115 956 989 ÷ 2 = 9 099 272 659 393 557 978 494 + 1;
  • 9 099 272 659 393 557 978 494 ÷ 2 = 4 549 636 329 696 778 989 247 + 0;
  • 4 549 636 329 696 778 989 247 ÷ 2 = 2 274 818 164 848 389 494 623 + 1;
  • 2 274 818 164 848 389 494 623 ÷ 2 = 1 137 409 082 424 194 747 311 + 1;
  • 1 137 409 082 424 194 747 311 ÷ 2 = 568 704 541 212 097 373 655 + 1;
  • 568 704 541 212 097 373 655 ÷ 2 = 284 352 270 606 048 686 827 + 1;
  • 284 352 270 606 048 686 827 ÷ 2 = 142 176 135 303 024 343 413 + 1;
  • 142 176 135 303 024 343 413 ÷ 2 = 71 088 067 651 512 171 706 + 1;
  • 71 088 067 651 512 171 706 ÷ 2 = 35 544 033 825 756 085 853 + 0;
  • 35 544 033 825 756 085 853 ÷ 2 = 17 772 016 912 878 042 926 + 1;
  • 17 772 016 912 878 042 926 ÷ 2 = 8 886 008 456 439 021 463 + 0;
  • 8 886 008 456 439 021 463 ÷ 2 = 4 443 004 228 219 510 731 + 1;
  • 4 443 004 228 219 510 731 ÷ 2 = 2 221 502 114 109 755 365 + 1;
  • 2 221 502 114 109 755 365 ÷ 2 = 1 110 751 057 054 877 682 + 1;
  • 1 110 751 057 054 877 682 ÷ 2 = 555 375 528 527 438 841 + 0;
  • 555 375 528 527 438 841 ÷ 2 = 277 687 764 263 719 420 + 1;
  • 277 687 764 263 719 420 ÷ 2 = 138 843 882 131 859 710 + 0;
  • 138 843 882 131 859 710 ÷ 2 = 69 421 941 065 929 855 + 0;
  • 69 421 941 065 929 855 ÷ 2 = 34 710 970 532 964 927 + 1;
  • 34 710 970 532 964 927 ÷ 2 = 17 355 485 266 482 463 + 1;
  • 17 355 485 266 482 463 ÷ 2 = 8 677 742 633 241 231 + 1;
  • 8 677 742 633 241 231 ÷ 2 = 4 338 871 316 620 615 + 1;
  • 4 338 871 316 620 615 ÷ 2 = 2 169 435 658 310 307 + 1;
  • 2 169 435 658 310 307 ÷ 2 = 1 084 717 829 155 153 + 1;
  • 1 084 717 829 155 153 ÷ 2 = 542 358 914 577 576 + 1;
  • 542 358 914 577 576 ÷ 2 = 271 179 457 288 788 + 0;
  • 271 179 457 288 788 ÷ 2 = 135 589 728 644 394 + 0;
  • 135 589 728 644 394 ÷ 2 = 67 794 864 322 197 + 0;
  • 67 794 864 322 197 ÷ 2 = 33 897 432 161 098 + 1;
  • 33 897 432 161 098 ÷ 2 = 16 948 716 080 549 + 0;
  • 16 948 716 080 549 ÷ 2 = 8 474 358 040 274 + 1;
  • 8 474 358 040 274 ÷ 2 = 4 237 179 020 137 + 0;
  • 4 237 179 020 137 ÷ 2 = 2 118 589 510 068 + 1;
  • 2 118 589 510 068 ÷ 2 = 1 059 294 755 034 + 0;
  • 1 059 294 755 034 ÷ 2 = 529 647 377 517 + 0;
  • 529 647 377 517 ÷ 2 = 264 823 688 758 + 1;
  • 264 823 688 758 ÷ 2 = 132 411 844 379 + 0;
  • 132 411 844 379 ÷ 2 = 66 205 922 189 + 1;
  • 66 205 922 189 ÷ 2 = 33 102 961 094 + 1;
  • 33 102 961 094 ÷ 2 = 16 551 480 547 + 0;
  • 16 551 480 547 ÷ 2 = 8 275 740 273 + 1;
  • 8 275 740 273 ÷ 2 = 4 137 870 136 + 1;
  • 4 137 870 136 ÷ 2 = 2 068 935 068 + 0;
  • 2 068 935 068 ÷ 2 = 1 034 467 534 + 0;
  • 1 034 467 534 ÷ 2 = 517 233 767 + 0;
  • 517 233 767 ÷ 2 = 258 616 883 + 1;
  • 258 616 883 ÷ 2 = 129 308 441 + 1;
  • 129 308 441 ÷ 2 = 64 654 220 + 1;
  • 64 654 220 ÷ 2 = 32 327 110 + 0;
  • 32 327 110 ÷ 2 = 16 163 555 + 0;
  • 16 163 555 ÷ 2 = 8 081 777 + 1;
  • 8 081 777 ÷ 2 = 4 040 888 + 1;
  • 4 040 888 ÷ 2 = 2 020 444 + 0;
  • 2 020 444 ÷ 2 = 1 010 222 + 0;
  • 1 010 222 ÷ 2 = 505 111 + 0;
  • 505 111 ÷ 2 = 252 555 + 1;
  • 252 555 ÷ 2 = 126 277 + 1;
  • 126 277 ÷ 2 = 63 138 + 1;
  • 63 138 ÷ 2 = 31 569 + 0;
  • 31 569 ÷ 2 = 15 784 + 1;
  • 15 784 ÷ 2 = 7 892 + 0;
  • 7 892 ÷ 2 = 3 946 + 0;
  • 3 946 ÷ 2 = 1 973 + 0;
  • 1 973 ÷ 2 = 986 + 1;
  • 986 ÷ 2 = 493 + 0;
  • 493 ÷ 2 = 246 + 1;
  • 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;

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.

11 000 345 657 654 343 567 545 456 545 454 345 643 564 301 111(10) =


1 1110 1101 0100 0101 1100 0110 0111 0001 1011 0100 1010 1000 1111 1110 0101 1101 0111 1110 1110 0100 1001 0000 0001 1110 1100 0001 1111 0101 1010 0001 1110 0011 1100 1001 0111 1011 0011 0111(2)


3. Convert to the binary (base 2) the fractional part: 0.099 945 643 564 356 543 458 82.

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) 0.099 945 643 564 356 543 458 82 × 2 = 0 + 0.199 891 287 128 713 086 917 64;
  • 2) 0.199 891 287 128 713 086 917 64 × 2 = 0 + 0.399 782 574 257 426 173 835 28;
  • 3) 0.399 782 574 257 426 173 835 28 × 2 = 0 + 0.799 565 148 514 852 347 670 56;
  • 4) 0.799 565 148 514 852 347 670 56 × 2 = 1 + 0.599 130 297 029 704 695 341 12;
  • 5) 0.599 130 297 029 704 695 341 12 × 2 = 1 + 0.198 260 594 059 409 390 682 24;
  • 6) 0.198 260 594 059 409 390 682 24 × 2 = 0 + 0.396 521 188 118 818 781 364 48;
  • 7) 0.396 521 188 118 818 781 364 48 × 2 = 0 + 0.793 042 376 237 637 562 728 96;
  • 8) 0.793 042 376 237 637 562 728 96 × 2 = 1 + 0.586 084 752 475 275 125 457 92;
  • 9) 0.586 084 752 475 275 125 457 92 × 2 = 1 + 0.172 169 504 950 550 250 915 84;
  • 10) 0.172 169 504 950 550 250 915 84 × 2 = 0 + 0.344 339 009 901 100 501 831 68;
  • 11) 0.344 339 009 901 100 501 831 68 × 2 = 0 + 0.688 678 019 802 201 003 663 36;
  • 12) 0.688 678 019 802 201 003 663 36 × 2 = 1 + 0.377 356 039 604 402 007 326 72;
  • 13) 0.377 356 039 604 402 007 326 72 × 2 = 0 + 0.754 712 079 208 804 014 653 44;
  • 14) 0.754 712 079 208 804 014 653 44 × 2 = 1 + 0.509 424 158 417 608 029 306 88;
  • 15) 0.509 424 158 417 608 029 306 88 × 2 = 1 + 0.018 848 316 835 216 058 613 76;
  • 16) 0.018 848 316 835 216 058 613 76 × 2 = 0 + 0.037 696 633 670 432 117 227 52;
  • 17) 0.037 696 633 670 432 117 227 52 × 2 = 0 + 0.075 393 267 340 864 234 455 04;
  • 18) 0.075 393 267 340 864 234 455 04 × 2 = 0 + 0.150 786 534 681 728 468 910 08;
  • 19) 0.150 786 534 681 728 468 910 08 × 2 = 0 + 0.301 573 069 363 456 937 820 16;
  • 20) 0.301 573 069 363 456 937 820 16 × 2 = 0 + 0.603 146 138 726 913 875 640 32;
  • 21) 0.603 146 138 726 913 875 640 32 × 2 = 1 + 0.206 292 277 453 827 751 280 64;
  • 22) 0.206 292 277 453 827 751 280 64 × 2 = 0 + 0.412 584 554 907 655 502 561 28;
  • 23) 0.412 584 554 907 655 502 561 28 × 2 = 0 + 0.825 169 109 815 311 005 122 56;
  • 24) 0.825 169 109 815 311 005 122 56 × 2 = 1 + 0.650 338 219 630 622 010 245 12;
  • 25) 0.650 338 219 630 622 010 245 12 × 2 = 1 + 0.300 676 439 261 244 020 490 24;
  • 26) 0.300 676 439 261 244 020 490 24 × 2 = 0 + 0.601 352 878 522 488 040 980 48;
  • 27) 0.601 352 878 522 488 040 980 48 × 2 = 1 + 0.202 705 757 044 976 081 960 96;
  • 28) 0.202 705 757 044 976 081 960 96 × 2 = 0 + 0.405 411 514 089 952 163 921 92;
  • 29) 0.405 411 514 089 952 163 921 92 × 2 = 0 + 0.810 823 028 179 904 327 843 84;
  • 30) 0.810 823 028 179 904 327 843 84 × 2 = 1 + 0.621 646 056 359 808 655 687 68;
  • 31) 0.621 646 056 359 808 655 687 68 × 2 = 1 + 0.243 292 112 719 617 311 375 36;
  • 32) 0.243 292 112 719 617 311 375 36 × 2 = 0 + 0.486 584 225 439 234 622 750 72;
  • 33) 0.486 584 225 439 234 622 750 72 × 2 = 0 + 0.973 168 450 878 469 245 501 44;
  • 34) 0.973 168 450 878 469 245 501 44 × 2 = 1 + 0.946 336 901 756 938 491 002 88;
  • 35) 0.946 336 901 756 938 491 002 88 × 2 = 1 + 0.892 673 803 513 876 982 005 76;
  • 36) 0.892 673 803 513 876 982 005 76 × 2 = 1 + 0.785 347 607 027 753 964 011 52;
  • 37) 0.785 347 607 027 753 964 011 52 × 2 = 1 + 0.570 695 214 055 507 928 023 04;
  • 38) 0.570 695 214 055 507 928 023 04 × 2 = 1 + 0.141 390 428 111 015 856 046 08;
  • 39) 0.141 390 428 111 015 856 046 08 × 2 = 0 + 0.282 780 856 222 031 712 092 16;
  • 40) 0.282 780 856 222 031 712 092 16 × 2 = 0 + 0.565 561 712 444 063 424 184 32;
  • 41) 0.565 561 712 444 063 424 184 32 × 2 = 1 + 0.131 123 424 888 126 848 368 64;
  • 42) 0.131 123 424 888 126 848 368 64 × 2 = 0 + 0.262 246 849 776 253 696 737 28;
  • 43) 0.262 246 849 776 253 696 737 28 × 2 = 0 + 0.524 493 699 552 507 393 474 56;
  • 44) 0.524 493 699 552 507 393 474 56 × 2 = 1 + 0.048 987 399 105 014 786 949 12;
  • 45) 0.048 987 399 105 014 786 949 12 × 2 = 0 + 0.097 974 798 210 029 573 898 24;
  • 46) 0.097 974 798 210 029 573 898 24 × 2 = 0 + 0.195 949 596 420 059 147 796 48;
  • 47) 0.195 949 596 420 059 147 796 48 × 2 = 0 + 0.391 899 192 840 118 295 592 96;
  • 48) 0.391 899 192 840 118 295 592 96 × 2 = 0 + 0.783 798 385 680 236 591 185 92;
  • 49) 0.783 798 385 680 236 591 185 92 × 2 = 1 + 0.567 596 771 360 473 182 371 84;
  • 50) 0.567 596 771 360 473 182 371 84 × 2 = 1 + 0.135 193 542 720 946 364 743 68;
  • 51) 0.135 193 542 720 946 364 743 68 × 2 = 0 + 0.270 387 085 441 892 729 487 36;
  • 52) 0.270 387 085 441 892 729 487 36 × 2 = 0 + 0.540 774 170 883 785 458 974 72;
  • 53) 0.540 774 170 883 785 458 974 72 × 2 = 1 + 0.081 548 341 767 570 917 949 44;

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...)


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:

0.099 945 643 564 356 543 458 82(10) =


0.0001 1001 1001 0110 0000 1001 1010 0110 0111 1100 1001 0000 1100 1(2)


5. Positive number before normalization:

11 000 345 657 654 343 567 545 456 545 454 345 643 564 301 111.099 945 643 564 356 543 458 82(10) =


1 1110 1101 0100 0101 1100 0110 0111 0001 1011 0100 1010 1000 1111 1110 0101 1101 0111 1110 1110 0100 1001 0000 0001 1110 1100 0001 1111 0101 1010 0001 1110 0011 1100 1001 0111 1011 0011 0111.0001 1001 1001 0110 0000 1001 1010 0110 0111 1100 1001 0000 1100 1(2)


6. Normalize the binary representation of the number.

Shift the decimal mark 152 positions to the left so that only one non zero digit remains to the left of it:

11 000 345 657 654 343 567 545 456 545 454 345 643 564 301 111.099 945 643 564 356 543 458 82(10) =


1 1110 1101 0100 0101 1100 0110 0111 0001 1011 0100 1010 1000 1111 1110 0101 1101 0111 1110 1110 0100 1001 0000 0001 1110 1100 0001 1111 0101 1010 0001 1110 0011 1100 1001 0111 1011 0011 0111.0001 1001 1001 0110 0000 1001 1010 0110 0111 1100 1001 0000 1100 1(2) =


1 1110 1101 0100 0101 1100 0110 0111 0001 1011 0100 1010 1000 1111 1110 0101 1101 0111 1110 1110 0100 1001 0000 0001 1110 1100 0001 1111 0101 1010 0001 1110 0011 1100 1001 0111 1011 0011 0111.0001 1001 1001 0110 0000 1001 1010 0110 0111 1100 1001 0000 1100 1(2) × 20 =


1.1110 1101 0100 0101 1100 0110 0111 0001 1011 0100 1010 1000 1111 1110 0101 1101 0111 1110 1110 0100 1001 0000 0001 1110 1100 0001 1111 0101 1010 0001 1110 0011 1100 1001 0111 1011 0011 0111 0001 1001 1001 0110 0000 1001 1010 0110 0111 1100 1001 0000 1100 1(2) × 2152


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): 152


Mantissa (not normalized):
1.1110 1101 0100 0101 1100 0110 0111 0001 1011 0100 1010 1000 1111 1110 0101 1101 0111 1110 1110 0100 1001 0000 0001 1110 1100 0001 1111 0101 1010 0001 1110 0011 1100 1001 0111 1011 0011 0111 0001 1001 1001 0110 0000 1001 1010 0110 0111 1100 1001 0000 1100 1


8. Adjust the exponent.

Use the 11 bit excess/bias notation:

Exponent (adjusted) =


Exponent (unadjusted) + 2(11-1) - 1 =


152 + 2(11-1) - 1 =


(152 + 1 023)(10) =


1 175(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;
  • 1 175 ÷ 2 = 587 + 1;
  • 587 ÷ 2 = 293 + 1;
  • 293 ÷ 2 = 146 + 1;
  • 146 ÷ 2 = 73 + 0;
  • 73 ÷ 2 = 36 + 1;
  • 36 ÷ 2 = 18 + 0;
  • 18 ÷ 2 = 9 + 0;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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) =


1175(10) =


100 1001 0111(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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).

Mantissa (normalized) =


1. 1110 1101 0100 0101 1100 0110 0111 0001 1011 0100 1010 1000 1111 1 1100 1011 1010 1111 1101 1100 1001 0010 0000 0011 1101 1000 0011 1110 1011 0100 0011 1100 0111 1001 0010 1111 0110 0110 1110 0011 0011 0010 1100 0001 0011 0100 1100 1111 1001 0010 0001 1001 =


1110 1101 0100 0101 1100 0110 0111 0001 1011 0100 1010 1000 1111


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) =
100 1001 0111


Mantissa (52 bits) =
1110 1101 0100 0101 1100 0110 0111 0001 1011 0100 1010 1000 1111


Number 11 000 345 657 654 343 567 545 456 545 454 345 643 564 301 111.099 945 643 564 356 543 458 82 converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point:
0 - 100 1001 0111 - 1110 1101 0100 0101 1100 0110 0111 0001 1011 0100 1010 1000 1111

(64 bits IEEE 754)
  • Sign (1 bit):

    • 0

      63
  • Exponent (11 bits):

    • 1

      62
    • 0

      61
    • 0

      60
    • 1

      59
    • 0

      58
    • 0

      57
    • 1

      56
    • 0

      55
    • 1

      54
    • 1

      53
    • 1

      52
  • Mantissa (52 bits):

    • 1

      51
    • 1

      50
    • 1

      49
    • 0

      48
    • 1

      47
    • 1

      46
    • 0

      45
    • 1

      44
    • 0

      43
    • 1

      42
    • 0

      41
    • 0

      40
    • 0

      39
    • 1

      38
    • 0

      37
    • 1

      36
    • 1

      35
    • 1

      34
    • 0

      33
    • 0

      32
    • 0

      31
    • 1

      30
    • 1

      29
    • 0

      28
    • 0

      27
    • 1

      26
    • 1

      25
    • 1

      24
    • 0

      23
    • 0

      22
    • 0

      21
    • 1

      20
    • 1

      19
    • 0

      18
    • 1

      17
    • 1

      16
    • 0

      15
    • 1

      14
    • 0

      13
    • 0

      12
    • 1

      11
    • 0

      10
    • 1

      9
    • 0

      8
    • 1

      7
    • 0

      6
    • 0

      5
    • 0

      4
    • 1

      3
    • 1

      2
    • 1

      1
    • 1

      0

More operations of this kind:

11 000 345 657 654 343 567 545 456 545 454 345 643 564 301 111.099 945 643 564 356 543 458 81 = ? ... 11 000 345 657 654 343 567 545 456 545 454 345 643 564 301 111.099 945 643 564 356 543 458 83 = ?


Convert to 64 bit double precision IEEE 754 binary floating point standard

A number in 64 bit double precision IEEE 754 binary floating point standard representation requires three building elements: sign (it takes one bit and it's either 0 for positive or 1 for negative numbers), exponent (11 bits), mantissa (52 bits)

Latest decimal numbers converted from base ten to 64 bit double precision IEEE 754 floating point binary standard representation

11 000 345 657 654 343 567 545 456 545 454 345 643 564 301 111.099 945 643 564 356 543 458 82 to 64 bit double precision IEEE 754 binary floating point = ? Mar 03 03:01 UTC (GMT)
470.982 46 to 64 bit double precision IEEE 754 binary floating point = ? Mar 03 03:00 UTC (GMT)
1 010 094 to 64 bit double precision IEEE 754 binary floating point = ? Mar 03 03:00 UTC (GMT)
17.802 4 to 64 bit double precision IEEE 754 binary floating point = ? Mar 03 03:00 UTC (GMT)
124.48 to 64 bit double precision IEEE 754 binary floating point = ? Mar 03 03:00 UTC (GMT)
110 to 64 bit double precision IEEE 754 binary floating point = ? Mar 03 03:00 UTC (GMT)
1 550.453 5 to 64 bit double precision IEEE 754 binary floating point = ? Mar 03 03:00 UTC (GMT)
228.13 to 64 bit double precision IEEE 754 binary floating point = ? Mar 03 03:00 UTC (GMT)
10.666 4 to 64 bit double precision IEEE 754 binary floating point = ? Mar 03 02:59 UTC (GMT)
0.000 000 347 222 222 224 to 64 bit double precision IEEE 754 binary floating point = ? Mar 03 02:59 UTC (GMT)
1.234 567 87 to 64 bit double precision IEEE 754 binary floating point = ? Mar 03 02:59 UTC (GMT)
329.36 to 64 bit double precision IEEE 754 binary floating point = ? Mar 03 02:59 UTC (GMT)
0.066 4 to 64 bit double precision IEEE 754 binary floating point = ? Mar 03 02:59 UTC (GMT)
All base ten decimal numbers converted to 64 bit double precision IEEE 754 binary floating point

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