64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: 199 999 999 999 999 999 999 999 999 999 999 999 999 999 999 987 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number 199 999 999 999 999 999 999 999 999 999 999 999 999 999 999 987(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. 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;
  • 199 999 999 999 999 999 999 999 999 999 999 999 999 999 999 987 ÷ 2 = 99 999 999 999 999 999 999 999 999 999 999 999 999 999 999 993 + 1;
  • 99 999 999 999 999 999 999 999 999 999 999 999 999 999 999 993 ÷ 2 = 49 999 999 999 999 999 999 999 999 999 999 999 999 999 999 996 + 1;
  • 49 999 999 999 999 999 999 999 999 999 999 999 999 999 999 996 ÷ 2 = 24 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 + 0;
  • 24 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 ÷ 2 = 12 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 + 0;
  • 12 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 6 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 6 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 3 124 999 999 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 3 124 999 999 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 1 562 499 999 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 1 562 499 999 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 781 249 999 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 781 249 999 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 390 624 999 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 390 624 999 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 195 312 499 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 195 312 499 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 97 656 249 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 97 656 249 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 48 828 124 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 48 828 124 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 24 414 062 499 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 24 414 062 499 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 12 207 031 249 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 12 207 031 249 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 6 103 515 624 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 6 103 515 624 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 3 051 757 812 499 999 999 999 999 999 999 999 999 999 999 + 1;
  • 3 051 757 812 499 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 1 525 878 906 249 999 999 999 999 999 999 999 999 999 999 + 1;
  • 1 525 878 906 249 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 762 939 453 124 999 999 999 999 999 999 999 999 999 999 + 1;
  • 762 939 453 124 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 381 469 726 562 499 999 999 999 999 999 999 999 999 999 + 1;
  • 381 469 726 562 499 999 999 999 999 999 999 999 999 999 ÷ 2 = 190 734 863 281 249 999 999 999 999 999 999 999 999 999 + 1;
  • 190 734 863 281 249 999 999 999 999 999 999 999 999 999 ÷ 2 = 95 367 431 640 624 999 999 999 999 999 999 999 999 999 + 1;
  • 95 367 431 640 624 999 999 999 999 999 999 999 999 999 ÷ 2 = 47 683 715 820 312 499 999 999 999 999 999 999 999 999 + 1;
  • 47 683 715 820 312 499 999 999 999 999 999 999 999 999 ÷ 2 = 23 841 857 910 156 249 999 999 999 999 999 999 999 999 + 1;
  • 23 841 857 910 156 249 999 999 999 999 999 999 999 999 ÷ 2 = 11 920 928 955 078 124 999 999 999 999 999 999 999 999 + 1;
  • 11 920 928 955 078 124 999 999 999 999 999 999 999 999 ÷ 2 = 5 960 464 477 539 062 499 999 999 999 999 999 999 999 + 1;
  • 5 960 464 477 539 062 499 999 999 999 999 999 999 999 ÷ 2 = 2 980 232 238 769 531 249 999 999 999 999 999 999 999 + 1;
  • 2 980 232 238 769 531 249 999 999 999 999 999 999 999 ÷ 2 = 1 490 116 119 384 765 624 999 999 999 999 999 999 999 + 1;
  • 1 490 116 119 384 765 624 999 999 999 999 999 999 999 ÷ 2 = 745 058 059 692 382 812 499 999 999 999 999 999 999 + 1;
  • 745 058 059 692 382 812 499 999 999 999 999 999 999 ÷ 2 = 372 529 029 846 191 406 249 999 999 999 999 999 999 + 1;
  • 372 529 029 846 191 406 249 999 999 999 999 999 999 ÷ 2 = 186 264 514 923 095 703 124 999 999 999 999 999 999 + 1;
  • 186 264 514 923 095 703 124 999 999 999 999 999 999 ÷ 2 = 93 132 257 461 547 851 562 499 999 999 999 999 999 + 1;
  • 93 132 257 461 547 851 562 499 999 999 999 999 999 ÷ 2 = 46 566 128 730 773 925 781 249 999 999 999 999 999 + 1;
  • 46 566 128 730 773 925 781 249 999 999 999 999 999 ÷ 2 = 23 283 064 365 386 962 890 624 999 999 999 999 999 + 1;
  • 23 283 064 365 386 962 890 624 999 999 999 999 999 ÷ 2 = 11 641 532 182 693 481 445 312 499 999 999 999 999 + 1;
  • 11 641 532 182 693 481 445 312 499 999 999 999 999 ÷ 2 = 5 820 766 091 346 740 722 656 249 999 999 999 999 + 1;
  • 5 820 766 091 346 740 722 656 249 999 999 999 999 ÷ 2 = 2 910 383 045 673 370 361 328 124 999 999 999 999 + 1;
  • 2 910 383 045 673 370 361 328 124 999 999 999 999 ÷ 2 = 1 455 191 522 836 685 180 664 062 499 999 999 999 + 1;
  • 1 455 191 522 836 685 180 664 062 499 999 999 999 ÷ 2 = 727 595 761 418 342 590 332 031 249 999 999 999 + 1;
  • 727 595 761 418 342 590 332 031 249 999 999 999 ÷ 2 = 363 797 880 709 171 295 166 015 624 999 999 999 + 1;
  • 363 797 880 709 171 295 166 015 624 999 999 999 ÷ 2 = 181 898 940 354 585 647 583 007 812 499 999 999 + 1;
  • 181 898 940 354 585 647 583 007 812 499 999 999 ÷ 2 = 90 949 470 177 292 823 791 503 906 249 999 999 + 1;
  • 90 949 470 177 292 823 791 503 906 249 999 999 ÷ 2 = 45 474 735 088 646 411 895 751 953 124 999 999 + 1;
  • 45 474 735 088 646 411 895 751 953 124 999 999 ÷ 2 = 22 737 367 544 323 205 947 875 976 562 499 999 + 1;
  • 22 737 367 544 323 205 947 875 976 562 499 999 ÷ 2 = 11 368 683 772 161 602 973 937 988 281 249 999 + 1;
  • 11 368 683 772 161 602 973 937 988 281 249 999 ÷ 2 = 5 684 341 886 080 801 486 968 994 140 624 999 + 1;
  • 5 684 341 886 080 801 486 968 994 140 624 999 ÷ 2 = 2 842 170 943 040 400 743 484 497 070 312 499 + 1;
  • 2 842 170 943 040 400 743 484 497 070 312 499 ÷ 2 = 1 421 085 471 520 200 371 742 248 535 156 249 + 1;
  • 1 421 085 471 520 200 371 742 248 535 156 249 ÷ 2 = 710 542 735 760 100 185 871 124 267 578 124 + 1;
  • 710 542 735 760 100 185 871 124 267 578 124 ÷ 2 = 355 271 367 880 050 092 935 562 133 789 062 + 0;
  • 355 271 367 880 050 092 935 562 133 789 062 ÷ 2 = 177 635 683 940 025 046 467 781 066 894 531 + 0;
  • 177 635 683 940 025 046 467 781 066 894 531 ÷ 2 = 88 817 841 970 012 523 233 890 533 447 265 + 1;
  • 88 817 841 970 012 523 233 890 533 447 265 ÷ 2 = 44 408 920 985 006 261 616 945 266 723 632 + 1;
  • 44 408 920 985 006 261 616 945 266 723 632 ÷ 2 = 22 204 460 492 503 130 808 472 633 361 816 + 0;
  • 22 204 460 492 503 130 808 472 633 361 816 ÷ 2 = 11 102 230 246 251 565 404 236 316 680 908 + 0;
  • 11 102 230 246 251 565 404 236 316 680 908 ÷ 2 = 5 551 115 123 125 782 702 118 158 340 454 + 0;
  • 5 551 115 123 125 782 702 118 158 340 454 ÷ 2 = 2 775 557 561 562 891 351 059 079 170 227 + 0;
  • 2 775 557 561 562 891 351 059 079 170 227 ÷ 2 = 1 387 778 780 781 445 675 529 539 585 113 + 1;
  • 1 387 778 780 781 445 675 529 539 585 113 ÷ 2 = 693 889 390 390 722 837 764 769 792 556 + 1;
  • 693 889 390 390 722 837 764 769 792 556 ÷ 2 = 346 944 695 195 361 418 882 384 896 278 + 0;
  • 346 944 695 195 361 418 882 384 896 278 ÷ 2 = 173 472 347 597 680 709 441 192 448 139 + 0;
  • 173 472 347 597 680 709 441 192 448 139 ÷ 2 = 86 736 173 798 840 354 720 596 224 069 + 1;
  • 86 736 173 798 840 354 720 596 224 069 ÷ 2 = 43 368 086 899 420 177 360 298 112 034 + 1;
  • 43 368 086 899 420 177 360 298 112 034 ÷ 2 = 21 684 043 449 710 088 680 149 056 017 + 0;
  • 21 684 043 449 710 088 680 149 056 017 ÷ 2 = 10 842 021 724 855 044 340 074 528 008 + 1;
  • 10 842 021 724 855 044 340 074 528 008 ÷ 2 = 5 421 010 862 427 522 170 037 264 004 + 0;
  • 5 421 010 862 427 522 170 037 264 004 ÷ 2 = 2 710 505 431 213 761 085 018 632 002 + 0;
  • 2 710 505 431 213 761 085 018 632 002 ÷ 2 = 1 355 252 715 606 880 542 509 316 001 + 0;
  • 1 355 252 715 606 880 542 509 316 001 ÷ 2 = 677 626 357 803 440 271 254 658 000 + 1;
  • 677 626 357 803 440 271 254 658 000 ÷ 2 = 338 813 178 901 720 135 627 329 000 + 0;
  • 338 813 178 901 720 135 627 329 000 ÷ 2 = 169 406 589 450 860 067 813 664 500 + 0;
  • 169 406 589 450 860 067 813 664 500 ÷ 2 = 84 703 294 725 430 033 906 832 250 + 0;
  • 84 703 294 725 430 033 906 832 250 ÷ 2 = 42 351 647 362 715 016 953 416 125 + 0;
  • 42 351 647 362 715 016 953 416 125 ÷ 2 = 21 175 823 681 357 508 476 708 062 + 1;
  • 21 175 823 681 357 508 476 708 062 ÷ 2 = 10 587 911 840 678 754 238 354 031 + 0;
  • 10 587 911 840 678 754 238 354 031 ÷ 2 = 5 293 955 920 339 377 119 177 015 + 1;
  • 5 293 955 920 339 377 119 177 015 ÷ 2 = 2 646 977 960 169 688 559 588 507 + 1;
  • 2 646 977 960 169 688 559 588 507 ÷ 2 = 1 323 488 980 084 844 279 794 253 + 1;
  • 1 323 488 980 084 844 279 794 253 ÷ 2 = 661 744 490 042 422 139 897 126 + 1;
  • 661 744 490 042 422 139 897 126 ÷ 2 = 330 872 245 021 211 069 948 563 + 0;
  • 330 872 245 021 211 069 948 563 ÷ 2 = 165 436 122 510 605 534 974 281 + 1;
  • 165 436 122 510 605 534 974 281 ÷ 2 = 82 718 061 255 302 767 487 140 + 1;
  • 82 718 061 255 302 767 487 140 ÷ 2 = 41 359 030 627 651 383 743 570 + 0;
  • 41 359 030 627 651 383 743 570 ÷ 2 = 20 679 515 313 825 691 871 785 + 0;
  • 20 679 515 313 825 691 871 785 ÷ 2 = 10 339 757 656 912 845 935 892 + 1;
  • 10 339 757 656 912 845 935 892 ÷ 2 = 5 169 878 828 456 422 967 946 + 0;
  • 5 169 878 828 456 422 967 946 ÷ 2 = 2 584 939 414 228 211 483 973 + 0;
  • 2 584 939 414 228 211 483 973 ÷ 2 = 1 292 469 707 114 105 741 986 + 1;
  • 1 292 469 707 114 105 741 986 ÷ 2 = 646 234 853 557 052 870 993 + 0;
  • 646 234 853 557 052 870 993 ÷ 2 = 323 117 426 778 526 435 496 + 1;
  • 323 117 426 778 526 435 496 ÷ 2 = 161 558 713 389 263 217 748 + 0;
  • 161 558 713 389 263 217 748 ÷ 2 = 80 779 356 694 631 608 874 + 0;
  • 80 779 356 694 631 608 874 ÷ 2 = 40 389 678 347 315 804 437 + 0;
  • 40 389 678 347 315 804 437 ÷ 2 = 20 194 839 173 657 902 218 + 1;
  • 20 194 839 173 657 902 218 ÷ 2 = 10 097 419 586 828 951 109 + 0;
  • 10 097 419 586 828 951 109 ÷ 2 = 5 048 709 793 414 475 554 + 1;
  • 5 048 709 793 414 475 554 ÷ 2 = 2 524 354 896 707 237 777 + 0;
  • 2 524 354 896 707 237 777 ÷ 2 = 1 262 177 448 353 618 888 + 1;
  • 1 262 177 448 353 618 888 ÷ 2 = 631 088 724 176 809 444 + 0;
  • 631 088 724 176 809 444 ÷ 2 = 315 544 362 088 404 722 + 0;
  • 315 544 362 088 404 722 ÷ 2 = 157 772 181 044 202 361 + 0;
  • 157 772 181 044 202 361 ÷ 2 = 78 886 090 522 101 180 + 1;
  • 78 886 090 522 101 180 ÷ 2 = 39 443 045 261 050 590 + 0;
  • 39 443 045 261 050 590 ÷ 2 = 19 721 522 630 525 295 + 0;
  • 19 721 522 630 525 295 ÷ 2 = 9 860 761 315 262 647 + 1;
  • 9 860 761 315 262 647 ÷ 2 = 4 930 380 657 631 323 + 1;
  • 4 930 380 657 631 323 ÷ 2 = 2 465 190 328 815 661 + 1;
  • 2 465 190 328 815 661 ÷ 2 = 1 232 595 164 407 830 + 1;
  • 1 232 595 164 407 830 ÷ 2 = 616 297 582 203 915 + 0;
  • 616 297 582 203 915 ÷ 2 = 308 148 791 101 957 + 1;
  • 308 148 791 101 957 ÷ 2 = 154 074 395 550 978 + 1;
  • 154 074 395 550 978 ÷ 2 = 77 037 197 775 489 + 0;
  • 77 037 197 775 489 ÷ 2 = 38 518 598 887 744 + 1;
  • 38 518 598 887 744 ÷ 2 = 19 259 299 443 872 + 0;
  • 19 259 299 443 872 ÷ 2 = 9 629 649 721 936 + 0;
  • 9 629 649 721 936 ÷ 2 = 4 814 824 860 968 + 0;
  • 4 814 824 860 968 ÷ 2 = 2 407 412 430 484 + 0;
  • 2 407 412 430 484 ÷ 2 = 1 203 706 215 242 + 0;
  • 1 203 706 215 242 ÷ 2 = 601 853 107 621 + 0;
  • 601 853 107 621 ÷ 2 = 300 926 553 810 + 1;
  • 300 926 553 810 ÷ 2 = 150 463 276 905 + 0;
  • 150 463 276 905 ÷ 2 = 75 231 638 452 + 1;
  • 75 231 638 452 ÷ 2 = 37 615 819 226 + 0;
  • 37 615 819 226 ÷ 2 = 18 807 909 613 + 0;
  • 18 807 909 613 ÷ 2 = 9 403 954 806 + 1;
  • 9 403 954 806 ÷ 2 = 4 701 977 403 + 0;
  • 4 701 977 403 ÷ 2 = 2 350 988 701 + 1;
  • 2 350 988 701 ÷ 2 = 1 175 494 350 + 1;
  • 1 175 494 350 ÷ 2 = 587 747 175 + 0;
  • 587 747 175 ÷ 2 = 293 873 587 + 1;
  • 293 873 587 ÷ 2 = 146 936 793 + 1;
  • 146 936 793 ÷ 2 = 73 468 396 + 1;
  • 73 468 396 ÷ 2 = 36 734 198 + 0;
  • 36 734 198 ÷ 2 = 18 367 099 + 0;
  • 18 367 099 ÷ 2 = 9 183 549 + 1;
  • 9 183 549 ÷ 2 = 4 591 774 + 1;
  • 4 591 774 ÷ 2 = 2 295 887 + 0;
  • 2 295 887 ÷ 2 = 1 147 943 + 1;
  • 1 147 943 ÷ 2 = 573 971 + 1;
  • 573 971 ÷ 2 = 286 985 + 1;
  • 286 985 ÷ 2 = 143 492 + 1;
  • 143 492 ÷ 2 = 71 746 + 0;
  • 71 746 ÷ 2 = 35 873 + 0;
  • 35 873 ÷ 2 = 17 936 + 1;
  • 17 936 ÷ 2 = 8 968 + 0;
  • 8 968 ÷ 2 = 4 484 + 0;
  • 4 484 ÷ 2 = 2 242 + 0;
  • 2 242 ÷ 2 = 1 121 + 0;
  • 1 121 ÷ 2 = 560 + 1;
  • 560 ÷ 2 = 280 + 0;
  • 280 ÷ 2 = 140 + 0;
  • 140 ÷ 2 = 70 + 0;
  • 70 ÷ 2 = 35 + 0;
  • 35 ÷ 2 = 17 + 1;
  • 17 ÷ 2 = 8 + 1;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the positive number.

Take all the remainders starting from the bottom of the list constructed above.


199 999 999 999 999 999 999 999 999 999 999 999 999 999 999 987(10) =


10 0011 0000 1000 0100 1111 0110 0111 0110 1001 0100 0000 1011 0111 1001 0001 0101 0001 0100 1001 1011 1101 0000 1000 1011 0011 0000 1100 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0011(2)


3. Normalize the binary representation of the number.

Shift the decimal mark 157 positions to the left, so that only one non zero digit remains to the left of it:


199 999 999 999 999 999 999 999 999 999 999 999 999 999 999 987(10) =


10 0011 0000 1000 0100 1111 0110 0111 0110 1001 0100 0000 1011 0111 1001 0001 0101 0001 0100 1001 1011 1101 0000 1000 1011 0011 0000 1100 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0011(2) =


10 0011 0000 1000 0100 1111 0110 0111 0110 1001 0100 0000 1011 0111 1001 0001 0101 0001 0100 1001 1011 1101 0000 1000 1011 0011 0000 1100 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0011(2) × 20 =


1.0001 1000 0100 0010 0111 1011 0011 1011 0100 1010 0000 0101 1011 1100 1000 1010 1000 1010 0100 1101 1110 1000 0100 0101 1001 1000 0110 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1001 1(2) × 2157


4. 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): 157


Mantissa (not normalized):
1.0001 1000 0100 0010 0111 1011 0011 1011 0100 1010 0000 0101 1011 1100 1000 1010 1000 1010 0100 1101 1110 1000 0100 0101 1001 1000 0110 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1001 1


5. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


157 + 2(11-1) - 1 =


(157 + 1 023)(10) =


1 180(10)


6. 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;
  • 1 180 ÷ 2 = 590 + 0;
  • 590 ÷ 2 = 295 + 0;
  • 295 ÷ 2 = 147 + 1;
  • 147 ÷ 2 = 73 + 1;
  • 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;

7. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =


1180(10) =


100 1001 1100(2)


8. 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. 0001 1000 0100 0010 0111 1011 0011 1011 0100 1010 0000 0101 1011 1 1001 0001 0101 0001 0100 1001 1011 1101 0000 1000 1011 0011 0000 1100 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0011 =


0001 1000 0100 0010 0111 1011 0011 1011 0100 1010 0000 0101 1011


9. 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 1100


Mantissa (52 bits) =
0001 1000 0100 0010 0111 1011 0011 1011 0100 1010 0000 0101 1011


The base ten decimal number 199 999 999 999 999 999 999 999 999 999 999 999 999 999 999 987 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 100 1001 1100 - 0001 1000 0100 0010 0111 1011 0011 1011 0100 1010 0000 0101 1011

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

Number 737 195 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard May 21 13:39 UTC (GMT)
Number 8 352 356 132 311 793 590 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard May 21 13:39 UTC (GMT)
Number 5 052 023 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard May 21 13:39 UTC (GMT)
Number -78.125 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard May 21 13:39 UTC (GMT)
Number 1 721 212 122 115 241 421 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard May 21 13:39 UTC (GMT)
Number -0.154 999 999 999 94 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard May 21 13:39 UTC (GMT)
Number 5 052 023 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard May 21 13:39 UTC (GMT)
Number -78.125 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard May 21 13:38 UTC (GMT)
Number 1 721 212 122 115 241 421 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard May 21 13:38 UTC (GMT)
Number 129 995 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard May 21 13:38 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