0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 351 4 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 351 4(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
0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 351 4(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: 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 351 4.

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) 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 351 4 × 2 = 0 + 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 702 8;
  • 2) 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 702 8 × 2 = 0 + 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 405 6;
  • 3) 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 405 6 × 2 = 0 + 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 811 2;
  • 4) 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 811 2 × 2 = 0 + 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 622 4;
  • 5) 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 622 4 × 2 = 0 + 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 244 8;
  • 6) 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 244 8 × 2 = 0 + 0.000 000 000 000 000 790 124 025 252 133 724 078 653 462 489 6;
  • 7) 0.000 000 000 000 000 790 124 025 252 133 724 078 653 462 489 6 × 2 = 0 + 0.000 000 000 000 001 580 248 050 504 267 448 157 306 924 979 2;
  • 8) 0.000 000 000 000 001 580 248 050 504 267 448 157 306 924 979 2 × 2 = 0 + 0.000 000 000 000 003 160 496 101 008 534 896 314 613 849 958 4;
  • 9) 0.000 000 000 000 003 160 496 101 008 534 896 314 613 849 958 4 × 2 = 0 + 0.000 000 000 000 006 320 992 202 017 069 792 629 227 699 916 8;
  • 10) 0.000 000 000 000 006 320 992 202 017 069 792 629 227 699 916 8 × 2 = 0 + 0.000 000 000 000 012 641 984 404 034 139 585 258 455 399 833 6;
  • 11) 0.000 000 000 000 012 641 984 404 034 139 585 258 455 399 833 6 × 2 = 0 + 0.000 000 000 000 025 283 968 808 068 279 170 516 910 799 667 2;
  • 12) 0.000 000 000 000 025 283 968 808 068 279 170 516 910 799 667 2 × 2 = 0 + 0.000 000 000 000 050 567 937 616 136 558 341 033 821 599 334 4;
  • 13) 0.000 000 000 000 050 567 937 616 136 558 341 033 821 599 334 4 × 2 = 0 + 0.000 000 000 000 101 135 875 232 273 116 682 067 643 198 668 8;
  • 14) 0.000 000 000 000 101 135 875 232 273 116 682 067 643 198 668 8 × 2 = 0 + 0.000 000 000 000 202 271 750 464 546 233 364 135 286 397 337 6;
  • 15) 0.000 000 000 000 202 271 750 464 546 233 364 135 286 397 337 6 × 2 = 0 + 0.000 000 000 000 404 543 500 929 092 466 728 270 572 794 675 2;
  • 16) 0.000 000 000 000 404 543 500 929 092 466 728 270 572 794 675 2 × 2 = 0 + 0.000 000 000 000 809 087 001 858 184 933 456 541 145 589 350 4;
  • 17) 0.000 000 000 000 809 087 001 858 184 933 456 541 145 589 350 4 × 2 = 0 + 0.000 000 000 001 618 174 003 716 369 866 913 082 291 178 700 8;
  • 18) 0.000 000 000 001 618 174 003 716 369 866 913 082 291 178 700 8 × 2 = 0 + 0.000 000 000 003 236 348 007 432 739 733 826 164 582 357 401 6;
  • 19) 0.000 000 000 003 236 348 007 432 739 733 826 164 582 357 401 6 × 2 = 0 + 0.000 000 000 006 472 696 014 865 479 467 652 329 164 714 803 2;
  • 20) 0.000 000 000 006 472 696 014 865 479 467 652 329 164 714 803 2 × 2 = 0 + 0.000 000 000 012 945 392 029 730 958 935 304 658 329 429 606 4;
  • 21) 0.000 000 000 012 945 392 029 730 958 935 304 658 329 429 606 4 × 2 = 0 + 0.000 000 000 025 890 784 059 461 917 870 609 316 658 859 212 8;
  • 22) 0.000 000 000 025 890 784 059 461 917 870 609 316 658 859 212 8 × 2 = 0 + 0.000 000 000 051 781 568 118 923 835 741 218 633 317 718 425 6;
  • 23) 0.000 000 000 051 781 568 118 923 835 741 218 633 317 718 425 6 × 2 = 0 + 0.000 000 000 103 563 136 237 847 671 482 437 266 635 436 851 2;
  • 24) 0.000 000 000 103 563 136 237 847 671 482 437 266 635 436 851 2 × 2 = 0 + 0.000 000 000 207 126 272 475 695 342 964 874 533 270 873 702 4;
  • 25) 0.000 000 000 207 126 272 475 695 342 964 874 533 270 873 702 4 × 2 = 0 + 0.000 000 000 414 252 544 951 390 685 929 749 066 541 747 404 8;
  • 26) 0.000 000 000 414 252 544 951 390 685 929 749 066 541 747 404 8 × 2 = 0 + 0.000 000 000 828 505 089 902 781 371 859 498 133 083 494 809 6;
  • 27) 0.000 000 000 828 505 089 902 781 371 859 498 133 083 494 809 6 × 2 = 0 + 0.000 000 001 657 010 179 805 562 743 718 996 266 166 989 619 2;
  • 28) 0.000 000 001 657 010 179 805 562 743 718 996 266 166 989 619 2 × 2 = 0 + 0.000 000 003 314 020 359 611 125 487 437 992 532 333 979 238 4;
  • 29) 0.000 000 003 314 020 359 611 125 487 437 992 532 333 979 238 4 × 2 = 0 + 0.000 000 006 628 040 719 222 250 974 875 985 064 667 958 476 8;
  • 30) 0.000 000 006 628 040 719 222 250 974 875 985 064 667 958 476 8 × 2 = 0 + 0.000 000 013 256 081 438 444 501 949 751 970 129 335 916 953 6;
  • 31) 0.000 000 013 256 081 438 444 501 949 751 970 129 335 916 953 6 × 2 = 0 + 0.000 000 026 512 162 876 889 003 899 503 940 258 671 833 907 2;
  • 32) 0.000 000 026 512 162 876 889 003 899 503 940 258 671 833 907 2 × 2 = 0 + 0.000 000 053 024 325 753 778 007 799 007 880 517 343 667 814 4;
  • 33) 0.000 000 053 024 325 753 778 007 799 007 880 517 343 667 814 4 × 2 = 0 + 0.000 000 106 048 651 507 556 015 598 015 761 034 687 335 628 8;
  • 34) 0.000 000 106 048 651 507 556 015 598 015 761 034 687 335 628 8 × 2 = 0 + 0.000 000 212 097 303 015 112 031 196 031 522 069 374 671 257 6;
  • 35) 0.000 000 212 097 303 015 112 031 196 031 522 069 374 671 257 6 × 2 = 0 + 0.000 000 424 194 606 030 224 062 392 063 044 138 749 342 515 2;
  • 36) 0.000 000 424 194 606 030 224 062 392 063 044 138 749 342 515 2 × 2 = 0 + 0.000 000 848 389 212 060 448 124 784 126 088 277 498 685 030 4;
  • 37) 0.000 000 848 389 212 060 448 124 784 126 088 277 498 685 030 4 × 2 = 0 + 0.000 001 696 778 424 120 896 249 568 252 176 554 997 370 060 8;
  • 38) 0.000 001 696 778 424 120 896 249 568 252 176 554 997 370 060 8 × 2 = 0 + 0.000 003 393 556 848 241 792 499 136 504 353 109 994 740 121 6;
  • 39) 0.000 003 393 556 848 241 792 499 136 504 353 109 994 740 121 6 × 2 = 0 + 0.000 006 787 113 696 483 584 998 273 008 706 219 989 480 243 2;
  • 40) 0.000 006 787 113 696 483 584 998 273 008 706 219 989 480 243 2 × 2 = 0 + 0.000 013 574 227 392 967 169 996 546 017 412 439 978 960 486 4;
  • 41) 0.000 013 574 227 392 967 169 996 546 017 412 439 978 960 486 4 × 2 = 0 + 0.000 027 148 454 785 934 339 993 092 034 824 879 957 920 972 8;
  • 42) 0.000 027 148 454 785 934 339 993 092 034 824 879 957 920 972 8 × 2 = 0 + 0.000 054 296 909 571 868 679 986 184 069 649 759 915 841 945 6;
  • 43) 0.000 054 296 909 571 868 679 986 184 069 649 759 915 841 945 6 × 2 = 0 + 0.000 108 593 819 143 737 359 972 368 139 299 519 831 683 891 2;
  • 44) 0.000 108 593 819 143 737 359 972 368 139 299 519 831 683 891 2 × 2 = 0 + 0.000 217 187 638 287 474 719 944 736 278 599 039 663 367 782 4;
  • 45) 0.000 217 187 638 287 474 719 944 736 278 599 039 663 367 782 4 × 2 = 0 + 0.000 434 375 276 574 949 439 889 472 557 198 079 326 735 564 8;
  • 46) 0.000 434 375 276 574 949 439 889 472 557 198 079 326 735 564 8 × 2 = 0 + 0.000 868 750 553 149 898 879 778 945 114 396 158 653 471 129 6;
  • 47) 0.000 868 750 553 149 898 879 778 945 114 396 158 653 471 129 6 × 2 = 0 + 0.001 737 501 106 299 797 759 557 890 228 792 317 306 942 259 2;
  • 48) 0.001 737 501 106 299 797 759 557 890 228 792 317 306 942 259 2 × 2 = 0 + 0.003 475 002 212 599 595 519 115 780 457 584 634 613 884 518 4;
  • 49) 0.003 475 002 212 599 595 519 115 780 457 584 634 613 884 518 4 × 2 = 0 + 0.006 950 004 425 199 191 038 231 560 915 169 269 227 769 036 8;
  • 50) 0.006 950 004 425 199 191 038 231 560 915 169 269 227 769 036 8 × 2 = 0 + 0.013 900 008 850 398 382 076 463 121 830 338 538 455 538 073 6;
  • 51) 0.013 900 008 850 398 382 076 463 121 830 338 538 455 538 073 6 × 2 = 0 + 0.027 800 017 700 796 764 152 926 243 660 677 076 911 076 147 2;
  • 52) 0.027 800 017 700 796 764 152 926 243 660 677 076 911 076 147 2 × 2 = 0 + 0.055 600 035 401 593 528 305 852 487 321 354 153 822 152 294 4;
  • 53) 0.055 600 035 401 593 528 305 852 487 321 354 153 822 152 294 4 × 2 = 0 + 0.111 200 070 803 187 056 611 704 974 642 708 307 644 304 588 8;
  • 54) 0.111 200 070 803 187 056 611 704 974 642 708 307 644 304 588 8 × 2 = 0 + 0.222 400 141 606 374 113 223 409 949 285 416 615 288 609 177 6;
  • 55) 0.222 400 141 606 374 113 223 409 949 285 416 615 288 609 177 6 × 2 = 0 + 0.444 800 283 212 748 226 446 819 898 570 833 230 577 218 355 2;
  • 56) 0.444 800 283 212 748 226 446 819 898 570 833 230 577 218 355 2 × 2 = 0 + 0.889 600 566 425 496 452 893 639 797 141 666 461 154 436 710 4;
  • 57) 0.889 600 566 425 496 452 893 639 797 141 666 461 154 436 710 4 × 2 = 1 + 0.779 201 132 850 992 905 787 279 594 283 332 922 308 873 420 8;
  • 58) 0.779 201 132 850 992 905 787 279 594 283 332 922 308 873 420 8 × 2 = 1 + 0.558 402 265 701 985 811 574 559 188 566 665 844 617 746 841 6;
  • 59) 0.558 402 265 701 985 811 574 559 188 566 665 844 617 746 841 6 × 2 = 1 + 0.116 804 531 403 971 623 149 118 377 133 331 689 235 493 683 2;
  • 60) 0.116 804 531 403 971 623 149 118 377 133 331 689 235 493 683 2 × 2 = 0 + 0.233 609 062 807 943 246 298 236 754 266 663 378 470 987 366 4;
  • 61) 0.233 609 062 807 943 246 298 236 754 266 663 378 470 987 366 4 × 2 = 0 + 0.467 218 125 615 886 492 596 473 508 533 326 756 941 974 732 8;
  • 62) 0.467 218 125 615 886 492 596 473 508 533 326 756 941 974 732 8 × 2 = 0 + 0.934 436 251 231 772 985 192 947 017 066 653 513 883 949 465 6;
  • 63) 0.934 436 251 231 772 985 192 947 017 066 653 513 883 949 465 6 × 2 = 1 + 0.868 872 502 463 545 970 385 894 034 133 307 027 767 898 931 2;
  • 64) 0.868 872 502 463 545 970 385 894 034 133 307 027 767 898 931 2 × 2 = 1 + 0.737 745 004 927 091 940 771 788 068 266 614 055 535 797 862 4;
  • 65) 0.737 745 004 927 091 940 771 788 068 266 614 055 535 797 862 4 × 2 = 1 + 0.475 490 009 854 183 881 543 576 136 533 228 111 071 595 724 8;
  • 66) 0.475 490 009 854 183 881 543 576 136 533 228 111 071 595 724 8 × 2 = 0 + 0.950 980 019 708 367 763 087 152 273 066 456 222 143 191 449 6;
  • 67) 0.950 980 019 708 367 763 087 152 273 066 456 222 143 191 449 6 × 2 = 1 + 0.901 960 039 416 735 526 174 304 546 132 912 444 286 382 899 2;
  • 68) 0.901 960 039 416 735 526 174 304 546 132 912 444 286 382 899 2 × 2 = 1 + 0.803 920 078 833 471 052 348 609 092 265 824 888 572 765 798 4;
  • 69) 0.803 920 078 833 471 052 348 609 092 265 824 888 572 765 798 4 × 2 = 1 + 0.607 840 157 666 942 104 697 218 184 531 649 777 145 531 596 8;
  • 70) 0.607 840 157 666 942 104 697 218 184 531 649 777 145 531 596 8 × 2 = 1 + 0.215 680 315 333 884 209 394 436 369 063 299 554 291 063 193 6;
  • 71) 0.215 680 315 333 884 209 394 436 369 063 299 554 291 063 193 6 × 2 = 0 + 0.431 360 630 667 768 418 788 872 738 126 599 108 582 126 387 2;
  • 72) 0.431 360 630 667 768 418 788 872 738 126 599 108 582 126 387 2 × 2 = 0 + 0.862 721 261 335 536 837 577 745 476 253 198 217 164 252 774 4;
  • 73) 0.862 721 261 335 536 837 577 745 476 253 198 217 164 252 774 4 × 2 = 1 + 0.725 442 522 671 073 675 155 490 952 506 396 434 328 505 548 8;
  • 74) 0.725 442 522 671 073 675 155 490 952 506 396 434 328 505 548 8 × 2 = 1 + 0.450 885 045 342 147 350 310 981 905 012 792 868 657 011 097 6;
  • 75) 0.450 885 045 342 147 350 310 981 905 012 792 868 657 011 097 6 × 2 = 0 + 0.901 770 090 684 294 700 621 963 810 025 585 737 314 022 195 2;
  • 76) 0.901 770 090 684 294 700 621 963 810 025 585 737 314 022 195 2 × 2 = 1 + 0.803 540 181 368 589 401 243 927 620 051 171 474 628 044 390 4;
  • 77) 0.803 540 181 368 589 401 243 927 620 051 171 474 628 044 390 4 × 2 = 1 + 0.607 080 362 737 178 802 487 855 240 102 342 949 256 088 780 8;
  • 78) 0.607 080 362 737 178 802 487 855 240 102 342 949 256 088 780 8 × 2 = 1 + 0.214 160 725 474 357 604 975 710 480 204 685 898 512 177 561 6;
  • 79) 0.214 160 725 474 357 604 975 710 480 204 685 898 512 177 561 6 × 2 = 0 + 0.428 321 450 948 715 209 951 420 960 409 371 797 024 355 123 2;
  • 80) 0.428 321 450 948 715 209 951 420 960 409 371 797 024 355 123 2 × 2 = 0 + 0.856 642 901 897 430 419 902 841 920 818 743 594 048 710 246 4;
  • 81) 0.856 642 901 897 430 419 902 841 920 818 743 594 048 710 246 4 × 2 = 1 + 0.713 285 803 794 860 839 805 683 841 637 487 188 097 420 492 8;
  • 82) 0.713 285 803 794 860 839 805 683 841 637 487 188 097 420 492 8 × 2 = 1 + 0.426 571 607 589 721 679 611 367 683 274 974 376 194 840 985 6;
  • 83) 0.426 571 607 589 721 679 611 367 683 274 974 376 194 840 985 6 × 2 = 0 + 0.853 143 215 179 443 359 222 735 366 549 948 752 389 681 971 2;
  • 84) 0.853 143 215 179 443 359 222 735 366 549 948 752 389 681 971 2 × 2 = 1 + 0.706 286 430 358 886 718 445 470 733 099 897 504 779 363 942 4;
  • 85) 0.706 286 430 358 886 718 445 470 733 099 897 504 779 363 942 4 × 2 = 1 + 0.412 572 860 717 773 436 890 941 466 199 795 009 558 727 884 8;
  • 86) 0.412 572 860 717 773 436 890 941 466 199 795 009 558 727 884 8 × 2 = 0 + 0.825 145 721 435 546 873 781 882 932 399 590 019 117 455 769 6;
  • 87) 0.825 145 721 435 546 873 781 882 932 399 590 019 117 455 769 6 × 2 = 1 + 0.650 291 442 871 093 747 563 765 864 799 180 038 234 911 539 2;
  • 88) 0.650 291 442 871 093 747 563 765 864 799 180 038 234 911 539 2 × 2 = 1 + 0.300 582 885 742 187 495 127 531 729 598 360 076 469 823 078 4;
  • 89) 0.300 582 885 742 187 495 127 531 729 598 360 076 469 823 078 4 × 2 = 0 + 0.601 165 771 484 374 990 255 063 459 196 720 152 939 646 156 8;
  • 90) 0.601 165 771 484 374 990 255 063 459 196 720 152 939 646 156 8 × 2 = 1 + 0.202 331 542 968 749 980 510 126 918 393 440 305 879 292 313 6;
  • 91) 0.202 331 542 968 749 980 510 126 918 393 440 305 879 292 313 6 × 2 = 0 + 0.404 663 085 937 499 961 020 253 836 786 880 611 758 584 627 2;
  • 92) 0.404 663 085 937 499 961 020 253 836 786 880 611 758 584 627 2 × 2 = 0 + 0.809 326 171 874 999 922 040 507 673 573 761 223 517 169 254 4;
  • 93) 0.809 326 171 874 999 922 040 507 673 573 761 223 517 169 254 4 × 2 = 1 + 0.618 652 343 749 999 844 081 015 347 147 522 447 034 338 508 8;
  • 94) 0.618 652 343 749 999 844 081 015 347 147 522 447 034 338 508 8 × 2 = 1 + 0.237 304 687 499 999 688 162 030 694 295 044 894 068 677 017 6;
  • 95) 0.237 304 687 499 999 688 162 030 694 295 044 894 068 677 017 6 × 2 = 0 + 0.474 609 374 999 999 376 324 061 388 590 089 788 137 354 035 2;
  • 96) 0.474 609 374 999 999 376 324 061 388 590 089 788 137 354 035 2 × 2 = 0 + 0.949 218 749 999 998 752 648 122 777 180 179 576 274 708 070 4;
  • 97) 0.949 218 749 999 998 752 648 122 777 180 179 576 274 708 070 4 × 2 = 1 + 0.898 437 499 999 997 505 296 245 554 360 359 152 549 416 140 8;
  • 98) 0.898 437 499 999 997 505 296 245 554 360 359 152 549 416 140 8 × 2 = 1 + 0.796 874 999 999 995 010 592 491 108 720 718 305 098 832 281 6;
  • 99) 0.796 874 999 999 995 010 592 491 108 720 718 305 098 832 281 6 × 2 = 1 + 0.593 749 999 999 990 021 184 982 217 441 436 610 197 664 563 2;
  • 100) 0.593 749 999 999 990 021 184 982 217 441 436 610 197 664 563 2 × 2 = 1 + 0.187 499 999 999 980 042 369 964 434 882 873 220 395 329 126 4;
  • 101) 0.187 499 999 999 980 042 369 964 434 882 873 220 395 329 126 4 × 2 = 0 + 0.374 999 999 999 960 084 739 928 869 765 746 440 790 658 252 8;
  • 102) 0.374 999 999 999 960 084 739 928 869 765 746 440 790 658 252 8 × 2 = 0 + 0.749 999 999 999 920 169 479 857 739 531 492 881 581 316 505 6;
  • 103) 0.749 999 999 999 920 169 479 857 739 531 492 881 581 316 505 6 × 2 = 1 + 0.499 999 999 999 840 338 959 715 479 062 985 763 162 633 011 2;
  • 104) 0.499 999 999 999 840 338 959 715 479 062 985 763 162 633 011 2 × 2 = 0 + 0.999 999 999 999 680 677 919 430 958 125 971 526 325 266 022 4;
  • 105) 0.999 999 999 999 680 677 919 430 958 125 971 526 325 266 022 4 × 2 = 1 + 0.999 999 999 999 361 355 838 861 916 251 943 052 650 532 044 8;
  • 106) 0.999 999 999 999 361 355 838 861 916 251 943 052 650 532 044 8 × 2 = 1 + 0.999 999 999 998 722 711 677 723 832 503 886 105 301 064 089 6;
  • 107) 0.999 999 999 998 722 711 677 723 832 503 886 105 301 064 089 6 × 2 = 1 + 0.999 999 999 997 445 423 355 447 665 007 772 210 602 128 179 2;
  • 108) 0.999 999 999 997 445 423 355 447 665 007 772 210 602 128 179 2 × 2 = 1 + 0.999 999 999 994 890 846 710 895 330 015 544 421 204 256 358 4;
  • 109) 0.999 999 999 994 890 846 710 895 330 015 544 421 204 256 358 4 × 2 = 1 + 0.999 999 999 989 781 693 421 790 660 031 088 842 408 512 716 8;

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:


0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 351 4(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110 0011 1011 1100 1101 1100 1101 1011 0100 1100 1111 0010 1111 1(2)

5. Positive number before normalization:

0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 351 4(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110 0011 1011 1100 1101 1100 1101 1011 0100 1100 1111 0010 1111 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 57 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 351 4(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110 0011 1011 1100 1101 1100 1101 1011 0100 1100 1111 0010 1111 1(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110 0011 1011 1100 1101 1100 1101 1011 0100 1100 1111 0010 1111 1(2) × 20 =

1.1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 1111(2) × 2-57


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): -57

Mantissa (not normalized):
1.1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 1111


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-57 + 2(11-1) - 1 =

(-57 + 1 023)(10) =

966(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;
  • 966 ÷ 2 = 483 + 0;
  • 483 ÷ 2 = 241 + 1;
  • 241 ÷ 2 = 120 + 1;
  • 120 ÷ 2 = 60 + 0;
  • 60 ÷ 2 = 30 + 0;
  • 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) =

966(10) =

011 1100 0110(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. 1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 1111 =

1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 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) =
011 1100 0110

Mantissa (52 bits) =
1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 1111


Decimal number 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 351 4 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1100 0110 - 1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 1111

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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