0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 366 94 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 366 94(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
0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 366 94(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: 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 366 94.

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) 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 366 94 × 2 = 0 + 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 733 88;
  • 2) 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 733 88 × 2 = 0 + 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 467 76;
  • 3) 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 467 76 × 2 = 0 + 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 935 52;
  • 4) 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 935 52 × 2 = 0 + 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 871 04;
  • 5) 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 871 04 × 2 = 0 + 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 742 08;
  • 6) 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 742 08 × 2 = 0 + 0.000 000 000 000 000 790 124 025 252 133 724 078 653 463 484 16;
  • 7) 0.000 000 000 000 000 790 124 025 252 133 724 078 653 463 484 16 × 2 = 0 + 0.000 000 000 000 001 580 248 050 504 267 448 157 306 926 968 32;
  • 8) 0.000 000 000 000 001 580 248 050 504 267 448 157 306 926 968 32 × 2 = 0 + 0.000 000 000 000 003 160 496 101 008 534 896 314 613 853 936 64;
  • 9) 0.000 000 000 000 003 160 496 101 008 534 896 314 613 853 936 64 × 2 = 0 + 0.000 000 000 000 006 320 992 202 017 069 792 629 227 707 873 28;
  • 10) 0.000 000 000 000 006 320 992 202 017 069 792 629 227 707 873 28 × 2 = 0 + 0.000 000 000 000 012 641 984 404 034 139 585 258 455 415 746 56;
  • 11) 0.000 000 000 000 012 641 984 404 034 139 585 258 455 415 746 56 × 2 = 0 + 0.000 000 000 000 025 283 968 808 068 279 170 516 910 831 493 12;
  • 12) 0.000 000 000 000 025 283 968 808 068 279 170 516 910 831 493 12 × 2 = 0 + 0.000 000 000 000 050 567 937 616 136 558 341 033 821 662 986 24;
  • 13) 0.000 000 000 000 050 567 937 616 136 558 341 033 821 662 986 24 × 2 = 0 + 0.000 000 000 000 101 135 875 232 273 116 682 067 643 325 972 48;
  • 14) 0.000 000 000 000 101 135 875 232 273 116 682 067 643 325 972 48 × 2 = 0 + 0.000 000 000 000 202 271 750 464 546 233 364 135 286 651 944 96;
  • 15) 0.000 000 000 000 202 271 750 464 546 233 364 135 286 651 944 96 × 2 = 0 + 0.000 000 000 000 404 543 500 929 092 466 728 270 573 303 889 92;
  • 16) 0.000 000 000 000 404 543 500 929 092 466 728 270 573 303 889 92 × 2 = 0 + 0.000 000 000 000 809 087 001 858 184 933 456 541 146 607 779 84;
  • 17) 0.000 000 000 000 809 087 001 858 184 933 456 541 146 607 779 84 × 2 = 0 + 0.000 000 000 001 618 174 003 716 369 866 913 082 293 215 559 68;
  • 18) 0.000 000 000 001 618 174 003 716 369 866 913 082 293 215 559 68 × 2 = 0 + 0.000 000 000 003 236 348 007 432 739 733 826 164 586 431 119 36;
  • 19) 0.000 000 000 003 236 348 007 432 739 733 826 164 586 431 119 36 × 2 = 0 + 0.000 000 000 006 472 696 014 865 479 467 652 329 172 862 238 72;
  • 20) 0.000 000 000 006 472 696 014 865 479 467 652 329 172 862 238 72 × 2 = 0 + 0.000 000 000 012 945 392 029 730 958 935 304 658 345 724 477 44;
  • 21) 0.000 000 000 012 945 392 029 730 958 935 304 658 345 724 477 44 × 2 = 0 + 0.000 000 000 025 890 784 059 461 917 870 609 316 691 448 954 88;
  • 22) 0.000 000 000 025 890 784 059 461 917 870 609 316 691 448 954 88 × 2 = 0 + 0.000 000 000 051 781 568 118 923 835 741 218 633 382 897 909 76;
  • 23) 0.000 000 000 051 781 568 118 923 835 741 218 633 382 897 909 76 × 2 = 0 + 0.000 000 000 103 563 136 237 847 671 482 437 266 765 795 819 52;
  • 24) 0.000 000 000 103 563 136 237 847 671 482 437 266 765 795 819 52 × 2 = 0 + 0.000 000 000 207 126 272 475 695 342 964 874 533 531 591 639 04;
  • 25) 0.000 000 000 207 126 272 475 695 342 964 874 533 531 591 639 04 × 2 = 0 + 0.000 000 000 414 252 544 951 390 685 929 749 067 063 183 278 08;
  • 26) 0.000 000 000 414 252 544 951 390 685 929 749 067 063 183 278 08 × 2 = 0 + 0.000 000 000 828 505 089 902 781 371 859 498 134 126 366 556 16;
  • 27) 0.000 000 000 828 505 089 902 781 371 859 498 134 126 366 556 16 × 2 = 0 + 0.000 000 001 657 010 179 805 562 743 718 996 268 252 733 112 32;
  • 28) 0.000 000 001 657 010 179 805 562 743 718 996 268 252 733 112 32 × 2 = 0 + 0.000 000 003 314 020 359 611 125 487 437 992 536 505 466 224 64;
  • 29) 0.000 000 003 314 020 359 611 125 487 437 992 536 505 466 224 64 × 2 = 0 + 0.000 000 006 628 040 719 222 250 974 875 985 073 010 932 449 28;
  • 30) 0.000 000 006 628 040 719 222 250 974 875 985 073 010 932 449 28 × 2 = 0 + 0.000 000 013 256 081 438 444 501 949 751 970 146 021 864 898 56;
  • 31) 0.000 000 013 256 081 438 444 501 949 751 970 146 021 864 898 56 × 2 = 0 + 0.000 000 026 512 162 876 889 003 899 503 940 292 043 729 797 12;
  • 32) 0.000 000 026 512 162 876 889 003 899 503 940 292 043 729 797 12 × 2 = 0 + 0.000 000 053 024 325 753 778 007 799 007 880 584 087 459 594 24;
  • 33) 0.000 000 053 024 325 753 778 007 799 007 880 584 087 459 594 24 × 2 = 0 + 0.000 000 106 048 651 507 556 015 598 015 761 168 174 919 188 48;
  • 34) 0.000 000 106 048 651 507 556 015 598 015 761 168 174 919 188 48 × 2 = 0 + 0.000 000 212 097 303 015 112 031 196 031 522 336 349 838 376 96;
  • 35) 0.000 000 212 097 303 015 112 031 196 031 522 336 349 838 376 96 × 2 = 0 + 0.000 000 424 194 606 030 224 062 392 063 044 672 699 676 753 92;
  • 36) 0.000 000 424 194 606 030 224 062 392 063 044 672 699 676 753 92 × 2 = 0 + 0.000 000 848 389 212 060 448 124 784 126 089 345 399 353 507 84;
  • 37) 0.000 000 848 389 212 060 448 124 784 126 089 345 399 353 507 84 × 2 = 0 + 0.000 001 696 778 424 120 896 249 568 252 178 690 798 707 015 68;
  • 38) 0.000 001 696 778 424 120 896 249 568 252 178 690 798 707 015 68 × 2 = 0 + 0.000 003 393 556 848 241 792 499 136 504 357 381 597 414 031 36;
  • 39) 0.000 003 393 556 848 241 792 499 136 504 357 381 597 414 031 36 × 2 = 0 + 0.000 006 787 113 696 483 584 998 273 008 714 763 194 828 062 72;
  • 40) 0.000 006 787 113 696 483 584 998 273 008 714 763 194 828 062 72 × 2 = 0 + 0.000 013 574 227 392 967 169 996 546 017 429 526 389 656 125 44;
  • 41) 0.000 013 574 227 392 967 169 996 546 017 429 526 389 656 125 44 × 2 = 0 + 0.000 027 148 454 785 934 339 993 092 034 859 052 779 312 250 88;
  • 42) 0.000 027 148 454 785 934 339 993 092 034 859 052 779 312 250 88 × 2 = 0 + 0.000 054 296 909 571 868 679 986 184 069 718 105 558 624 501 76;
  • 43) 0.000 054 296 909 571 868 679 986 184 069 718 105 558 624 501 76 × 2 = 0 + 0.000 108 593 819 143 737 359 972 368 139 436 211 117 249 003 52;
  • 44) 0.000 108 593 819 143 737 359 972 368 139 436 211 117 249 003 52 × 2 = 0 + 0.000 217 187 638 287 474 719 944 736 278 872 422 234 498 007 04;
  • 45) 0.000 217 187 638 287 474 719 944 736 278 872 422 234 498 007 04 × 2 = 0 + 0.000 434 375 276 574 949 439 889 472 557 744 844 468 996 014 08;
  • 46) 0.000 434 375 276 574 949 439 889 472 557 744 844 468 996 014 08 × 2 = 0 + 0.000 868 750 553 149 898 879 778 945 115 489 688 937 992 028 16;
  • 47) 0.000 868 750 553 149 898 879 778 945 115 489 688 937 992 028 16 × 2 = 0 + 0.001 737 501 106 299 797 759 557 890 230 979 377 875 984 056 32;
  • 48) 0.001 737 501 106 299 797 759 557 890 230 979 377 875 984 056 32 × 2 = 0 + 0.003 475 002 212 599 595 519 115 780 461 958 755 751 968 112 64;
  • 49) 0.003 475 002 212 599 595 519 115 780 461 958 755 751 968 112 64 × 2 = 0 + 0.006 950 004 425 199 191 038 231 560 923 917 511 503 936 225 28;
  • 50) 0.006 950 004 425 199 191 038 231 560 923 917 511 503 936 225 28 × 2 = 0 + 0.013 900 008 850 398 382 076 463 121 847 835 023 007 872 450 56;
  • 51) 0.013 900 008 850 398 382 076 463 121 847 835 023 007 872 450 56 × 2 = 0 + 0.027 800 017 700 796 764 152 926 243 695 670 046 015 744 901 12;
  • 52) 0.027 800 017 700 796 764 152 926 243 695 670 046 015 744 901 12 × 2 = 0 + 0.055 600 035 401 593 528 305 852 487 391 340 092 031 489 802 24;
  • 53) 0.055 600 035 401 593 528 305 852 487 391 340 092 031 489 802 24 × 2 = 0 + 0.111 200 070 803 187 056 611 704 974 782 680 184 062 979 604 48;
  • 54) 0.111 200 070 803 187 056 611 704 974 782 680 184 062 979 604 48 × 2 = 0 + 0.222 400 141 606 374 113 223 409 949 565 360 368 125 959 208 96;
  • 55) 0.222 400 141 606 374 113 223 409 949 565 360 368 125 959 208 96 × 2 = 0 + 0.444 800 283 212 748 226 446 819 899 130 720 736 251 918 417 92;
  • 56) 0.444 800 283 212 748 226 446 819 899 130 720 736 251 918 417 92 × 2 = 0 + 0.889 600 566 425 496 452 893 639 798 261 441 472 503 836 835 84;
  • 57) 0.889 600 566 425 496 452 893 639 798 261 441 472 503 836 835 84 × 2 = 1 + 0.779 201 132 850 992 905 787 279 596 522 882 945 007 673 671 68;
  • 58) 0.779 201 132 850 992 905 787 279 596 522 882 945 007 673 671 68 × 2 = 1 + 0.558 402 265 701 985 811 574 559 193 045 765 890 015 347 343 36;
  • 59) 0.558 402 265 701 985 811 574 559 193 045 765 890 015 347 343 36 × 2 = 1 + 0.116 804 531 403 971 623 149 118 386 091 531 780 030 694 686 72;
  • 60) 0.116 804 531 403 971 623 149 118 386 091 531 780 030 694 686 72 × 2 = 0 + 0.233 609 062 807 943 246 298 236 772 183 063 560 061 389 373 44;
  • 61) 0.233 609 062 807 943 246 298 236 772 183 063 560 061 389 373 44 × 2 = 0 + 0.467 218 125 615 886 492 596 473 544 366 127 120 122 778 746 88;
  • 62) 0.467 218 125 615 886 492 596 473 544 366 127 120 122 778 746 88 × 2 = 0 + 0.934 436 251 231 772 985 192 947 088 732 254 240 245 557 493 76;
  • 63) 0.934 436 251 231 772 985 192 947 088 732 254 240 245 557 493 76 × 2 = 1 + 0.868 872 502 463 545 970 385 894 177 464 508 480 491 114 987 52;
  • 64) 0.868 872 502 463 545 970 385 894 177 464 508 480 491 114 987 52 × 2 = 1 + 0.737 745 004 927 091 940 771 788 354 929 016 960 982 229 975 04;
  • 65) 0.737 745 004 927 091 940 771 788 354 929 016 960 982 229 975 04 × 2 = 1 + 0.475 490 009 854 183 881 543 576 709 858 033 921 964 459 950 08;
  • 66) 0.475 490 009 854 183 881 543 576 709 858 033 921 964 459 950 08 × 2 = 0 + 0.950 980 019 708 367 763 087 153 419 716 067 843 928 919 900 16;
  • 67) 0.950 980 019 708 367 763 087 153 419 716 067 843 928 919 900 16 × 2 = 1 + 0.901 960 039 416 735 526 174 306 839 432 135 687 857 839 800 32;
  • 68) 0.901 960 039 416 735 526 174 306 839 432 135 687 857 839 800 32 × 2 = 1 + 0.803 920 078 833 471 052 348 613 678 864 271 375 715 679 600 64;
  • 69) 0.803 920 078 833 471 052 348 613 678 864 271 375 715 679 600 64 × 2 = 1 + 0.607 840 157 666 942 104 697 227 357 728 542 751 431 359 201 28;
  • 70) 0.607 840 157 666 942 104 697 227 357 728 542 751 431 359 201 28 × 2 = 1 + 0.215 680 315 333 884 209 394 454 715 457 085 502 862 718 402 56;
  • 71) 0.215 680 315 333 884 209 394 454 715 457 085 502 862 718 402 56 × 2 = 0 + 0.431 360 630 667 768 418 788 909 430 914 171 005 725 436 805 12;
  • 72) 0.431 360 630 667 768 418 788 909 430 914 171 005 725 436 805 12 × 2 = 0 + 0.862 721 261 335 536 837 577 818 861 828 342 011 450 873 610 24;
  • 73) 0.862 721 261 335 536 837 577 818 861 828 342 011 450 873 610 24 × 2 = 1 + 0.725 442 522 671 073 675 155 637 723 656 684 022 901 747 220 48;
  • 74) 0.725 442 522 671 073 675 155 637 723 656 684 022 901 747 220 48 × 2 = 1 + 0.450 885 045 342 147 350 311 275 447 313 368 045 803 494 440 96;
  • 75) 0.450 885 045 342 147 350 311 275 447 313 368 045 803 494 440 96 × 2 = 0 + 0.901 770 090 684 294 700 622 550 894 626 736 091 606 988 881 92;
  • 76) 0.901 770 090 684 294 700 622 550 894 626 736 091 606 988 881 92 × 2 = 1 + 0.803 540 181 368 589 401 245 101 789 253 472 183 213 977 763 84;
  • 77) 0.803 540 181 368 589 401 245 101 789 253 472 183 213 977 763 84 × 2 = 1 + 0.607 080 362 737 178 802 490 203 578 506 944 366 427 955 527 68;
  • 78) 0.607 080 362 737 178 802 490 203 578 506 944 366 427 955 527 68 × 2 = 1 + 0.214 160 725 474 357 604 980 407 157 013 888 732 855 911 055 36;
  • 79) 0.214 160 725 474 357 604 980 407 157 013 888 732 855 911 055 36 × 2 = 0 + 0.428 321 450 948 715 209 960 814 314 027 777 465 711 822 110 72;
  • 80) 0.428 321 450 948 715 209 960 814 314 027 777 465 711 822 110 72 × 2 = 0 + 0.856 642 901 897 430 419 921 628 628 055 554 931 423 644 221 44;
  • 81) 0.856 642 901 897 430 419 921 628 628 055 554 931 423 644 221 44 × 2 = 1 + 0.713 285 803 794 860 839 843 257 256 111 109 862 847 288 442 88;
  • 82) 0.713 285 803 794 860 839 843 257 256 111 109 862 847 288 442 88 × 2 = 1 + 0.426 571 607 589 721 679 686 514 512 222 219 725 694 576 885 76;
  • 83) 0.426 571 607 589 721 679 686 514 512 222 219 725 694 576 885 76 × 2 = 0 + 0.853 143 215 179 443 359 373 029 024 444 439 451 389 153 771 52;
  • 84) 0.853 143 215 179 443 359 373 029 024 444 439 451 389 153 771 52 × 2 = 1 + 0.706 286 430 358 886 718 746 058 048 888 878 902 778 307 543 04;
  • 85) 0.706 286 430 358 886 718 746 058 048 888 878 902 778 307 543 04 × 2 = 1 + 0.412 572 860 717 773 437 492 116 097 777 757 805 556 615 086 08;
  • 86) 0.412 572 860 717 773 437 492 116 097 777 757 805 556 615 086 08 × 2 = 0 + 0.825 145 721 435 546 874 984 232 195 555 515 611 113 230 172 16;
  • 87) 0.825 145 721 435 546 874 984 232 195 555 515 611 113 230 172 16 × 2 = 1 + 0.650 291 442 871 093 749 968 464 391 111 031 222 226 460 344 32;
  • 88) 0.650 291 442 871 093 749 968 464 391 111 031 222 226 460 344 32 × 2 = 1 + 0.300 582 885 742 187 499 936 928 782 222 062 444 452 920 688 64;
  • 89) 0.300 582 885 742 187 499 936 928 782 222 062 444 452 920 688 64 × 2 = 0 + 0.601 165 771 484 374 999 873 857 564 444 124 888 905 841 377 28;
  • 90) 0.601 165 771 484 374 999 873 857 564 444 124 888 905 841 377 28 × 2 = 1 + 0.202 331 542 968 749 999 747 715 128 888 249 777 811 682 754 56;
  • 91) 0.202 331 542 968 749 999 747 715 128 888 249 777 811 682 754 56 × 2 = 0 + 0.404 663 085 937 499 999 495 430 257 776 499 555 623 365 509 12;
  • 92) 0.404 663 085 937 499 999 495 430 257 776 499 555 623 365 509 12 × 2 = 0 + 0.809 326 171 874 999 998 990 860 515 552 999 111 246 731 018 24;
  • 93) 0.809 326 171 874 999 998 990 860 515 552 999 111 246 731 018 24 × 2 = 1 + 0.618 652 343 749 999 997 981 721 031 105 998 222 493 462 036 48;
  • 94) 0.618 652 343 749 999 997 981 721 031 105 998 222 493 462 036 48 × 2 = 1 + 0.237 304 687 499 999 995 963 442 062 211 996 444 986 924 072 96;
  • 95) 0.237 304 687 499 999 995 963 442 062 211 996 444 986 924 072 96 × 2 = 0 + 0.474 609 374 999 999 991 926 884 124 423 992 889 973 848 145 92;
  • 96) 0.474 609 374 999 999 991 926 884 124 423 992 889 973 848 145 92 × 2 = 0 + 0.949 218 749 999 999 983 853 768 248 847 985 779 947 696 291 84;
  • 97) 0.949 218 749 999 999 983 853 768 248 847 985 779 947 696 291 84 × 2 = 1 + 0.898 437 499 999 999 967 707 536 497 695 971 559 895 392 583 68;
  • 98) 0.898 437 499 999 999 967 707 536 497 695 971 559 895 392 583 68 × 2 = 1 + 0.796 874 999 999 999 935 415 072 995 391 943 119 790 785 167 36;
  • 99) 0.796 874 999 999 999 935 415 072 995 391 943 119 790 785 167 36 × 2 = 1 + 0.593 749 999 999 999 870 830 145 990 783 886 239 581 570 334 72;
  • 100) 0.593 749 999 999 999 870 830 145 990 783 886 239 581 570 334 72 × 2 = 1 + 0.187 499 999 999 999 741 660 291 981 567 772 479 163 140 669 44;
  • 101) 0.187 499 999 999 999 741 660 291 981 567 772 479 163 140 669 44 × 2 = 0 + 0.374 999 999 999 999 483 320 583 963 135 544 958 326 281 338 88;
  • 102) 0.374 999 999 999 999 483 320 583 963 135 544 958 326 281 338 88 × 2 = 0 + 0.749 999 999 999 998 966 641 167 926 271 089 916 652 562 677 76;
  • 103) 0.749 999 999 999 998 966 641 167 926 271 089 916 652 562 677 76 × 2 = 1 + 0.499 999 999 999 997 933 282 335 852 542 179 833 305 125 355 52;
  • 104) 0.499 999 999 999 997 933 282 335 852 542 179 833 305 125 355 52 × 2 = 0 + 0.999 999 999 999 995 866 564 671 705 084 359 666 610 250 711 04;
  • 105) 0.999 999 999 999 995 866 564 671 705 084 359 666 610 250 711 04 × 2 = 1 + 0.999 999 999 999 991 733 129 343 410 168 719 333 220 501 422 08;
  • 106) 0.999 999 999 999 991 733 129 343 410 168 719 333 220 501 422 08 × 2 = 1 + 0.999 999 999 999 983 466 258 686 820 337 438 666 441 002 844 16;
  • 107) 0.999 999 999 999 983 466 258 686 820 337 438 666 441 002 844 16 × 2 = 1 + 0.999 999 999 999 966 932 517 373 640 674 877 332 882 005 688 32;
  • 108) 0.999 999 999 999 966 932 517 373 640 674 877 332 882 005 688 32 × 2 = 1 + 0.999 999 999 999 933 865 034 747 281 349 754 665 764 011 376 64;
  • 109) 0.999 999 999 999 933 865 034 747 281 349 754 665 764 011 376 64 × 2 = 1 + 0.999 999 999 999 867 730 069 494 562 699 509 331 528 022 753 28;

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:


0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 366 94(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:

0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 366 94(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:


0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 366 94(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 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 366 94 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