0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 357 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 357 9(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
0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 357 9(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: 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 357 9.

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) 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 357 9 × 2 = 0 + 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 715 8;
  • 2) 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 715 8 × 2 = 0 + 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 431 6;
  • 3) 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 431 6 × 2 = 0 + 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 863 2;
  • 4) 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 863 2 × 2 = 0 + 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 726 4;
  • 5) 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 726 4 × 2 = 0 + 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 452 8;
  • 6) 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 452 8 × 2 = 0 + 0.000 000 000 000 000 790 124 025 252 133 724 078 653 462 905 6;
  • 7) 0.000 000 000 000 000 790 124 025 252 133 724 078 653 462 905 6 × 2 = 0 + 0.000 000 000 000 001 580 248 050 504 267 448 157 306 925 811 2;
  • 8) 0.000 000 000 000 001 580 248 050 504 267 448 157 306 925 811 2 × 2 = 0 + 0.000 000 000 000 003 160 496 101 008 534 896 314 613 851 622 4;
  • 9) 0.000 000 000 000 003 160 496 101 008 534 896 314 613 851 622 4 × 2 = 0 + 0.000 000 000 000 006 320 992 202 017 069 792 629 227 703 244 8;
  • 10) 0.000 000 000 000 006 320 992 202 017 069 792 629 227 703 244 8 × 2 = 0 + 0.000 000 000 000 012 641 984 404 034 139 585 258 455 406 489 6;
  • 11) 0.000 000 000 000 012 641 984 404 034 139 585 258 455 406 489 6 × 2 = 0 + 0.000 000 000 000 025 283 968 808 068 279 170 516 910 812 979 2;
  • 12) 0.000 000 000 000 025 283 968 808 068 279 170 516 910 812 979 2 × 2 = 0 + 0.000 000 000 000 050 567 937 616 136 558 341 033 821 625 958 4;
  • 13) 0.000 000 000 000 050 567 937 616 136 558 341 033 821 625 958 4 × 2 = 0 + 0.000 000 000 000 101 135 875 232 273 116 682 067 643 251 916 8;
  • 14) 0.000 000 000 000 101 135 875 232 273 116 682 067 643 251 916 8 × 2 = 0 + 0.000 000 000 000 202 271 750 464 546 233 364 135 286 503 833 6;
  • 15) 0.000 000 000 000 202 271 750 464 546 233 364 135 286 503 833 6 × 2 = 0 + 0.000 000 000 000 404 543 500 929 092 466 728 270 573 007 667 2;
  • 16) 0.000 000 000 000 404 543 500 929 092 466 728 270 573 007 667 2 × 2 = 0 + 0.000 000 000 000 809 087 001 858 184 933 456 541 146 015 334 4;
  • 17) 0.000 000 000 000 809 087 001 858 184 933 456 541 146 015 334 4 × 2 = 0 + 0.000 000 000 001 618 174 003 716 369 866 913 082 292 030 668 8;
  • 18) 0.000 000 000 001 618 174 003 716 369 866 913 082 292 030 668 8 × 2 = 0 + 0.000 000 000 003 236 348 007 432 739 733 826 164 584 061 337 6;
  • 19) 0.000 000 000 003 236 348 007 432 739 733 826 164 584 061 337 6 × 2 = 0 + 0.000 000 000 006 472 696 014 865 479 467 652 329 168 122 675 2;
  • 20) 0.000 000 000 006 472 696 014 865 479 467 652 329 168 122 675 2 × 2 = 0 + 0.000 000 000 012 945 392 029 730 958 935 304 658 336 245 350 4;
  • 21) 0.000 000 000 012 945 392 029 730 958 935 304 658 336 245 350 4 × 2 = 0 + 0.000 000 000 025 890 784 059 461 917 870 609 316 672 490 700 8;
  • 22) 0.000 000 000 025 890 784 059 461 917 870 609 316 672 490 700 8 × 2 = 0 + 0.000 000 000 051 781 568 118 923 835 741 218 633 344 981 401 6;
  • 23) 0.000 000 000 051 781 568 118 923 835 741 218 633 344 981 401 6 × 2 = 0 + 0.000 000 000 103 563 136 237 847 671 482 437 266 689 962 803 2;
  • 24) 0.000 000 000 103 563 136 237 847 671 482 437 266 689 962 803 2 × 2 = 0 + 0.000 000 000 207 126 272 475 695 342 964 874 533 379 925 606 4;
  • 25) 0.000 000 000 207 126 272 475 695 342 964 874 533 379 925 606 4 × 2 = 0 + 0.000 000 000 414 252 544 951 390 685 929 749 066 759 851 212 8;
  • 26) 0.000 000 000 414 252 544 951 390 685 929 749 066 759 851 212 8 × 2 = 0 + 0.000 000 000 828 505 089 902 781 371 859 498 133 519 702 425 6;
  • 27) 0.000 000 000 828 505 089 902 781 371 859 498 133 519 702 425 6 × 2 = 0 + 0.000 000 001 657 010 179 805 562 743 718 996 267 039 404 851 2;
  • 28) 0.000 000 001 657 010 179 805 562 743 718 996 267 039 404 851 2 × 2 = 0 + 0.000 000 003 314 020 359 611 125 487 437 992 534 078 809 702 4;
  • 29) 0.000 000 003 314 020 359 611 125 487 437 992 534 078 809 702 4 × 2 = 0 + 0.000 000 006 628 040 719 222 250 974 875 985 068 157 619 404 8;
  • 30) 0.000 000 006 628 040 719 222 250 974 875 985 068 157 619 404 8 × 2 = 0 + 0.000 000 013 256 081 438 444 501 949 751 970 136 315 238 809 6;
  • 31) 0.000 000 013 256 081 438 444 501 949 751 970 136 315 238 809 6 × 2 = 0 + 0.000 000 026 512 162 876 889 003 899 503 940 272 630 477 619 2;
  • 32) 0.000 000 026 512 162 876 889 003 899 503 940 272 630 477 619 2 × 2 = 0 + 0.000 000 053 024 325 753 778 007 799 007 880 545 260 955 238 4;
  • 33) 0.000 000 053 024 325 753 778 007 799 007 880 545 260 955 238 4 × 2 = 0 + 0.000 000 106 048 651 507 556 015 598 015 761 090 521 910 476 8;
  • 34) 0.000 000 106 048 651 507 556 015 598 015 761 090 521 910 476 8 × 2 = 0 + 0.000 000 212 097 303 015 112 031 196 031 522 181 043 820 953 6;
  • 35) 0.000 000 212 097 303 015 112 031 196 031 522 181 043 820 953 6 × 2 = 0 + 0.000 000 424 194 606 030 224 062 392 063 044 362 087 641 907 2;
  • 36) 0.000 000 424 194 606 030 224 062 392 063 044 362 087 641 907 2 × 2 = 0 + 0.000 000 848 389 212 060 448 124 784 126 088 724 175 283 814 4;
  • 37) 0.000 000 848 389 212 060 448 124 784 126 088 724 175 283 814 4 × 2 = 0 + 0.000 001 696 778 424 120 896 249 568 252 177 448 350 567 628 8;
  • 38) 0.000 001 696 778 424 120 896 249 568 252 177 448 350 567 628 8 × 2 = 0 + 0.000 003 393 556 848 241 792 499 136 504 354 896 701 135 257 6;
  • 39) 0.000 003 393 556 848 241 792 499 136 504 354 896 701 135 257 6 × 2 = 0 + 0.000 006 787 113 696 483 584 998 273 008 709 793 402 270 515 2;
  • 40) 0.000 006 787 113 696 483 584 998 273 008 709 793 402 270 515 2 × 2 = 0 + 0.000 013 574 227 392 967 169 996 546 017 419 586 804 541 030 4;
  • 41) 0.000 013 574 227 392 967 169 996 546 017 419 586 804 541 030 4 × 2 = 0 + 0.000 027 148 454 785 934 339 993 092 034 839 173 609 082 060 8;
  • 42) 0.000 027 148 454 785 934 339 993 092 034 839 173 609 082 060 8 × 2 = 0 + 0.000 054 296 909 571 868 679 986 184 069 678 347 218 164 121 6;
  • 43) 0.000 054 296 909 571 868 679 986 184 069 678 347 218 164 121 6 × 2 = 0 + 0.000 108 593 819 143 737 359 972 368 139 356 694 436 328 243 2;
  • 44) 0.000 108 593 819 143 737 359 972 368 139 356 694 436 328 243 2 × 2 = 0 + 0.000 217 187 638 287 474 719 944 736 278 713 388 872 656 486 4;
  • 45) 0.000 217 187 638 287 474 719 944 736 278 713 388 872 656 486 4 × 2 = 0 + 0.000 434 375 276 574 949 439 889 472 557 426 777 745 312 972 8;
  • 46) 0.000 434 375 276 574 949 439 889 472 557 426 777 745 312 972 8 × 2 = 0 + 0.000 868 750 553 149 898 879 778 945 114 853 555 490 625 945 6;
  • 47) 0.000 868 750 553 149 898 879 778 945 114 853 555 490 625 945 6 × 2 = 0 + 0.001 737 501 106 299 797 759 557 890 229 707 110 981 251 891 2;
  • 48) 0.001 737 501 106 299 797 759 557 890 229 707 110 981 251 891 2 × 2 = 0 + 0.003 475 002 212 599 595 519 115 780 459 414 221 962 503 782 4;
  • 49) 0.003 475 002 212 599 595 519 115 780 459 414 221 962 503 782 4 × 2 = 0 + 0.006 950 004 425 199 191 038 231 560 918 828 443 925 007 564 8;
  • 50) 0.006 950 004 425 199 191 038 231 560 918 828 443 925 007 564 8 × 2 = 0 + 0.013 900 008 850 398 382 076 463 121 837 656 887 850 015 129 6;
  • 51) 0.013 900 008 850 398 382 076 463 121 837 656 887 850 015 129 6 × 2 = 0 + 0.027 800 017 700 796 764 152 926 243 675 313 775 700 030 259 2;
  • 52) 0.027 800 017 700 796 764 152 926 243 675 313 775 700 030 259 2 × 2 = 0 + 0.055 600 035 401 593 528 305 852 487 350 627 551 400 060 518 4;
  • 53) 0.055 600 035 401 593 528 305 852 487 350 627 551 400 060 518 4 × 2 = 0 + 0.111 200 070 803 187 056 611 704 974 701 255 102 800 121 036 8;
  • 54) 0.111 200 070 803 187 056 611 704 974 701 255 102 800 121 036 8 × 2 = 0 + 0.222 400 141 606 374 113 223 409 949 402 510 205 600 242 073 6;
  • 55) 0.222 400 141 606 374 113 223 409 949 402 510 205 600 242 073 6 × 2 = 0 + 0.444 800 283 212 748 226 446 819 898 805 020 411 200 484 147 2;
  • 56) 0.444 800 283 212 748 226 446 819 898 805 020 411 200 484 147 2 × 2 = 0 + 0.889 600 566 425 496 452 893 639 797 610 040 822 400 968 294 4;
  • 57) 0.889 600 566 425 496 452 893 639 797 610 040 822 400 968 294 4 × 2 = 1 + 0.779 201 132 850 992 905 787 279 595 220 081 644 801 936 588 8;
  • 58) 0.779 201 132 850 992 905 787 279 595 220 081 644 801 936 588 8 × 2 = 1 + 0.558 402 265 701 985 811 574 559 190 440 163 289 603 873 177 6;
  • 59) 0.558 402 265 701 985 811 574 559 190 440 163 289 603 873 177 6 × 2 = 1 + 0.116 804 531 403 971 623 149 118 380 880 326 579 207 746 355 2;
  • 60) 0.116 804 531 403 971 623 149 118 380 880 326 579 207 746 355 2 × 2 = 0 + 0.233 609 062 807 943 246 298 236 761 760 653 158 415 492 710 4;
  • 61) 0.233 609 062 807 943 246 298 236 761 760 653 158 415 492 710 4 × 2 = 0 + 0.467 218 125 615 886 492 596 473 523 521 306 316 830 985 420 8;
  • 62) 0.467 218 125 615 886 492 596 473 523 521 306 316 830 985 420 8 × 2 = 0 + 0.934 436 251 231 772 985 192 947 047 042 612 633 661 970 841 6;
  • 63) 0.934 436 251 231 772 985 192 947 047 042 612 633 661 970 841 6 × 2 = 1 + 0.868 872 502 463 545 970 385 894 094 085 225 267 323 941 683 2;
  • 64) 0.868 872 502 463 545 970 385 894 094 085 225 267 323 941 683 2 × 2 = 1 + 0.737 745 004 927 091 940 771 788 188 170 450 534 647 883 366 4;
  • 65) 0.737 745 004 927 091 940 771 788 188 170 450 534 647 883 366 4 × 2 = 1 + 0.475 490 009 854 183 881 543 576 376 340 901 069 295 766 732 8;
  • 66) 0.475 490 009 854 183 881 543 576 376 340 901 069 295 766 732 8 × 2 = 0 + 0.950 980 019 708 367 763 087 152 752 681 802 138 591 533 465 6;
  • 67) 0.950 980 019 708 367 763 087 152 752 681 802 138 591 533 465 6 × 2 = 1 + 0.901 960 039 416 735 526 174 305 505 363 604 277 183 066 931 2;
  • 68) 0.901 960 039 416 735 526 174 305 505 363 604 277 183 066 931 2 × 2 = 1 + 0.803 920 078 833 471 052 348 611 010 727 208 554 366 133 862 4;
  • 69) 0.803 920 078 833 471 052 348 611 010 727 208 554 366 133 862 4 × 2 = 1 + 0.607 840 157 666 942 104 697 222 021 454 417 108 732 267 724 8;
  • 70) 0.607 840 157 666 942 104 697 222 021 454 417 108 732 267 724 8 × 2 = 1 + 0.215 680 315 333 884 209 394 444 042 908 834 217 464 535 449 6;
  • 71) 0.215 680 315 333 884 209 394 444 042 908 834 217 464 535 449 6 × 2 = 0 + 0.431 360 630 667 768 418 788 888 085 817 668 434 929 070 899 2;
  • 72) 0.431 360 630 667 768 418 788 888 085 817 668 434 929 070 899 2 × 2 = 0 + 0.862 721 261 335 536 837 577 776 171 635 336 869 858 141 798 4;
  • 73) 0.862 721 261 335 536 837 577 776 171 635 336 869 858 141 798 4 × 2 = 1 + 0.725 442 522 671 073 675 155 552 343 270 673 739 716 283 596 8;
  • 74) 0.725 442 522 671 073 675 155 552 343 270 673 739 716 283 596 8 × 2 = 1 + 0.450 885 045 342 147 350 311 104 686 541 347 479 432 567 193 6;
  • 75) 0.450 885 045 342 147 350 311 104 686 541 347 479 432 567 193 6 × 2 = 0 + 0.901 770 090 684 294 700 622 209 373 082 694 958 865 134 387 2;
  • 76) 0.901 770 090 684 294 700 622 209 373 082 694 958 865 134 387 2 × 2 = 1 + 0.803 540 181 368 589 401 244 418 746 165 389 917 730 268 774 4;
  • 77) 0.803 540 181 368 589 401 244 418 746 165 389 917 730 268 774 4 × 2 = 1 + 0.607 080 362 737 178 802 488 837 492 330 779 835 460 537 548 8;
  • 78) 0.607 080 362 737 178 802 488 837 492 330 779 835 460 537 548 8 × 2 = 1 + 0.214 160 725 474 357 604 977 674 984 661 559 670 921 075 097 6;
  • 79) 0.214 160 725 474 357 604 977 674 984 661 559 670 921 075 097 6 × 2 = 0 + 0.428 321 450 948 715 209 955 349 969 323 119 341 842 150 195 2;
  • 80) 0.428 321 450 948 715 209 955 349 969 323 119 341 842 150 195 2 × 2 = 0 + 0.856 642 901 897 430 419 910 699 938 646 238 683 684 300 390 4;
  • 81) 0.856 642 901 897 430 419 910 699 938 646 238 683 684 300 390 4 × 2 = 1 + 0.713 285 803 794 860 839 821 399 877 292 477 367 368 600 780 8;
  • 82) 0.713 285 803 794 860 839 821 399 877 292 477 367 368 600 780 8 × 2 = 1 + 0.426 571 607 589 721 679 642 799 754 584 954 734 737 201 561 6;
  • 83) 0.426 571 607 589 721 679 642 799 754 584 954 734 737 201 561 6 × 2 = 0 + 0.853 143 215 179 443 359 285 599 509 169 909 469 474 403 123 2;
  • 84) 0.853 143 215 179 443 359 285 599 509 169 909 469 474 403 123 2 × 2 = 1 + 0.706 286 430 358 886 718 571 199 018 339 818 938 948 806 246 4;
  • 85) 0.706 286 430 358 886 718 571 199 018 339 818 938 948 806 246 4 × 2 = 1 + 0.412 572 860 717 773 437 142 398 036 679 637 877 897 612 492 8;
  • 86) 0.412 572 860 717 773 437 142 398 036 679 637 877 897 612 492 8 × 2 = 0 + 0.825 145 721 435 546 874 284 796 073 359 275 755 795 224 985 6;
  • 87) 0.825 145 721 435 546 874 284 796 073 359 275 755 795 224 985 6 × 2 = 1 + 0.650 291 442 871 093 748 569 592 146 718 551 511 590 449 971 2;
  • 88) 0.650 291 442 871 093 748 569 592 146 718 551 511 590 449 971 2 × 2 = 1 + 0.300 582 885 742 187 497 139 184 293 437 103 023 180 899 942 4;
  • 89) 0.300 582 885 742 187 497 139 184 293 437 103 023 180 899 942 4 × 2 = 0 + 0.601 165 771 484 374 994 278 368 586 874 206 046 361 799 884 8;
  • 90) 0.601 165 771 484 374 994 278 368 586 874 206 046 361 799 884 8 × 2 = 1 + 0.202 331 542 968 749 988 556 737 173 748 412 092 723 599 769 6;
  • 91) 0.202 331 542 968 749 988 556 737 173 748 412 092 723 599 769 6 × 2 = 0 + 0.404 663 085 937 499 977 113 474 347 496 824 185 447 199 539 2;
  • 92) 0.404 663 085 937 499 977 113 474 347 496 824 185 447 199 539 2 × 2 = 0 + 0.809 326 171 874 999 954 226 948 694 993 648 370 894 399 078 4;
  • 93) 0.809 326 171 874 999 954 226 948 694 993 648 370 894 399 078 4 × 2 = 1 + 0.618 652 343 749 999 908 453 897 389 987 296 741 788 798 156 8;
  • 94) 0.618 652 343 749 999 908 453 897 389 987 296 741 788 798 156 8 × 2 = 1 + 0.237 304 687 499 999 816 907 794 779 974 593 483 577 596 313 6;
  • 95) 0.237 304 687 499 999 816 907 794 779 974 593 483 577 596 313 6 × 2 = 0 + 0.474 609 374 999 999 633 815 589 559 949 186 967 155 192 627 2;
  • 96) 0.474 609 374 999 999 633 815 589 559 949 186 967 155 192 627 2 × 2 = 0 + 0.949 218 749 999 999 267 631 179 119 898 373 934 310 385 254 4;
  • 97) 0.949 218 749 999 999 267 631 179 119 898 373 934 310 385 254 4 × 2 = 1 + 0.898 437 499 999 998 535 262 358 239 796 747 868 620 770 508 8;
  • 98) 0.898 437 499 999 998 535 262 358 239 796 747 868 620 770 508 8 × 2 = 1 + 0.796 874 999 999 997 070 524 716 479 593 495 737 241 541 017 6;
  • 99) 0.796 874 999 999 997 070 524 716 479 593 495 737 241 541 017 6 × 2 = 1 + 0.593 749 999 999 994 141 049 432 959 186 991 474 483 082 035 2;
  • 100) 0.593 749 999 999 994 141 049 432 959 186 991 474 483 082 035 2 × 2 = 1 + 0.187 499 999 999 988 282 098 865 918 373 982 948 966 164 070 4;
  • 101) 0.187 499 999 999 988 282 098 865 918 373 982 948 966 164 070 4 × 2 = 0 + 0.374 999 999 999 976 564 197 731 836 747 965 897 932 328 140 8;
  • 102) 0.374 999 999 999 976 564 197 731 836 747 965 897 932 328 140 8 × 2 = 0 + 0.749 999 999 999 953 128 395 463 673 495 931 795 864 656 281 6;
  • 103) 0.749 999 999 999 953 128 395 463 673 495 931 795 864 656 281 6 × 2 = 1 + 0.499 999 999 999 906 256 790 927 346 991 863 591 729 312 563 2;
  • 104) 0.499 999 999 999 906 256 790 927 346 991 863 591 729 312 563 2 × 2 = 0 + 0.999 999 999 999 812 513 581 854 693 983 727 183 458 625 126 4;
  • 105) 0.999 999 999 999 812 513 581 854 693 983 727 183 458 625 126 4 × 2 = 1 + 0.999 999 999 999 625 027 163 709 387 967 454 366 917 250 252 8;
  • 106) 0.999 999 999 999 625 027 163 709 387 967 454 366 917 250 252 8 × 2 = 1 + 0.999 999 999 999 250 054 327 418 775 934 908 733 834 500 505 6;
  • 107) 0.999 999 999 999 250 054 327 418 775 934 908 733 834 500 505 6 × 2 = 1 + 0.999 999 999 998 500 108 654 837 551 869 817 467 669 001 011 2;
  • 108) 0.999 999 999 998 500 108 654 837 551 869 817 467 669 001 011 2 × 2 = 1 + 0.999 999 999 997 000 217 309 675 103 739 634 935 338 002 022 4;
  • 109) 0.999 999 999 997 000 217 309 675 103 739 634 935 338 002 022 4 × 2 = 1 + 0.999 999 999 994 000 434 619 350 207 479 269 870 676 004 044 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:


0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 357 9(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:

0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 357 9(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:


0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 357 9(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 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 357 9 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