0.000 000 000 000 000 012 345 687 894 564 589 438 730 1 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 012 345 687 894 564 589 438 730 1(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
0.000 000 000 000 000 012 345 687 894 564 589 438 730 1(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: 0.000 000 000 000 000 012 345 687 894 564 589 438 730 1.

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) 0.000 000 000 000 000 012 345 687 894 564 589 438 730 1 × 2 = 0 + 0.000 000 000 000 000 024 691 375 789 129 178 877 460 2;
  • 2) 0.000 000 000 000 000 024 691 375 789 129 178 877 460 2 × 2 = 0 + 0.000 000 000 000 000 049 382 751 578 258 357 754 920 4;
  • 3) 0.000 000 000 000 000 049 382 751 578 258 357 754 920 4 × 2 = 0 + 0.000 000 000 000 000 098 765 503 156 516 715 509 840 8;
  • 4) 0.000 000 000 000 000 098 765 503 156 516 715 509 840 8 × 2 = 0 + 0.000 000 000 000 000 197 531 006 313 033 431 019 681 6;
  • 5) 0.000 000 000 000 000 197 531 006 313 033 431 019 681 6 × 2 = 0 + 0.000 000 000 000 000 395 062 012 626 066 862 039 363 2;
  • 6) 0.000 000 000 000 000 395 062 012 626 066 862 039 363 2 × 2 = 0 + 0.000 000 000 000 000 790 124 025 252 133 724 078 726 4;
  • 7) 0.000 000 000 000 000 790 124 025 252 133 724 078 726 4 × 2 = 0 + 0.000 000 000 000 001 580 248 050 504 267 448 157 452 8;
  • 8) 0.000 000 000 000 001 580 248 050 504 267 448 157 452 8 × 2 = 0 + 0.000 000 000 000 003 160 496 101 008 534 896 314 905 6;
  • 9) 0.000 000 000 000 003 160 496 101 008 534 896 314 905 6 × 2 = 0 + 0.000 000 000 000 006 320 992 202 017 069 792 629 811 2;
  • 10) 0.000 000 000 000 006 320 992 202 017 069 792 629 811 2 × 2 = 0 + 0.000 000 000 000 012 641 984 404 034 139 585 259 622 4;
  • 11) 0.000 000 000 000 012 641 984 404 034 139 585 259 622 4 × 2 = 0 + 0.000 000 000 000 025 283 968 808 068 279 170 519 244 8;
  • 12) 0.000 000 000 000 025 283 968 808 068 279 170 519 244 8 × 2 = 0 + 0.000 000 000 000 050 567 937 616 136 558 341 038 489 6;
  • 13) 0.000 000 000 000 050 567 937 616 136 558 341 038 489 6 × 2 = 0 + 0.000 000 000 000 101 135 875 232 273 116 682 076 979 2;
  • 14) 0.000 000 000 000 101 135 875 232 273 116 682 076 979 2 × 2 = 0 + 0.000 000 000 000 202 271 750 464 546 233 364 153 958 4;
  • 15) 0.000 000 000 000 202 271 750 464 546 233 364 153 958 4 × 2 = 0 + 0.000 000 000 000 404 543 500 929 092 466 728 307 916 8;
  • 16) 0.000 000 000 000 404 543 500 929 092 466 728 307 916 8 × 2 = 0 + 0.000 000 000 000 809 087 001 858 184 933 456 615 833 6;
  • 17) 0.000 000 000 000 809 087 001 858 184 933 456 615 833 6 × 2 = 0 + 0.000 000 000 001 618 174 003 716 369 866 913 231 667 2;
  • 18) 0.000 000 000 001 618 174 003 716 369 866 913 231 667 2 × 2 = 0 + 0.000 000 000 003 236 348 007 432 739 733 826 463 334 4;
  • 19) 0.000 000 000 003 236 348 007 432 739 733 826 463 334 4 × 2 = 0 + 0.000 000 000 006 472 696 014 865 479 467 652 926 668 8;
  • 20) 0.000 000 000 006 472 696 014 865 479 467 652 926 668 8 × 2 = 0 + 0.000 000 000 012 945 392 029 730 958 935 305 853 337 6;
  • 21) 0.000 000 000 012 945 392 029 730 958 935 305 853 337 6 × 2 = 0 + 0.000 000 000 025 890 784 059 461 917 870 611 706 675 2;
  • 22) 0.000 000 000 025 890 784 059 461 917 870 611 706 675 2 × 2 = 0 + 0.000 000 000 051 781 568 118 923 835 741 223 413 350 4;
  • 23) 0.000 000 000 051 781 568 118 923 835 741 223 413 350 4 × 2 = 0 + 0.000 000 000 103 563 136 237 847 671 482 446 826 700 8;
  • 24) 0.000 000 000 103 563 136 237 847 671 482 446 826 700 8 × 2 = 0 + 0.000 000 000 207 126 272 475 695 342 964 893 653 401 6;
  • 25) 0.000 000 000 207 126 272 475 695 342 964 893 653 401 6 × 2 = 0 + 0.000 000 000 414 252 544 951 390 685 929 787 306 803 2;
  • 26) 0.000 000 000 414 252 544 951 390 685 929 787 306 803 2 × 2 = 0 + 0.000 000 000 828 505 089 902 781 371 859 574 613 606 4;
  • 27) 0.000 000 000 828 505 089 902 781 371 859 574 613 606 4 × 2 = 0 + 0.000 000 001 657 010 179 805 562 743 719 149 227 212 8;
  • 28) 0.000 000 001 657 010 179 805 562 743 719 149 227 212 8 × 2 = 0 + 0.000 000 003 314 020 359 611 125 487 438 298 454 425 6;
  • 29) 0.000 000 003 314 020 359 611 125 487 438 298 454 425 6 × 2 = 0 + 0.000 000 006 628 040 719 222 250 974 876 596 908 851 2;
  • 30) 0.000 000 006 628 040 719 222 250 974 876 596 908 851 2 × 2 = 0 + 0.000 000 013 256 081 438 444 501 949 753 193 817 702 4;
  • 31) 0.000 000 013 256 081 438 444 501 949 753 193 817 702 4 × 2 = 0 + 0.000 000 026 512 162 876 889 003 899 506 387 635 404 8;
  • 32) 0.000 000 026 512 162 876 889 003 899 506 387 635 404 8 × 2 = 0 + 0.000 000 053 024 325 753 778 007 799 012 775 270 809 6;
  • 33) 0.000 000 053 024 325 753 778 007 799 012 775 270 809 6 × 2 = 0 + 0.000 000 106 048 651 507 556 015 598 025 550 541 619 2;
  • 34) 0.000 000 106 048 651 507 556 015 598 025 550 541 619 2 × 2 = 0 + 0.000 000 212 097 303 015 112 031 196 051 101 083 238 4;
  • 35) 0.000 000 212 097 303 015 112 031 196 051 101 083 238 4 × 2 = 0 + 0.000 000 424 194 606 030 224 062 392 102 202 166 476 8;
  • 36) 0.000 000 424 194 606 030 224 062 392 102 202 166 476 8 × 2 = 0 + 0.000 000 848 389 212 060 448 124 784 204 404 332 953 6;
  • 37) 0.000 000 848 389 212 060 448 124 784 204 404 332 953 6 × 2 = 0 + 0.000 001 696 778 424 120 896 249 568 408 808 665 907 2;
  • 38) 0.000 001 696 778 424 120 896 249 568 408 808 665 907 2 × 2 = 0 + 0.000 003 393 556 848 241 792 499 136 817 617 331 814 4;
  • 39) 0.000 003 393 556 848 241 792 499 136 817 617 331 814 4 × 2 = 0 + 0.000 006 787 113 696 483 584 998 273 635 234 663 628 8;
  • 40) 0.000 006 787 113 696 483 584 998 273 635 234 663 628 8 × 2 = 0 + 0.000 013 574 227 392 967 169 996 547 270 469 327 257 6;
  • 41) 0.000 013 574 227 392 967 169 996 547 270 469 327 257 6 × 2 = 0 + 0.000 027 148 454 785 934 339 993 094 540 938 654 515 2;
  • 42) 0.000 027 148 454 785 934 339 993 094 540 938 654 515 2 × 2 = 0 + 0.000 054 296 909 571 868 679 986 189 081 877 309 030 4;
  • 43) 0.000 054 296 909 571 868 679 986 189 081 877 309 030 4 × 2 = 0 + 0.000 108 593 819 143 737 359 972 378 163 754 618 060 8;
  • 44) 0.000 108 593 819 143 737 359 972 378 163 754 618 060 8 × 2 = 0 + 0.000 217 187 638 287 474 719 944 756 327 509 236 121 6;
  • 45) 0.000 217 187 638 287 474 719 944 756 327 509 236 121 6 × 2 = 0 + 0.000 434 375 276 574 949 439 889 512 655 018 472 243 2;
  • 46) 0.000 434 375 276 574 949 439 889 512 655 018 472 243 2 × 2 = 0 + 0.000 868 750 553 149 898 879 779 025 310 036 944 486 4;
  • 47) 0.000 868 750 553 149 898 879 779 025 310 036 944 486 4 × 2 = 0 + 0.001 737 501 106 299 797 759 558 050 620 073 888 972 8;
  • 48) 0.001 737 501 106 299 797 759 558 050 620 073 888 972 8 × 2 = 0 + 0.003 475 002 212 599 595 519 116 101 240 147 777 945 6;
  • 49) 0.003 475 002 212 599 595 519 116 101 240 147 777 945 6 × 2 = 0 + 0.006 950 004 425 199 191 038 232 202 480 295 555 891 2;
  • 50) 0.006 950 004 425 199 191 038 232 202 480 295 555 891 2 × 2 = 0 + 0.013 900 008 850 398 382 076 464 404 960 591 111 782 4;
  • 51) 0.013 900 008 850 398 382 076 464 404 960 591 111 782 4 × 2 = 0 + 0.027 800 017 700 796 764 152 928 809 921 182 223 564 8;
  • 52) 0.027 800 017 700 796 764 152 928 809 921 182 223 564 8 × 2 = 0 + 0.055 600 035 401 593 528 305 857 619 842 364 447 129 6;
  • 53) 0.055 600 035 401 593 528 305 857 619 842 364 447 129 6 × 2 = 0 + 0.111 200 070 803 187 056 611 715 239 684 728 894 259 2;
  • 54) 0.111 200 070 803 187 056 611 715 239 684 728 894 259 2 × 2 = 0 + 0.222 400 141 606 374 113 223 430 479 369 457 788 518 4;
  • 55) 0.222 400 141 606 374 113 223 430 479 369 457 788 518 4 × 2 = 0 + 0.444 800 283 212 748 226 446 860 958 738 915 577 036 8;
  • 56) 0.444 800 283 212 748 226 446 860 958 738 915 577 036 8 × 2 = 0 + 0.889 600 566 425 496 452 893 721 917 477 831 154 073 6;
  • 57) 0.889 600 566 425 496 452 893 721 917 477 831 154 073 6 × 2 = 1 + 0.779 201 132 850 992 905 787 443 834 955 662 308 147 2;
  • 58) 0.779 201 132 850 992 905 787 443 834 955 662 308 147 2 × 2 = 1 + 0.558 402 265 701 985 811 574 887 669 911 324 616 294 4;
  • 59) 0.558 402 265 701 985 811 574 887 669 911 324 616 294 4 × 2 = 1 + 0.116 804 531 403 971 623 149 775 339 822 649 232 588 8;
  • 60) 0.116 804 531 403 971 623 149 775 339 822 649 232 588 8 × 2 = 0 + 0.233 609 062 807 943 246 299 550 679 645 298 465 177 6;
  • 61) 0.233 609 062 807 943 246 299 550 679 645 298 465 177 6 × 2 = 0 + 0.467 218 125 615 886 492 599 101 359 290 596 930 355 2;
  • 62) 0.467 218 125 615 886 492 599 101 359 290 596 930 355 2 × 2 = 0 + 0.934 436 251 231 772 985 198 202 718 581 193 860 710 4;
  • 63) 0.934 436 251 231 772 985 198 202 718 581 193 860 710 4 × 2 = 1 + 0.868 872 502 463 545 970 396 405 437 162 387 721 420 8;
  • 64) 0.868 872 502 463 545 970 396 405 437 162 387 721 420 8 × 2 = 1 + 0.737 745 004 927 091 940 792 810 874 324 775 442 841 6;
  • 65) 0.737 745 004 927 091 940 792 810 874 324 775 442 841 6 × 2 = 1 + 0.475 490 009 854 183 881 585 621 748 649 550 885 683 2;
  • 66) 0.475 490 009 854 183 881 585 621 748 649 550 885 683 2 × 2 = 0 + 0.950 980 019 708 367 763 171 243 497 299 101 771 366 4;
  • 67) 0.950 980 019 708 367 763 171 243 497 299 101 771 366 4 × 2 = 1 + 0.901 960 039 416 735 526 342 486 994 598 203 542 732 8;
  • 68) 0.901 960 039 416 735 526 342 486 994 598 203 542 732 8 × 2 = 1 + 0.803 920 078 833 471 052 684 973 989 196 407 085 465 6;
  • 69) 0.803 920 078 833 471 052 684 973 989 196 407 085 465 6 × 2 = 1 + 0.607 840 157 666 942 105 369 947 978 392 814 170 931 2;
  • 70) 0.607 840 157 666 942 105 369 947 978 392 814 170 931 2 × 2 = 1 + 0.215 680 315 333 884 210 739 895 956 785 628 341 862 4;
  • 71) 0.215 680 315 333 884 210 739 895 956 785 628 341 862 4 × 2 = 0 + 0.431 360 630 667 768 421 479 791 913 571 256 683 724 8;
  • 72) 0.431 360 630 667 768 421 479 791 913 571 256 683 724 8 × 2 = 0 + 0.862 721 261 335 536 842 959 583 827 142 513 367 449 6;
  • 73) 0.862 721 261 335 536 842 959 583 827 142 513 367 449 6 × 2 = 1 + 0.725 442 522 671 073 685 919 167 654 285 026 734 899 2;
  • 74) 0.725 442 522 671 073 685 919 167 654 285 026 734 899 2 × 2 = 1 + 0.450 885 045 342 147 371 838 335 308 570 053 469 798 4;
  • 75) 0.450 885 045 342 147 371 838 335 308 570 053 469 798 4 × 2 = 0 + 0.901 770 090 684 294 743 676 670 617 140 106 939 596 8;
  • 76) 0.901 770 090 684 294 743 676 670 617 140 106 939 596 8 × 2 = 1 + 0.803 540 181 368 589 487 353 341 234 280 213 879 193 6;
  • 77) 0.803 540 181 368 589 487 353 341 234 280 213 879 193 6 × 2 = 1 + 0.607 080 362 737 178 974 706 682 468 560 427 758 387 2;
  • 78) 0.607 080 362 737 178 974 706 682 468 560 427 758 387 2 × 2 = 1 + 0.214 160 725 474 357 949 413 364 937 120 855 516 774 4;
  • 79) 0.214 160 725 474 357 949 413 364 937 120 855 516 774 4 × 2 = 0 + 0.428 321 450 948 715 898 826 729 874 241 711 033 548 8;
  • 80) 0.428 321 450 948 715 898 826 729 874 241 711 033 548 8 × 2 = 0 + 0.856 642 901 897 431 797 653 459 748 483 422 067 097 6;
  • 81) 0.856 642 901 897 431 797 653 459 748 483 422 067 097 6 × 2 = 1 + 0.713 285 803 794 863 595 306 919 496 966 844 134 195 2;
  • 82) 0.713 285 803 794 863 595 306 919 496 966 844 134 195 2 × 2 = 1 + 0.426 571 607 589 727 190 613 838 993 933 688 268 390 4;
  • 83) 0.426 571 607 589 727 190 613 838 993 933 688 268 390 4 × 2 = 0 + 0.853 143 215 179 454 381 227 677 987 867 376 536 780 8;
  • 84) 0.853 143 215 179 454 381 227 677 987 867 376 536 780 8 × 2 = 1 + 0.706 286 430 358 908 762 455 355 975 734 753 073 561 6;
  • 85) 0.706 286 430 358 908 762 455 355 975 734 753 073 561 6 × 2 = 1 + 0.412 572 860 717 817 524 910 711 951 469 506 147 123 2;
  • 86) 0.412 572 860 717 817 524 910 711 951 469 506 147 123 2 × 2 = 0 + 0.825 145 721 435 635 049 821 423 902 939 012 294 246 4;
  • 87) 0.825 145 721 435 635 049 821 423 902 939 012 294 246 4 × 2 = 1 + 0.650 291 442 871 270 099 642 847 805 878 024 588 492 8;
  • 88) 0.650 291 442 871 270 099 642 847 805 878 024 588 492 8 × 2 = 1 + 0.300 582 885 742 540 199 285 695 611 756 049 176 985 6;
  • 89) 0.300 582 885 742 540 199 285 695 611 756 049 176 985 6 × 2 = 0 + 0.601 165 771 485 080 398 571 391 223 512 098 353 971 2;
  • 90) 0.601 165 771 485 080 398 571 391 223 512 098 353 971 2 × 2 = 1 + 0.202 331 542 970 160 797 142 782 447 024 196 707 942 4;
  • 91) 0.202 331 542 970 160 797 142 782 447 024 196 707 942 4 × 2 = 0 + 0.404 663 085 940 321 594 285 564 894 048 393 415 884 8;
  • 92) 0.404 663 085 940 321 594 285 564 894 048 393 415 884 8 × 2 = 0 + 0.809 326 171 880 643 188 571 129 788 096 786 831 769 6;
  • 93) 0.809 326 171 880 643 188 571 129 788 096 786 831 769 6 × 2 = 1 + 0.618 652 343 761 286 377 142 259 576 193 573 663 539 2;
  • 94) 0.618 652 343 761 286 377 142 259 576 193 573 663 539 2 × 2 = 1 + 0.237 304 687 522 572 754 284 519 152 387 147 327 078 4;
  • 95) 0.237 304 687 522 572 754 284 519 152 387 147 327 078 4 × 2 = 0 + 0.474 609 375 045 145 508 569 038 304 774 294 654 156 8;
  • 96) 0.474 609 375 045 145 508 569 038 304 774 294 654 156 8 × 2 = 0 + 0.949 218 750 090 291 017 138 076 609 548 589 308 313 6;
  • 97) 0.949 218 750 090 291 017 138 076 609 548 589 308 313 6 × 2 = 1 + 0.898 437 500 180 582 034 276 153 219 097 178 616 627 2;
  • 98) 0.898 437 500 180 582 034 276 153 219 097 178 616 627 2 × 2 = 1 + 0.796 875 000 361 164 068 552 306 438 194 357 233 254 4;
  • 99) 0.796 875 000 361 164 068 552 306 438 194 357 233 254 4 × 2 = 1 + 0.593 750 000 722 328 137 104 612 876 388 714 466 508 8;
  • 100) 0.593 750 000 722 328 137 104 612 876 388 714 466 508 8 × 2 = 1 + 0.187 500 001 444 656 274 209 225 752 777 428 933 017 6;
  • 101) 0.187 500 001 444 656 274 209 225 752 777 428 933 017 6 × 2 = 0 + 0.375 000 002 889 312 548 418 451 505 554 857 866 035 2;
  • 102) 0.375 000 002 889 312 548 418 451 505 554 857 866 035 2 × 2 = 0 + 0.750 000 005 778 625 096 836 903 011 109 715 732 070 4;
  • 103) 0.750 000 005 778 625 096 836 903 011 109 715 732 070 4 × 2 = 1 + 0.500 000 011 557 250 193 673 806 022 219 431 464 140 8;
  • 104) 0.500 000 011 557 250 193 673 806 022 219 431 464 140 8 × 2 = 1 + 0.000 000 023 114 500 387 347 612 044 438 862 928 281 6;
  • 105) 0.000 000 023 114 500 387 347 612 044 438 862 928 281 6 × 2 = 0 + 0.000 000 046 229 000 774 695 224 088 877 725 856 563 2;
  • 106) 0.000 000 046 229 000 774 695 224 088 877 725 856 563 2 × 2 = 0 + 0.000 000 092 458 001 549 390 448 177 755 451 713 126 4;
  • 107) 0.000 000 092 458 001 549 390 448 177 755 451 713 126 4 × 2 = 0 + 0.000 000 184 916 003 098 780 896 355 510 903 426 252 8;
  • 108) 0.000 000 184 916 003 098 780 896 355 510 903 426 252 8 × 2 = 0 + 0.000 000 369 832 006 197 561 792 711 021 806 852 505 6;
  • 109) 0.000 000 369 832 006 197 561 792 711 021 806 852 505 6 × 2 = 0 + 0.000 000 739 664 012 395 123 585 422 043 613 705 011 2;

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:


0.000 000 000 000 000 012 345 687 894 564 589 438 730 1(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 0011 0000 0(2)

5. Positive number before normalization:

0.000 000 000 000 000 012 345 687 894 564 589 438 730 1(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 0011 0000 0(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:


0.000 000 000 000 000 012 345 687 894 564 589 438 730 1(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 0011 0000 0(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 0011 0000 0(2) × 20 =

1.1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0110 0000(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 0110 0000


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 0110 0000 =

1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0110 0000


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 0110 0000


Decimal number 0.000 000 000 000 000 012 345 687 894 564 589 438 730 1 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 0110 0000

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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