0.000 000 000 000 000 000 054 210 108 624 275 96 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 054 210 108 624 275 96(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
0.000 000 000 000 000 000 054 210 108 624 275 96(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: 0.000 000 000 000 000 000 054 210 108 624 275 96.

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) 0.000 000 000 000 000 000 054 210 108 624 275 96 × 2 = 0 + 0.000 000 000 000 000 000 108 420 217 248 551 92;
  • 2) 0.000 000 000 000 000 000 108 420 217 248 551 92 × 2 = 0 + 0.000 000 000 000 000 000 216 840 434 497 103 84;
  • 3) 0.000 000 000 000 000 000 216 840 434 497 103 84 × 2 = 0 + 0.000 000 000 000 000 000 433 680 868 994 207 68;
  • 4) 0.000 000 000 000 000 000 433 680 868 994 207 68 × 2 = 0 + 0.000 000 000 000 000 000 867 361 737 988 415 36;
  • 5) 0.000 000 000 000 000 000 867 361 737 988 415 36 × 2 = 0 + 0.000 000 000 000 000 001 734 723 475 976 830 72;
  • 6) 0.000 000 000 000 000 001 734 723 475 976 830 72 × 2 = 0 + 0.000 000 000 000 000 003 469 446 951 953 661 44;
  • 7) 0.000 000 000 000 000 003 469 446 951 953 661 44 × 2 = 0 + 0.000 000 000 000 000 006 938 893 903 907 322 88;
  • 8) 0.000 000 000 000 000 006 938 893 903 907 322 88 × 2 = 0 + 0.000 000 000 000 000 013 877 787 807 814 645 76;
  • 9) 0.000 000 000 000 000 013 877 787 807 814 645 76 × 2 = 0 + 0.000 000 000 000 000 027 755 575 615 629 291 52;
  • 10) 0.000 000 000 000 000 027 755 575 615 629 291 52 × 2 = 0 + 0.000 000 000 000 000 055 511 151 231 258 583 04;
  • 11) 0.000 000 000 000 000 055 511 151 231 258 583 04 × 2 = 0 + 0.000 000 000 000 000 111 022 302 462 517 166 08;
  • 12) 0.000 000 000 000 000 111 022 302 462 517 166 08 × 2 = 0 + 0.000 000 000 000 000 222 044 604 925 034 332 16;
  • 13) 0.000 000 000 000 000 222 044 604 925 034 332 16 × 2 = 0 + 0.000 000 000 000 000 444 089 209 850 068 664 32;
  • 14) 0.000 000 000 000 000 444 089 209 850 068 664 32 × 2 = 0 + 0.000 000 000 000 000 888 178 419 700 137 328 64;
  • 15) 0.000 000 000 000 000 888 178 419 700 137 328 64 × 2 = 0 + 0.000 000 000 000 001 776 356 839 400 274 657 28;
  • 16) 0.000 000 000 000 001 776 356 839 400 274 657 28 × 2 = 0 + 0.000 000 000 000 003 552 713 678 800 549 314 56;
  • 17) 0.000 000 000 000 003 552 713 678 800 549 314 56 × 2 = 0 + 0.000 000 000 000 007 105 427 357 601 098 629 12;
  • 18) 0.000 000 000 000 007 105 427 357 601 098 629 12 × 2 = 0 + 0.000 000 000 000 014 210 854 715 202 197 258 24;
  • 19) 0.000 000 000 000 014 210 854 715 202 197 258 24 × 2 = 0 + 0.000 000 000 000 028 421 709 430 404 394 516 48;
  • 20) 0.000 000 000 000 028 421 709 430 404 394 516 48 × 2 = 0 + 0.000 000 000 000 056 843 418 860 808 789 032 96;
  • 21) 0.000 000 000 000 056 843 418 860 808 789 032 96 × 2 = 0 + 0.000 000 000 000 113 686 837 721 617 578 065 92;
  • 22) 0.000 000 000 000 113 686 837 721 617 578 065 92 × 2 = 0 + 0.000 000 000 000 227 373 675 443 235 156 131 84;
  • 23) 0.000 000 000 000 227 373 675 443 235 156 131 84 × 2 = 0 + 0.000 000 000 000 454 747 350 886 470 312 263 68;
  • 24) 0.000 000 000 000 454 747 350 886 470 312 263 68 × 2 = 0 + 0.000 000 000 000 909 494 701 772 940 624 527 36;
  • 25) 0.000 000 000 000 909 494 701 772 940 624 527 36 × 2 = 0 + 0.000 000 000 001 818 989 403 545 881 249 054 72;
  • 26) 0.000 000 000 001 818 989 403 545 881 249 054 72 × 2 = 0 + 0.000 000 000 003 637 978 807 091 762 498 109 44;
  • 27) 0.000 000 000 003 637 978 807 091 762 498 109 44 × 2 = 0 + 0.000 000 000 007 275 957 614 183 524 996 218 88;
  • 28) 0.000 000 000 007 275 957 614 183 524 996 218 88 × 2 = 0 + 0.000 000 000 014 551 915 228 367 049 992 437 76;
  • 29) 0.000 000 000 014 551 915 228 367 049 992 437 76 × 2 = 0 + 0.000 000 000 029 103 830 456 734 099 984 875 52;
  • 30) 0.000 000 000 029 103 830 456 734 099 984 875 52 × 2 = 0 + 0.000 000 000 058 207 660 913 468 199 969 751 04;
  • 31) 0.000 000 000 058 207 660 913 468 199 969 751 04 × 2 = 0 + 0.000 000 000 116 415 321 826 936 399 939 502 08;
  • 32) 0.000 000 000 116 415 321 826 936 399 939 502 08 × 2 = 0 + 0.000 000 000 232 830 643 653 872 799 879 004 16;
  • 33) 0.000 000 000 232 830 643 653 872 799 879 004 16 × 2 = 0 + 0.000 000 000 465 661 287 307 745 599 758 008 32;
  • 34) 0.000 000 000 465 661 287 307 745 599 758 008 32 × 2 = 0 + 0.000 000 000 931 322 574 615 491 199 516 016 64;
  • 35) 0.000 000 000 931 322 574 615 491 199 516 016 64 × 2 = 0 + 0.000 000 001 862 645 149 230 982 399 032 033 28;
  • 36) 0.000 000 001 862 645 149 230 982 399 032 033 28 × 2 = 0 + 0.000 000 003 725 290 298 461 964 798 064 066 56;
  • 37) 0.000 000 003 725 290 298 461 964 798 064 066 56 × 2 = 0 + 0.000 000 007 450 580 596 923 929 596 128 133 12;
  • 38) 0.000 000 007 450 580 596 923 929 596 128 133 12 × 2 = 0 + 0.000 000 014 901 161 193 847 859 192 256 266 24;
  • 39) 0.000 000 014 901 161 193 847 859 192 256 266 24 × 2 = 0 + 0.000 000 029 802 322 387 695 718 384 512 532 48;
  • 40) 0.000 000 029 802 322 387 695 718 384 512 532 48 × 2 = 0 + 0.000 000 059 604 644 775 391 436 769 025 064 96;
  • 41) 0.000 000 059 604 644 775 391 436 769 025 064 96 × 2 = 0 + 0.000 000 119 209 289 550 782 873 538 050 129 92;
  • 42) 0.000 000 119 209 289 550 782 873 538 050 129 92 × 2 = 0 + 0.000 000 238 418 579 101 565 747 076 100 259 84;
  • 43) 0.000 000 238 418 579 101 565 747 076 100 259 84 × 2 = 0 + 0.000 000 476 837 158 203 131 494 152 200 519 68;
  • 44) 0.000 000 476 837 158 203 131 494 152 200 519 68 × 2 = 0 + 0.000 000 953 674 316 406 262 988 304 401 039 36;
  • 45) 0.000 000 953 674 316 406 262 988 304 401 039 36 × 2 = 0 + 0.000 001 907 348 632 812 525 976 608 802 078 72;
  • 46) 0.000 001 907 348 632 812 525 976 608 802 078 72 × 2 = 0 + 0.000 003 814 697 265 625 051 953 217 604 157 44;
  • 47) 0.000 003 814 697 265 625 051 953 217 604 157 44 × 2 = 0 + 0.000 007 629 394 531 250 103 906 435 208 314 88;
  • 48) 0.000 007 629 394 531 250 103 906 435 208 314 88 × 2 = 0 + 0.000 015 258 789 062 500 207 812 870 416 629 76;
  • 49) 0.000 015 258 789 062 500 207 812 870 416 629 76 × 2 = 0 + 0.000 030 517 578 125 000 415 625 740 833 259 52;
  • 50) 0.000 030 517 578 125 000 415 625 740 833 259 52 × 2 = 0 + 0.000 061 035 156 250 000 831 251 481 666 519 04;
  • 51) 0.000 061 035 156 250 000 831 251 481 666 519 04 × 2 = 0 + 0.000 122 070 312 500 001 662 502 963 333 038 08;
  • 52) 0.000 122 070 312 500 001 662 502 963 333 038 08 × 2 = 0 + 0.000 244 140 625 000 003 325 005 926 666 076 16;
  • 53) 0.000 244 140 625 000 003 325 005 926 666 076 16 × 2 = 0 + 0.000 488 281 250 000 006 650 011 853 332 152 32;
  • 54) 0.000 488 281 250 000 006 650 011 853 332 152 32 × 2 = 0 + 0.000 976 562 500 000 013 300 023 706 664 304 64;
  • 55) 0.000 976 562 500 000 013 300 023 706 664 304 64 × 2 = 0 + 0.001 953 125 000 000 026 600 047 413 328 609 28;
  • 56) 0.001 953 125 000 000 026 600 047 413 328 609 28 × 2 = 0 + 0.003 906 250 000 000 053 200 094 826 657 218 56;
  • 57) 0.003 906 250 000 000 053 200 094 826 657 218 56 × 2 = 0 + 0.007 812 500 000 000 106 400 189 653 314 437 12;
  • 58) 0.007 812 500 000 000 106 400 189 653 314 437 12 × 2 = 0 + 0.015 625 000 000 000 212 800 379 306 628 874 24;
  • 59) 0.015 625 000 000 000 212 800 379 306 628 874 24 × 2 = 0 + 0.031 250 000 000 000 425 600 758 613 257 748 48;
  • 60) 0.031 250 000 000 000 425 600 758 613 257 748 48 × 2 = 0 + 0.062 500 000 000 000 851 201 517 226 515 496 96;
  • 61) 0.062 500 000 000 000 851 201 517 226 515 496 96 × 2 = 0 + 0.125 000 000 000 001 702 403 034 453 030 993 92;
  • 62) 0.125 000 000 000 001 702 403 034 453 030 993 92 × 2 = 0 + 0.250 000 000 000 003 404 806 068 906 061 987 84;
  • 63) 0.250 000 000 000 003 404 806 068 906 061 987 84 × 2 = 0 + 0.500 000 000 000 006 809 612 137 812 123 975 68;
  • 64) 0.500 000 000 000 006 809 612 137 812 123 975 68 × 2 = 1 + 0.000 000 000 000 013 619 224 275 624 247 951 36;
  • 65) 0.000 000 000 000 013 619 224 275 624 247 951 36 × 2 = 0 + 0.000 000 000 000 027 238 448 551 248 495 902 72;
  • 66) 0.000 000 000 000 027 238 448 551 248 495 902 72 × 2 = 0 + 0.000 000 000 000 054 476 897 102 496 991 805 44;
  • 67) 0.000 000 000 000 054 476 897 102 496 991 805 44 × 2 = 0 + 0.000 000 000 000 108 953 794 204 993 983 610 88;
  • 68) 0.000 000 000 000 108 953 794 204 993 983 610 88 × 2 = 0 + 0.000 000 000 000 217 907 588 409 987 967 221 76;
  • 69) 0.000 000 000 000 217 907 588 409 987 967 221 76 × 2 = 0 + 0.000 000 000 000 435 815 176 819 975 934 443 52;
  • 70) 0.000 000 000 000 435 815 176 819 975 934 443 52 × 2 = 0 + 0.000 000 000 000 871 630 353 639 951 868 887 04;
  • 71) 0.000 000 000 000 871 630 353 639 951 868 887 04 × 2 = 0 + 0.000 000 000 001 743 260 707 279 903 737 774 08;
  • 72) 0.000 000 000 001 743 260 707 279 903 737 774 08 × 2 = 0 + 0.000 000 000 003 486 521 414 559 807 475 548 16;
  • 73) 0.000 000 000 003 486 521 414 559 807 475 548 16 × 2 = 0 + 0.000 000 000 006 973 042 829 119 614 951 096 32;
  • 74) 0.000 000 000 006 973 042 829 119 614 951 096 32 × 2 = 0 + 0.000 000 000 013 946 085 658 239 229 902 192 64;
  • 75) 0.000 000 000 013 946 085 658 239 229 902 192 64 × 2 = 0 + 0.000 000 000 027 892 171 316 478 459 804 385 28;
  • 76) 0.000 000 000 027 892 171 316 478 459 804 385 28 × 2 = 0 + 0.000 000 000 055 784 342 632 956 919 608 770 56;
  • 77) 0.000 000 000 055 784 342 632 956 919 608 770 56 × 2 = 0 + 0.000 000 000 111 568 685 265 913 839 217 541 12;
  • 78) 0.000 000 000 111 568 685 265 913 839 217 541 12 × 2 = 0 + 0.000 000 000 223 137 370 531 827 678 435 082 24;
  • 79) 0.000 000 000 223 137 370 531 827 678 435 082 24 × 2 = 0 + 0.000 000 000 446 274 741 063 655 356 870 164 48;
  • 80) 0.000 000 000 446 274 741 063 655 356 870 164 48 × 2 = 0 + 0.000 000 000 892 549 482 127 310 713 740 328 96;
  • 81) 0.000 000 000 892 549 482 127 310 713 740 328 96 × 2 = 0 + 0.000 000 001 785 098 964 254 621 427 480 657 92;
  • 82) 0.000 000 001 785 098 964 254 621 427 480 657 92 × 2 = 0 + 0.000 000 003 570 197 928 509 242 854 961 315 84;
  • 83) 0.000 000 003 570 197 928 509 242 854 961 315 84 × 2 = 0 + 0.000 000 007 140 395 857 018 485 709 922 631 68;
  • 84) 0.000 000 007 140 395 857 018 485 709 922 631 68 × 2 = 0 + 0.000 000 014 280 791 714 036 971 419 845 263 36;
  • 85) 0.000 000 014 280 791 714 036 971 419 845 263 36 × 2 = 0 + 0.000 000 028 561 583 428 073 942 839 690 526 72;
  • 86) 0.000 000 028 561 583 428 073 942 839 690 526 72 × 2 = 0 + 0.000 000 057 123 166 856 147 885 679 381 053 44;
  • 87) 0.000 000 057 123 166 856 147 885 679 381 053 44 × 2 = 0 + 0.000 000 114 246 333 712 295 771 358 762 106 88;
  • 88) 0.000 000 114 246 333 712 295 771 358 762 106 88 × 2 = 0 + 0.000 000 228 492 667 424 591 542 717 524 213 76;
  • 89) 0.000 000 228 492 667 424 591 542 717 524 213 76 × 2 = 0 + 0.000 000 456 985 334 849 183 085 435 048 427 52;
  • 90) 0.000 000 456 985 334 849 183 085 435 048 427 52 × 2 = 0 + 0.000 000 913 970 669 698 366 170 870 096 855 04;
  • 91) 0.000 000 913 970 669 698 366 170 870 096 855 04 × 2 = 0 + 0.000 001 827 941 339 396 732 341 740 193 710 08;
  • 92) 0.000 001 827 941 339 396 732 341 740 193 710 08 × 2 = 0 + 0.000 003 655 882 678 793 464 683 480 387 420 16;
  • 93) 0.000 003 655 882 678 793 464 683 480 387 420 16 × 2 = 0 + 0.000 007 311 765 357 586 929 366 960 774 840 32;
  • 94) 0.000 007 311 765 357 586 929 366 960 774 840 32 × 2 = 0 + 0.000 014 623 530 715 173 858 733 921 549 680 64;
  • 95) 0.000 014 623 530 715 173 858 733 921 549 680 64 × 2 = 0 + 0.000 029 247 061 430 347 717 467 843 099 361 28;
  • 96) 0.000 029 247 061 430 347 717 467 843 099 361 28 × 2 = 0 + 0.000 058 494 122 860 695 434 935 686 198 722 56;
  • 97) 0.000 058 494 122 860 695 434 935 686 198 722 56 × 2 = 0 + 0.000 116 988 245 721 390 869 871 372 397 445 12;
  • 98) 0.000 116 988 245 721 390 869 871 372 397 445 12 × 2 = 0 + 0.000 233 976 491 442 781 739 742 744 794 890 24;
  • 99) 0.000 233 976 491 442 781 739 742 744 794 890 24 × 2 = 0 + 0.000 467 952 982 885 563 479 485 489 589 780 48;
  • 100) 0.000 467 952 982 885 563 479 485 489 589 780 48 × 2 = 0 + 0.000 935 905 965 771 126 958 970 979 179 560 96;
  • 101) 0.000 935 905 965 771 126 958 970 979 179 560 96 × 2 = 0 + 0.001 871 811 931 542 253 917 941 958 359 121 92;
  • 102) 0.001 871 811 931 542 253 917 941 958 359 121 92 × 2 = 0 + 0.003 743 623 863 084 507 835 883 916 718 243 84;
  • 103) 0.003 743 623 863 084 507 835 883 916 718 243 84 × 2 = 0 + 0.007 487 247 726 169 015 671 767 833 436 487 68;
  • 104) 0.007 487 247 726 169 015 671 767 833 436 487 68 × 2 = 0 + 0.014 974 495 452 338 031 343 535 666 872 975 36;
  • 105) 0.014 974 495 452 338 031 343 535 666 872 975 36 × 2 = 0 + 0.029 948 990 904 676 062 687 071 333 745 950 72;
  • 106) 0.029 948 990 904 676 062 687 071 333 745 950 72 × 2 = 0 + 0.059 897 981 809 352 125 374 142 667 491 901 44;
  • 107) 0.059 897 981 809 352 125 374 142 667 491 901 44 × 2 = 0 + 0.119 795 963 618 704 250 748 285 334 983 802 88;
  • 108) 0.119 795 963 618 704 250 748 285 334 983 802 88 × 2 = 0 + 0.239 591 927 237 408 501 496 570 669 967 605 76;
  • 109) 0.239 591 927 237 408 501 496 570 669 967 605 76 × 2 = 0 + 0.479 183 854 474 817 002 993 141 339 935 211 52;
  • 110) 0.479 183 854 474 817 002 993 141 339 935 211 52 × 2 = 0 + 0.958 367 708 949 634 005 986 282 679 870 423 04;
  • 111) 0.958 367 708 949 634 005 986 282 679 870 423 04 × 2 = 1 + 0.916 735 417 899 268 011 972 565 359 740 846 08;
  • 112) 0.916 735 417 899 268 011 972 565 359 740 846 08 × 2 = 1 + 0.833 470 835 798 536 023 945 130 719 481 692 16;
  • 113) 0.833 470 835 798 536 023 945 130 719 481 692 16 × 2 = 1 + 0.666 941 671 597 072 047 890 261 438 963 384 32;
  • 114) 0.666 941 671 597 072 047 890 261 438 963 384 32 × 2 = 1 + 0.333 883 343 194 144 095 780 522 877 926 768 64;
  • 115) 0.333 883 343 194 144 095 780 522 877 926 768 64 × 2 = 0 + 0.667 766 686 388 288 191 561 045 755 853 537 28;
  • 116) 0.667 766 686 388 288 191 561 045 755 853 537 28 × 2 = 1 + 0.335 533 372 776 576 383 122 091 511 707 074 56;

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:


0.000 000 000 000 000 000 054 210 108 624 275 96(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1101(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 054 210 108 624 275 96(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1101(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 64 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 054 210 108 624 275 96(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1101(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1101(2) × 20 =

1.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1101(2) × 2-64


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): -64

Mantissa (not normalized):
1.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1101


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-64 + 2(11-1) - 1 =

(-64 + 1 023)(10) =

959(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;
  • 959 ÷ 2 = 479 + 1;
  • 479 ÷ 2 = 239 + 1;
  • 239 ÷ 2 = 119 + 1;
  • 119 ÷ 2 = 59 + 1;
  • 59 ÷ 2 = 29 + 1;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 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) =

959(10) =

011 1011 1111(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. 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1101 =

0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1101


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 1011 1111

Mantissa (52 bits) =
0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1101


Decimal number 0.000 000 000 000 000 000 054 210 108 624 275 96 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1111 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1101

» Convert decimal 0.000 000 000 000 000 000 054 210 108 624 275 68 to 64 bit double precision IEEE 754 binary floating point representation standard

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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