0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 332 74 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 332 74(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 332 74(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 332 74.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 332 74 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 665 48;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 665 48 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 330 96;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 330 96 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 661 92;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 661 92 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 323 84;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 323 84 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 647 68;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 647 68 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 295 36;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 295 36 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 590 72;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 590 72 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 181 44;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 181 44 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 306 362 88;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 306 362 88 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 612 725 76;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 612 725 76 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 225 451 52;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 225 451 52 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 450 903 04;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 450 903 04 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 901 806 08;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 901 806 08 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 803 612 16;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 803 612 16 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 607 224 32;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 607 224 32 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 214 448 64;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 214 448 64 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 428 897 28;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 428 897 28 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 020 857 794 56;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 020 857 794 56 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 041 715 589 12;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 041 715 589 12 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 083 431 178 24;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 083 431 178 24 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 166 862 356 48;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 166 862 356 48 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 333 724 712 96;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 333 724 712 96 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 667 449 425 92;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 667 449 425 92 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 334 898 851 84;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 334 898 851 84 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 669 797 703 68;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 669 797 703 68 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 339 595 407 36;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 339 595 407 36 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 679 190 814 72;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 679 190 814 72 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 358 381 629 44;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 358 381 629 44 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 634 716 763 258 88;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 634 716 763 258 88 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 269 433 526 517 76;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 269 433 526 517 76 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 538 867 053 035 52;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 538 867 053 035 52 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 077 734 106 071 04;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 077 734 106 071 04 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 155 468 212 142 08;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 155 468 212 142 08 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 310 936 424 284 16;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 310 936 424 284 16 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 621 872 848 568 32;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 621 872 848 568 32 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 243 745 697 136 64;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 243 745 697 136 64 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 487 491 394 273 28;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 487 491 394 273 28 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 660 974 982 788 546 56;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 660 974 982 788 546 56 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 321 949 965 577 093 12;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 321 949 965 577 093 12 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 643 899 931 154 186 24;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 643 899 931 154 186 24 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 287 799 862 308 372 48;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 287 799 862 308 372 48 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 575 599 724 616 744 96;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 575 599 724 616 744 96 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 151 199 449 233 489 92;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 151 199 449 233 489 92 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 302 398 898 466 979 84;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 302 398 898 466 979 84 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 604 797 796 933 959 68;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 604 797 796 933 959 68 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 209 595 593 867 919 36;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 209 595 593 867 919 36 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 419 191 187 735 838 72;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 419 191 187 735 838 72 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 404 838 382 375 471 677 44;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 404 838 382 375 471 677 44 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 809 676 764 750 943 354 88;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 809 676 764 750 943 354 88 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 619 353 529 501 886 709 76;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 619 353 529 501 886 709 76 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 238 707 059 003 773 419 52;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 238 707 059 003 773 419 52 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 477 414 118 007 546 839 04;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 477 414 118 007 546 839 04 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 954 828 236 015 093 678 08;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 954 828 236 015 093 678 08 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 909 656 472 030 187 356 16;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 909 656 472 030 187 356 16 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 819 312 944 060 374 712 32;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 819 312 944 060 374 712 32 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 638 625 888 120 749 424 64;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 638 625 888 120 749 424 64 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 277 251 776 241 498 849 28;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 277 251 776 241 498 849 28 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 030 554 503 552 482 997 698 56;
  • 59) 0.417 311 472 119 763 493 537 902 832 030 554 503 552 482 997 698 56 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 061 109 007 104 965 995 397 12;
  • 60) 0.834 622 944 239 526 987 075 805 664 061 109 007 104 965 995 397 12 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 122 218 014 209 931 990 794 24;
  • 61) 0.669 245 888 479 053 974 151 611 328 122 218 014 209 931 990 794 24 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 244 436 028 419 863 981 588 48;
  • 62) 0.338 491 776 958 107 948 303 222 656 244 436 028 419 863 981 588 48 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 488 872 056 839 727 963 176 96;
  • 63) 0.676 983 553 916 215 896 606 445 312 488 872 056 839 727 963 176 96 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 977 744 113 679 455 926 353 92;
  • 64) 0.353 967 107 832 431 793 212 890 624 977 744 113 679 455 926 353 92 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 955 488 227 358 911 852 707 84;
  • 65) 0.707 934 215 664 863 586 425 781 249 955 488 227 358 911 852 707 84 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 910 976 454 717 823 705 415 68;
  • 66) 0.415 868 431 329 727 172 851 562 499 910 976 454 717 823 705 415 68 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 821 952 909 435 647 410 831 36;
  • 67) 0.831 736 862 659 454 345 703 124 999 821 952 909 435 647 410 831 36 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 643 905 818 871 294 821 662 72;
  • 68) 0.663 473 725 318 908 691 406 249 999 643 905 818 871 294 821 662 72 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 287 811 637 742 589 643 325 44;
  • 69) 0.326 947 450 637 817 382 812 499 999 287 811 637 742 589 643 325 44 × 2 = 0 + 0.653 894 901 275 634 765 624 999 998 575 623 275 485 179 286 650 88;
  • 70) 0.653 894 901 275 634 765 624 999 998 575 623 275 485 179 286 650 88 × 2 = 1 + 0.307 789 802 551 269 531 249 999 997 151 246 550 970 358 573 301 76;
  • 71) 0.307 789 802 551 269 531 249 999 997 151 246 550 970 358 573 301 76 × 2 = 0 + 0.615 579 605 102 539 062 499 999 994 302 493 101 940 717 146 603 52;
  • 72) 0.615 579 605 102 539 062 499 999 994 302 493 101 940 717 146 603 52 × 2 = 1 + 0.231 159 210 205 078 124 999 999 988 604 986 203 881 434 293 207 04;
  • 73) 0.231 159 210 205 078 124 999 999 988 604 986 203 881 434 293 207 04 × 2 = 0 + 0.462 318 420 410 156 249 999 999 977 209 972 407 762 868 586 414 08;
  • 74) 0.462 318 420 410 156 249 999 999 977 209 972 407 762 868 586 414 08 × 2 = 0 + 0.924 636 840 820 312 499 999 999 954 419 944 815 525 737 172 828 16;
  • 75) 0.924 636 840 820 312 499 999 999 954 419 944 815 525 737 172 828 16 × 2 = 1 + 0.849 273 681 640 624 999 999 999 908 839 889 631 051 474 345 656 32;
  • 76) 0.849 273 681 640 624 999 999 999 908 839 889 631 051 474 345 656 32 × 2 = 1 + 0.698 547 363 281 249 999 999 999 817 679 779 262 102 948 691 312 64;
  • 77) 0.698 547 363 281 249 999 999 999 817 679 779 262 102 948 691 312 64 × 2 = 1 + 0.397 094 726 562 499 999 999 999 635 359 558 524 205 897 382 625 28;
  • 78) 0.397 094 726 562 499 999 999 999 635 359 558 524 205 897 382 625 28 × 2 = 0 + 0.794 189 453 124 999 999 999 999 270 719 117 048 411 794 765 250 56;
  • 79) 0.794 189 453 124 999 999 999 999 270 719 117 048 411 794 765 250 56 × 2 = 1 + 0.588 378 906 249 999 999 999 998 541 438 234 096 823 589 530 501 12;
  • 80) 0.588 378 906 249 999 999 999 998 541 438 234 096 823 589 530 501 12 × 2 = 1 + 0.176 757 812 499 999 999 999 997 082 876 468 193 647 179 061 002 24;
  • 81) 0.176 757 812 499 999 999 999 997 082 876 468 193 647 179 061 002 24 × 2 = 0 + 0.353 515 624 999 999 999 999 994 165 752 936 387 294 358 122 004 48;
  • 82) 0.353 515 624 999 999 999 999 994 165 752 936 387 294 358 122 004 48 × 2 = 0 + 0.707 031 249 999 999 999 999 988 331 505 872 774 588 716 244 008 96;
  • 83) 0.707 031 249 999 999 999 999 988 331 505 872 774 588 716 244 008 96 × 2 = 1 + 0.414 062 499 999 999 999 999 976 663 011 745 549 177 432 488 017 92;
  • 84) 0.414 062 499 999 999 999 999 976 663 011 745 549 177 432 488 017 92 × 2 = 0 + 0.828 124 999 999 999 999 999 953 326 023 491 098 354 864 976 035 84;
  • 85) 0.828 124 999 999 999 999 999 953 326 023 491 098 354 864 976 035 84 × 2 = 1 + 0.656 249 999 999 999 999 999 906 652 046 982 196 709 729 952 071 68;
  • 86) 0.656 249 999 999 999 999 999 906 652 046 982 196 709 729 952 071 68 × 2 = 1 + 0.312 499 999 999 999 999 999 813 304 093 964 393 419 459 904 143 36;
  • 87) 0.312 499 999 999 999 999 999 813 304 093 964 393 419 459 904 143 36 × 2 = 0 + 0.624 999 999 999 999 999 999 626 608 187 928 786 838 919 808 286 72;
  • 88) 0.624 999 999 999 999 999 999 626 608 187 928 786 838 919 808 286 72 × 2 = 1 + 0.249 999 999 999 999 999 999 253 216 375 857 573 677 839 616 573 44;
  • 89) 0.249 999 999 999 999 999 999 253 216 375 857 573 677 839 616 573 44 × 2 = 0 + 0.499 999 999 999 999 999 998 506 432 751 715 147 355 679 233 146 88;
  • 90) 0.499 999 999 999 999 999 998 506 432 751 715 147 355 679 233 146 88 × 2 = 0 + 0.999 999 999 999 999 999 997 012 865 503 430 294 711 358 466 293 76;
  • 91) 0.999 999 999 999 999 999 997 012 865 503 430 294 711 358 466 293 76 × 2 = 1 + 0.999 999 999 999 999 999 994 025 731 006 860 589 422 716 932 587 52;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 332 74(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 332 74(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 39 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 332 74(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001(2) × 2-39


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): -39

Mantissa (not normalized):
1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-39 + 2(11-1) - 1 =

(-39 + 1 023)(10) =

984(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;
  • 984 ÷ 2 = 492 + 0;
  • 492 ÷ 2 = 246 + 0;
  • 246 ÷ 2 = 123 + 0;
  • 123 ÷ 2 = 61 + 1;
  • 61 ÷ 2 = 30 + 1;
  • 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) =

984(10) =

011 1101 1000(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. 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001 =

1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


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 1101 1000

Mantissa (52 bits) =
1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 332 74 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1101 1000 - 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001

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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