0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 147 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 147 8(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 147 8(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 147 8.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 147 8 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 295 6;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 295 6 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 591 2;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 591 2 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 182 4;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 182 4 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 364 8;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 364 8 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 729 6;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 729 6 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 459 2;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 459 2 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 898 918 4;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 898 918 4 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 797 836 8;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 797 836 8 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 595 673 6;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 595 673 6 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 191 347 2;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 191 347 2 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 382 694 4;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 382 694 4 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 765 388 8;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 765 388 8 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 530 777 6;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 530 777 6 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 061 555 2;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 061 555 2 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 123 110 4;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 123 110 4 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 246 220 8;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 246 220 8 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 744 492 441 6;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 744 492 441 6 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 488 984 883 2;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 488 984 883 2 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 977 969 766 4;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 977 969 766 4 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 955 939 532 8;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 955 939 532 8 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 911 879 065 6;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 911 879 065 6 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 823 758 131 2;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 823 758 131 2 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 647 516 262 4;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 647 516 262 4 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 295 032 524 8;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 295 032 524 8 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 590 065 049 6;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 590 065 049 6 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 180 130 099 2;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 180 130 099 2 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 360 260 198 4;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 360 260 198 4 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 004 720 520 396 8;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 004 720 520 396 8 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 009 441 040 793 6;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 009 441 040 793 6 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 018 882 081 587 2;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 018 882 081 587 2 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 037 764 163 174 4;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 037 764 163 174 4 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 075 528 326 348 8;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 075 528 326 348 8 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 151 056 652 697 6;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 151 056 652 697 6 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 302 113 305 395 2;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 302 113 305 395 2 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 604 226 610 790 4;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 604 226 610 790 4 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 208 453 221 580 8;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 208 453 221 580 8 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 818 416 906 443 161 6;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 818 416 906 443 161 6 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 636 833 812 886 323 2;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 636 833 812 886 323 2 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 273 667 625 772 646 4;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 273 667 625 772 646 4 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 547 335 251 545 292 8;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 547 335 251 545 292 8 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 094 670 503 090 585 6;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 094 670 503 090 585 6 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 189 341 006 181 171 2;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 189 341 006 181 171 2 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 378 682 012 362 342 4;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 378 682 012 362 342 4 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 757 364 024 724 684 8;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 757 364 024 724 684 8 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 514 728 049 449 369 6;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 514 728 049 449 369 6 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 029 456 098 898 739 2;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 029 456 098 898 739 2 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 058 912 197 797 478 4;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 058 912 197 797 478 4 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 516 117 824 395 594 956 8;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 516 117 824 395 594 956 8 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 032 235 648 791 189 913 6;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 032 235 648 791 189 913 6 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 064 471 297 582 379 827 2;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 064 471 297 582 379 827 2 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 128 942 595 164 759 654 4;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 128 942 595 164 759 654 4 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 257 885 190 329 519 308 8;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 257 885 190 329 519 308 8 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 515 770 380 659 038 617 6;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 515 770 380 659 038 617 6 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 031 540 761 318 077 235 2;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 031 540 761 318 077 235 2 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 063 081 522 636 154 470 4;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 063 081 522 636 154 470 4 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 126 163 045 272 308 940 8;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 126 163 045 272 308 940 8 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 252 326 090 544 617 881 6;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 252 326 090 544 617 881 6 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 248 504 652 181 089 235 763 2;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 248 504 652 181 089 235 763 2 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 497 009 304 362 178 471 526 4;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 497 009 304 362 178 471 526 4 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 994 018 608 724 356 943 052 8;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 994 018 608 724 356 943 052 8 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 988 037 217 448 713 886 105 6;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 988 037 217 448 713 886 105 6 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 976 074 434 897 427 772 211 2;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 976 074 434 897 427 772 211 2 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 952 148 869 794 855 544 422 4;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 952 148 869 794 855 544 422 4 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 904 297 739 589 711 088 844 8;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 904 297 739 589 711 088 844 8 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 808 595 479 179 422 177 689 6;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 808 595 479 179 422 177 689 6 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 617 190 958 358 844 355 379 2;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 617 190 958 358 844 355 379 2 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 234 381 916 717 688 710 758 4;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 234 381 916 717 688 710 758 4 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 998 468 763 833 435 377 421 516 8;
  • 69) 0.326 947 450 637 817 382 812 499 999 998 468 763 833 435 377 421 516 8 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 996 937 527 666 870 754 843 033 6;
  • 70) 0.653 894 901 275 634 765 624 999 999 996 937 527 666 870 754 843 033 6 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 993 875 055 333 741 509 686 067 2;
  • 71) 0.307 789 802 551 269 531 249 999 999 993 875 055 333 741 509 686 067 2 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 987 750 110 667 483 019 372 134 4;
  • 72) 0.615 579 605 102 539 062 499 999 999 987 750 110 667 483 019 372 134 4 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 975 500 221 334 966 038 744 268 8;
  • 73) 0.231 159 210 205 078 124 999 999 999 975 500 221 334 966 038 744 268 8 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 951 000 442 669 932 077 488 537 6;
  • 74) 0.462 318 420 410 156 249 999 999 999 951 000 442 669 932 077 488 537 6 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 902 000 885 339 864 154 977 075 2;
  • 75) 0.924 636 840 820 312 499 999 999 999 902 000 885 339 864 154 977 075 2 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 804 001 770 679 728 309 954 150 4;
  • 76) 0.849 273 681 640 624 999 999 999 999 804 001 770 679 728 309 954 150 4 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 608 003 541 359 456 619 908 300 8;
  • 77) 0.698 547 363 281 249 999 999 999 999 608 003 541 359 456 619 908 300 8 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 216 007 082 718 913 239 816 601 6;
  • 78) 0.397 094 726 562 499 999 999 999 999 216 007 082 718 913 239 816 601 6 × 2 = 0 + 0.794 189 453 124 999 999 999 999 998 432 014 165 437 826 479 633 203 2;
  • 79) 0.794 189 453 124 999 999 999 999 998 432 014 165 437 826 479 633 203 2 × 2 = 1 + 0.588 378 906 249 999 999 999 999 996 864 028 330 875 652 959 266 406 4;
  • 80) 0.588 378 906 249 999 999 999 999 996 864 028 330 875 652 959 266 406 4 × 2 = 1 + 0.176 757 812 499 999 999 999 999 993 728 056 661 751 305 918 532 812 8;
  • 81) 0.176 757 812 499 999 999 999 999 993 728 056 661 751 305 918 532 812 8 × 2 = 0 + 0.353 515 624 999 999 999 999 999 987 456 113 323 502 611 837 065 625 6;
  • 82) 0.353 515 624 999 999 999 999 999 987 456 113 323 502 611 837 065 625 6 × 2 = 0 + 0.707 031 249 999 999 999 999 999 974 912 226 647 005 223 674 131 251 2;
  • 83) 0.707 031 249 999 999 999 999 999 974 912 226 647 005 223 674 131 251 2 × 2 = 1 + 0.414 062 499 999 999 999 999 999 949 824 453 294 010 447 348 262 502 4;
  • 84) 0.414 062 499 999 999 999 999 999 949 824 453 294 010 447 348 262 502 4 × 2 = 0 + 0.828 124 999 999 999 999 999 999 899 648 906 588 020 894 696 525 004 8;
  • 85) 0.828 124 999 999 999 999 999 999 899 648 906 588 020 894 696 525 004 8 × 2 = 1 + 0.656 249 999 999 999 999 999 999 799 297 813 176 041 789 393 050 009 6;
  • 86) 0.656 249 999 999 999 999 999 999 799 297 813 176 041 789 393 050 009 6 × 2 = 1 + 0.312 499 999 999 999 999 999 999 598 595 626 352 083 578 786 100 019 2;
  • 87) 0.312 499 999 999 999 999 999 999 598 595 626 352 083 578 786 100 019 2 × 2 = 0 + 0.624 999 999 999 999 999 999 999 197 191 252 704 167 157 572 200 038 4;
  • 88) 0.624 999 999 999 999 999 999 999 197 191 252 704 167 157 572 200 038 4 × 2 = 1 + 0.249 999 999 999 999 999 999 998 394 382 505 408 334 315 144 400 076 8;
  • 89) 0.249 999 999 999 999 999 999 998 394 382 505 408 334 315 144 400 076 8 × 2 = 0 + 0.499 999 999 999 999 999 999 996 788 765 010 816 668 630 288 800 153 6;
  • 90) 0.499 999 999 999 999 999 999 996 788 765 010 816 668 630 288 800 153 6 × 2 = 0 + 0.999 999 999 999 999 999 999 993 577 530 021 633 337 260 577 600 307 2;
  • 91) 0.999 999 999 999 999 999 999 993 577 530 021 633 337 260 577 600 307 2 × 2 = 1 + 0.999 999 999 999 999 999 999 987 155 060 043 266 674 521 155 200 614 4;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 147 8(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 147 8(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 147 8(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 147 8 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

Share

» Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 9 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 07, 2026 [July]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: July - by our visitors

» Improve your social skills: Show, Don’t Tell, the Strategic Power of Action. Gemini Notebook YouTube Video of 9.21 minutes

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