64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 010 012 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 010 012(10) converted and written in 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: 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 010 012.

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) 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 010 012 × 2 = 0 + 0.000 000 000 001 511 680 022 220 202 202 020 200 220 000 000 000 002 200 002 020 022 020 024;
  • 2) 0.000 000 000 001 511 680 022 220 202 202 020 200 220 000 000 000 002 200 002 020 022 020 024 × 2 = 0 + 0.000 000 000 003 023 360 044 440 404 404 040 400 440 000 000 000 004 400 004 040 044 040 048;
  • 3) 0.000 000 000 003 023 360 044 440 404 404 040 400 440 000 000 000 004 400 004 040 044 040 048 × 2 = 0 + 0.000 000 000 006 046 720 088 880 808 808 080 800 880 000 000 000 008 800 008 080 088 080 096;
  • 4) 0.000 000 000 006 046 720 088 880 808 808 080 800 880 000 000 000 008 800 008 080 088 080 096 × 2 = 0 + 0.000 000 000 012 093 440 177 761 617 616 161 601 760 000 000 000 017 600 016 160 176 160 192;
  • 5) 0.000 000 000 012 093 440 177 761 617 616 161 601 760 000 000 000 017 600 016 160 176 160 192 × 2 = 0 + 0.000 000 000 024 186 880 355 523 235 232 323 203 520 000 000 000 035 200 032 320 352 320 384;
  • 6) 0.000 000 000 024 186 880 355 523 235 232 323 203 520 000 000 000 035 200 032 320 352 320 384 × 2 = 0 + 0.000 000 000 048 373 760 711 046 470 464 646 407 040 000 000 000 070 400 064 640 704 640 768;
  • 7) 0.000 000 000 048 373 760 711 046 470 464 646 407 040 000 000 000 070 400 064 640 704 640 768 × 2 = 0 + 0.000 000 000 096 747 521 422 092 940 929 292 814 080 000 000 000 140 800 129 281 409 281 536;
  • 8) 0.000 000 000 096 747 521 422 092 940 929 292 814 080 000 000 000 140 800 129 281 409 281 536 × 2 = 0 + 0.000 000 000 193 495 042 844 185 881 858 585 628 160 000 000 000 281 600 258 562 818 563 072;
  • 9) 0.000 000 000 193 495 042 844 185 881 858 585 628 160 000 000 000 281 600 258 562 818 563 072 × 2 = 0 + 0.000 000 000 386 990 085 688 371 763 717 171 256 320 000 000 000 563 200 517 125 637 126 144;
  • 10) 0.000 000 000 386 990 085 688 371 763 717 171 256 320 000 000 000 563 200 517 125 637 126 144 × 2 = 0 + 0.000 000 000 773 980 171 376 743 527 434 342 512 640 000 000 001 126 401 034 251 274 252 288;
  • 11) 0.000 000 000 773 980 171 376 743 527 434 342 512 640 000 000 001 126 401 034 251 274 252 288 × 2 = 0 + 0.000 000 001 547 960 342 753 487 054 868 685 025 280 000 000 002 252 802 068 502 548 504 576;
  • 12) 0.000 000 001 547 960 342 753 487 054 868 685 025 280 000 000 002 252 802 068 502 548 504 576 × 2 = 0 + 0.000 000 003 095 920 685 506 974 109 737 370 050 560 000 000 004 505 604 137 005 097 009 152;
  • 13) 0.000 000 003 095 920 685 506 974 109 737 370 050 560 000 000 004 505 604 137 005 097 009 152 × 2 = 0 + 0.000 000 006 191 841 371 013 948 219 474 740 101 120 000 000 009 011 208 274 010 194 018 304;
  • 14) 0.000 000 006 191 841 371 013 948 219 474 740 101 120 000 000 009 011 208 274 010 194 018 304 × 2 = 0 + 0.000 000 012 383 682 742 027 896 438 949 480 202 240 000 000 018 022 416 548 020 388 036 608;
  • 15) 0.000 000 012 383 682 742 027 896 438 949 480 202 240 000 000 018 022 416 548 020 388 036 608 × 2 = 0 + 0.000 000 024 767 365 484 055 792 877 898 960 404 480 000 000 036 044 833 096 040 776 073 216;
  • 16) 0.000 000 024 767 365 484 055 792 877 898 960 404 480 000 000 036 044 833 096 040 776 073 216 × 2 = 0 + 0.000 000 049 534 730 968 111 585 755 797 920 808 960 000 000 072 089 666 192 081 552 146 432;
  • 17) 0.000 000 049 534 730 968 111 585 755 797 920 808 960 000 000 072 089 666 192 081 552 146 432 × 2 = 0 + 0.000 000 099 069 461 936 223 171 511 595 841 617 920 000 000 144 179 332 384 163 104 292 864;
  • 18) 0.000 000 099 069 461 936 223 171 511 595 841 617 920 000 000 144 179 332 384 163 104 292 864 × 2 = 0 + 0.000 000 198 138 923 872 446 343 023 191 683 235 840 000 000 288 358 664 768 326 208 585 728;
  • 19) 0.000 000 198 138 923 872 446 343 023 191 683 235 840 000 000 288 358 664 768 326 208 585 728 × 2 = 0 + 0.000 000 396 277 847 744 892 686 046 383 366 471 680 000 000 576 717 329 536 652 417 171 456;
  • 20) 0.000 000 396 277 847 744 892 686 046 383 366 471 680 000 000 576 717 329 536 652 417 171 456 × 2 = 0 + 0.000 000 792 555 695 489 785 372 092 766 732 943 360 000 001 153 434 659 073 304 834 342 912;
  • 21) 0.000 000 792 555 695 489 785 372 092 766 732 943 360 000 001 153 434 659 073 304 834 342 912 × 2 = 0 + 0.000 001 585 111 390 979 570 744 185 533 465 886 720 000 002 306 869 318 146 609 668 685 824;
  • 22) 0.000 001 585 111 390 979 570 744 185 533 465 886 720 000 002 306 869 318 146 609 668 685 824 × 2 = 0 + 0.000 003 170 222 781 959 141 488 371 066 931 773 440 000 004 613 738 636 293 219 337 371 648;
  • 23) 0.000 003 170 222 781 959 141 488 371 066 931 773 440 000 004 613 738 636 293 219 337 371 648 × 2 = 0 + 0.000 006 340 445 563 918 282 976 742 133 863 546 880 000 009 227 477 272 586 438 674 743 296;
  • 24) 0.000 006 340 445 563 918 282 976 742 133 863 546 880 000 009 227 477 272 586 438 674 743 296 × 2 = 0 + 0.000 012 680 891 127 836 565 953 484 267 727 093 760 000 018 454 954 545 172 877 349 486 592;
  • 25) 0.000 012 680 891 127 836 565 953 484 267 727 093 760 000 018 454 954 545 172 877 349 486 592 × 2 = 0 + 0.000 025 361 782 255 673 131 906 968 535 454 187 520 000 036 909 909 090 345 754 698 973 184;
  • 26) 0.000 025 361 782 255 673 131 906 968 535 454 187 520 000 036 909 909 090 345 754 698 973 184 × 2 = 0 + 0.000 050 723 564 511 346 263 813 937 070 908 375 040 000 073 819 818 180 691 509 397 946 368;
  • 27) 0.000 050 723 564 511 346 263 813 937 070 908 375 040 000 073 819 818 180 691 509 397 946 368 × 2 = 0 + 0.000 101 447 129 022 692 527 627 874 141 816 750 080 000 147 639 636 361 383 018 795 892 736;
  • 28) 0.000 101 447 129 022 692 527 627 874 141 816 750 080 000 147 639 636 361 383 018 795 892 736 × 2 = 0 + 0.000 202 894 258 045 385 055 255 748 283 633 500 160 000 295 279 272 722 766 037 591 785 472;
  • 29) 0.000 202 894 258 045 385 055 255 748 283 633 500 160 000 295 279 272 722 766 037 591 785 472 × 2 = 0 + 0.000 405 788 516 090 770 110 511 496 567 267 000 320 000 590 558 545 445 532 075 183 570 944;
  • 30) 0.000 405 788 516 090 770 110 511 496 567 267 000 320 000 590 558 545 445 532 075 183 570 944 × 2 = 0 + 0.000 811 577 032 181 540 221 022 993 134 534 000 640 001 181 117 090 891 064 150 367 141 888;
  • 31) 0.000 811 577 032 181 540 221 022 993 134 534 000 640 001 181 117 090 891 064 150 367 141 888 × 2 = 0 + 0.001 623 154 064 363 080 442 045 986 269 068 001 280 002 362 234 181 782 128 300 734 283 776;
  • 32) 0.001 623 154 064 363 080 442 045 986 269 068 001 280 002 362 234 181 782 128 300 734 283 776 × 2 = 0 + 0.003 246 308 128 726 160 884 091 972 538 136 002 560 004 724 468 363 564 256 601 468 567 552;
  • 33) 0.003 246 308 128 726 160 884 091 972 538 136 002 560 004 724 468 363 564 256 601 468 567 552 × 2 = 0 + 0.006 492 616 257 452 321 768 183 945 076 272 005 120 009 448 936 727 128 513 202 937 135 104;
  • 34) 0.006 492 616 257 452 321 768 183 945 076 272 005 120 009 448 936 727 128 513 202 937 135 104 × 2 = 0 + 0.012 985 232 514 904 643 536 367 890 152 544 010 240 018 897 873 454 257 026 405 874 270 208;
  • 35) 0.012 985 232 514 904 643 536 367 890 152 544 010 240 018 897 873 454 257 026 405 874 270 208 × 2 = 0 + 0.025 970 465 029 809 287 072 735 780 305 088 020 480 037 795 746 908 514 052 811 748 540 416;
  • 36) 0.025 970 465 029 809 287 072 735 780 305 088 020 480 037 795 746 908 514 052 811 748 540 416 × 2 = 0 + 0.051 940 930 059 618 574 145 471 560 610 176 040 960 075 591 493 817 028 105 623 497 080 832;
  • 37) 0.051 940 930 059 618 574 145 471 560 610 176 040 960 075 591 493 817 028 105 623 497 080 832 × 2 = 0 + 0.103 881 860 119 237 148 290 943 121 220 352 081 920 151 182 987 634 056 211 246 994 161 664;
  • 38) 0.103 881 860 119 237 148 290 943 121 220 352 081 920 151 182 987 634 056 211 246 994 161 664 × 2 = 0 + 0.207 763 720 238 474 296 581 886 242 440 704 163 840 302 365 975 268 112 422 493 988 323 328;
  • 39) 0.207 763 720 238 474 296 581 886 242 440 704 163 840 302 365 975 268 112 422 493 988 323 328 × 2 = 0 + 0.415 527 440 476 948 593 163 772 484 881 408 327 680 604 731 950 536 224 844 987 976 646 656;
  • 40) 0.415 527 440 476 948 593 163 772 484 881 408 327 680 604 731 950 536 224 844 987 976 646 656 × 2 = 0 + 0.831 054 880 953 897 186 327 544 969 762 816 655 361 209 463 901 072 449 689 975 953 293 312;
  • 41) 0.831 054 880 953 897 186 327 544 969 762 816 655 361 209 463 901 072 449 689 975 953 293 312 × 2 = 1 + 0.662 109 761 907 794 372 655 089 939 525 633 310 722 418 927 802 144 899 379 951 906 586 624;
  • 42) 0.662 109 761 907 794 372 655 089 939 525 633 310 722 418 927 802 144 899 379 951 906 586 624 × 2 = 1 + 0.324 219 523 815 588 745 310 179 879 051 266 621 444 837 855 604 289 798 759 903 813 173 248;
  • 43) 0.324 219 523 815 588 745 310 179 879 051 266 621 444 837 855 604 289 798 759 903 813 173 248 × 2 = 0 + 0.648 439 047 631 177 490 620 359 758 102 533 242 889 675 711 208 579 597 519 807 626 346 496;
  • 44) 0.648 439 047 631 177 490 620 359 758 102 533 242 889 675 711 208 579 597 519 807 626 346 496 × 2 = 1 + 0.296 878 095 262 354 981 240 719 516 205 066 485 779 351 422 417 159 195 039 615 252 692 992;
  • 45) 0.296 878 095 262 354 981 240 719 516 205 066 485 779 351 422 417 159 195 039 615 252 692 992 × 2 = 0 + 0.593 756 190 524 709 962 481 439 032 410 132 971 558 702 844 834 318 390 079 230 505 385 984;
  • 46) 0.593 756 190 524 709 962 481 439 032 410 132 971 558 702 844 834 318 390 079 230 505 385 984 × 2 = 1 + 0.187 512 381 049 419 924 962 878 064 820 265 943 117 405 689 668 636 780 158 461 010 771 968;
  • 47) 0.187 512 381 049 419 924 962 878 064 820 265 943 117 405 689 668 636 780 158 461 010 771 968 × 2 = 0 + 0.375 024 762 098 839 849 925 756 129 640 531 886 234 811 379 337 273 560 316 922 021 543 936;
  • 48) 0.375 024 762 098 839 849 925 756 129 640 531 886 234 811 379 337 273 560 316 922 021 543 936 × 2 = 0 + 0.750 049 524 197 679 699 851 512 259 281 063 772 469 622 758 674 547 120 633 844 043 087 872;
  • 49) 0.750 049 524 197 679 699 851 512 259 281 063 772 469 622 758 674 547 120 633 844 043 087 872 × 2 = 1 + 0.500 099 048 395 359 399 703 024 518 562 127 544 939 245 517 349 094 241 267 688 086 175 744;
  • 50) 0.500 099 048 395 359 399 703 024 518 562 127 544 939 245 517 349 094 241 267 688 086 175 744 × 2 = 1 + 0.000 198 096 790 718 799 406 049 037 124 255 089 878 491 034 698 188 482 535 376 172 351 488;
  • 51) 0.000 198 096 790 718 799 406 049 037 124 255 089 878 491 034 698 188 482 535 376 172 351 488 × 2 = 0 + 0.000 396 193 581 437 598 812 098 074 248 510 179 756 982 069 396 376 965 070 752 344 702 976;
  • 52) 0.000 396 193 581 437 598 812 098 074 248 510 179 756 982 069 396 376 965 070 752 344 702 976 × 2 = 0 + 0.000 792 387 162 875 197 624 196 148 497 020 359 513 964 138 792 753 930 141 504 689 405 952;
  • 53) 0.000 792 387 162 875 197 624 196 148 497 020 359 513 964 138 792 753 930 141 504 689 405 952 × 2 = 0 + 0.001 584 774 325 750 395 248 392 296 994 040 719 027 928 277 585 507 860 283 009 378 811 904;
  • 54) 0.001 584 774 325 750 395 248 392 296 994 040 719 027 928 277 585 507 860 283 009 378 811 904 × 2 = 0 + 0.003 169 548 651 500 790 496 784 593 988 081 438 055 856 555 171 015 720 566 018 757 623 808;
  • 55) 0.003 169 548 651 500 790 496 784 593 988 081 438 055 856 555 171 015 720 566 018 757 623 808 × 2 = 0 + 0.006 339 097 303 001 580 993 569 187 976 162 876 111 713 110 342 031 441 132 037 515 247 616;
  • 56) 0.006 339 097 303 001 580 993 569 187 976 162 876 111 713 110 342 031 441 132 037 515 247 616 × 2 = 0 + 0.012 678 194 606 003 161 987 138 375 952 325 752 223 426 220 684 062 882 264 075 030 495 232;
  • 57) 0.012 678 194 606 003 161 987 138 375 952 325 752 223 426 220 684 062 882 264 075 030 495 232 × 2 = 0 + 0.025 356 389 212 006 323 974 276 751 904 651 504 446 852 441 368 125 764 528 150 060 990 464;
  • 58) 0.025 356 389 212 006 323 974 276 751 904 651 504 446 852 441 368 125 764 528 150 060 990 464 × 2 = 0 + 0.050 712 778 424 012 647 948 553 503 809 303 008 893 704 882 736 251 529 056 300 121 980 928;
  • 59) 0.050 712 778 424 012 647 948 553 503 809 303 008 893 704 882 736 251 529 056 300 121 980 928 × 2 = 0 + 0.101 425 556 848 025 295 897 107 007 618 606 017 787 409 765 472 503 058 112 600 243 961 856;
  • 60) 0.101 425 556 848 025 295 897 107 007 618 606 017 787 409 765 472 503 058 112 600 243 961 856 × 2 = 0 + 0.202 851 113 696 050 591 794 214 015 237 212 035 574 819 530 945 006 116 225 200 487 923 712;
  • 61) 0.202 851 113 696 050 591 794 214 015 237 212 035 574 819 530 945 006 116 225 200 487 923 712 × 2 = 0 + 0.405 702 227 392 101 183 588 428 030 474 424 071 149 639 061 890 012 232 450 400 975 847 424;
  • 62) 0.405 702 227 392 101 183 588 428 030 474 424 071 149 639 061 890 012 232 450 400 975 847 424 × 2 = 0 + 0.811 404 454 784 202 367 176 856 060 948 848 142 299 278 123 780 024 464 900 801 951 694 848;
  • 63) 0.811 404 454 784 202 367 176 856 060 948 848 142 299 278 123 780 024 464 900 801 951 694 848 × 2 = 1 + 0.622 808 909 568 404 734 353 712 121 897 696 284 598 556 247 560 048 929 801 603 903 389 696;
  • 64) 0.622 808 909 568 404 734 353 712 121 897 696 284 598 556 247 560 048 929 801 603 903 389 696 × 2 = 1 + 0.245 617 819 136 809 468 707 424 243 795 392 569 197 112 495 120 097 859 603 207 806 779 392;
  • 65) 0.245 617 819 136 809 468 707 424 243 795 392 569 197 112 495 120 097 859 603 207 806 779 392 × 2 = 0 + 0.491 235 638 273 618 937 414 848 487 590 785 138 394 224 990 240 195 719 206 415 613 558 784;
  • 66) 0.491 235 638 273 618 937 414 848 487 590 785 138 394 224 990 240 195 719 206 415 613 558 784 × 2 = 0 + 0.982 471 276 547 237 874 829 696 975 181 570 276 788 449 980 480 391 438 412 831 227 117 568;
  • 67) 0.982 471 276 547 237 874 829 696 975 181 570 276 788 449 980 480 391 438 412 831 227 117 568 × 2 = 1 + 0.964 942 553 094 475 749 659 393 950 363 140 553 576 899 960 960 782 876 825 662 454 235 136;
  • 68) 0.964 942 553 094 475 749 659 393 950 363 140 553 576 899 960 960 782 876 825 662 454 235 136 × 2 = 1 + 0.929 885 106 188 951 499 318 787 900 726 281 107 153 799 921 921 565 753 651 324 908 470 272;
  • 69) 0.929 885 106 188 951 499 318 787 900 726 281 107 153 799 921 921 565 753 651 324 908 470 272 × 2 = 1 + 0.859 770 212 377 902 998 637 575 801 452 562 214 307 599 843 843 131 507 302 649 816 940 544;
  • 70) 0.859 770 212 377 902 998 637 575 801 452 562 214 307 599 843 843 131 507 302 649 816 940 544 × 2 = 1 + 0.719 540 424 755 805 997 275 151 602 905 124 428 615 199 687 686 263 014 605 299 633 881 088;
  • 71) 0.719 540 424 755 805 997 275 151 602 905 124 428 615 199 687 686 263 014 605 299 633 881 088 × 2 = 1 + 0.439 080 849 511 611 994 550 303 205 810 248 857 230 399 375 372 526 029 210 599 267 762 176;
  • 72) 0.439 080 849 511 611 994 550 303 205 810 248 857 230 399 375 372 526 029 210 599 267 762 176 × 2 = 0 + 0.878 161 699 023 223 989 100 606 411 620 497 714 460 798 750 745 052 058 421 198 535 524 352;
  • 73) 0.878 161 699 023 223 989 100 606 411 620 497 714 460 798 750 745 052 058 421 198 535 524 352 × 2 = 1 + 0.756 323 398 046 447 978 201 212 823 240 995 428 921 597 501 490 104 116 842 397 071 048 704;
  • 74) 0.756 323 398 046 447 978 201 212 823 240 995 428 921 597 501 490 104 116 842 397 071 048 704 × 2 = 1 + 0.512 646 796 092 895 956 402 425 646 481 990 857 843 195 002 980 208 233 684 794 142 097 408;
  • 75) 0.512 646 796 092 895 956 402 425 646 481 990 857 843 195 002 980 208 233 684 794 142 097 408 × 2 = 1 + 0.025 293 592 185 791 912 804 851 292 963 981 715 686 390 005 960 416 467 369 588 284 194 816;
  • 76) 0.025 293 592 185 791 912 804 851 292 963 981 715 686 390 005 960 416 467 369 588 284 194 816 × 2 = 0 + 0.050 587 184 371 583 825 609 702 585 927 963 431 372 780 011 920 832 934 739 176 568 389 632;
  • 77) 0.050 587 184 371 583 825 609 702 585 927 963 431 372 780 011 920 832 934 739 176 568 389 632 × 2 = 0 + 0.101 174 368 743 167 651 219 405 171 855 926 862 745 560 023 841 665 869 478 353 136 779 264;
  • 78) 0.101 174 368 743 167 651 219 405 171 855 926 862 745 560 023 841 665 869 478 353 136 779 264 × 2 = 0 + 0.202 348 737 486 335 302 438 810 343 711 853 725 491 120 047 683 331 738 956 706 273 558 528;
  • 79) 0.202 348 737 486 335 302 438 810 343 711 853 725 491 120 047 683 331 738 956 706 273 558 528 × 2 = 0 + 0.404 697 474 972 670 604 877 620 687 423 707 450 982 240 095 366 663 477 913 412 547 117 056;
  • 80) 0.404 697 474 972 670 604 877 620 687 423 707 450 982 240 095 366 663 477 913 412 547 117 056 × 2 = 0 + 0.809 394 949 945 341 209 755 241 374 847 414 901 964 480 190 733 326 955 826 825 094 234 112;
  • 81) 0.809 394 949 945 341 209 755 241 374 847 414 901 964 480 190 733 326 955 826 825 094 234 112 × 2 = 1 + 0.618 789 899 890 682 419 510 482 749 694 829 803 928 960 381 466 653 911 653 650 188 468 224;
  • 82) 0.618 789 899 890 682 419 510 482 749 694 829 803 928 960 381 466 653 911 653 650 188 468 224 × 2 = 1 + 0.237 579 799 781 364 839 020 965 499 389 659 607 857 920 762 933 307 823 307 300 376 936 448;
  • 83) 0.237 579 799 781 364 839 020 965 499 389 659 607 857 920 762 933 307 823 307 300 376 936 448 × 2 = 0 + 0.475 159 599 562 729 678 041 930 998 779 319 215 715 841 525 866 615 646 614 600 753 872 896;
  • 84) 0.475 159 599 562 729 678 041 930 998 779 319 215 715 841 525 866 615 646 614 600 753 872 896 × 2 = 0 + 0.950 319 199 125 459 356 083 861 997 558 638 431 431 683 051 733 231 293 229 201 507 745 792;
  • 85) 0.950 319 199 125 459 356 083 861 997 558 638 431 431 683 051 733 231 293 229 201 507 745 792 × 2 = 1 + 0.900 638 398 250 918 712 167 723 995 117 276 862 863 366 103 466 462 586 458 403 015 491 584;
  • 86) 0.900 638 398 250 918 712 167 723 995 117 276 862 863 366 103 466 462 586 458 403 015 491 584 × 2 = 1 + 0.801 276 796 501 837 424 335 447 990 234 553 725 726 732 206 932 925 172 916 806 030 983 168;
  • 87) 0.801 276 796 501 837 424 335 447 990 234 553 725 726 732 206 932 925 172 916 806 030 983 168 × 2 = 1 + 0.602 553 593 003 674 848 670 895 980 469 107 451 453 464 413 865 850 345 833 612 061 966 336;
  • 88) 0.602 553 593 003 674 848 670 895 980 469 107 451 453 464 413 865 850 345 833 612 061 966 336 × 2 = 1 + 0.205 107 186 007 349 697 341 791 960 938 214 902 906 928 827 731 700 691 667 224 123 932 672;
  • 89) 0.205 107 186 007 349 697 341 791 960 938 214 902 906 928 827 731 700 691 667 224 123 932 672 × 2 = 0 + 0.410 214 372 014 699 394 683 583 921 876 429 805 813 857 655 463 401 383 334 448 247 865 344;
  • 90) 0.410 214 372 014 699 394 683 583 921 876 429 805 813 857 655 463 401 383 334 448 247 865 344 × 2 = 0 + 0.820 428 744 029 398 789 367 167 843 752 859 611 627 715 310 926 802 766 668 896 495 730 688;
  • 91) 0.820 428 744 029 398 789 367 167 843 752 859 611 627 715 310 926 802 766 668 896 495 730 688 × 2 = 1 + 0.640 857 488 058 797 578 734 335 687 505 719 223 255 430 621 853 605 533 337 792 991 461 376;
  • 92) 0.640 857 488 058 797 578 734 335 687 505 719 223 255 430 621 853 605 533 337 792 991 461 376 × 2 = 1 + 0.281 714 976 117 595 157 468 671 375 011 438 446 510 861 243 707 211 066 675 585 982 922 752;
  • 93) 0.281 714 976 117 595 157 468 671 375 011 438 446 510 861 243 707 211 066 675 585 982 922 752 × 2 = 0 + 0.563 429 952 235 190 314 937 342 750 022 876 893 021 722 487 414 422 133 351 171 965 845 504;

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...)


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:


0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 010 012(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1101 0100 1100 0000 0000 0011 0011 1110 1110 0000 1100 1111 0011 0(2)


5. Positive number before normalization:

0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 010 012(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1101 0100 1100 0000 0000 0011 0011 1110 1110 0000 1100 1111 0011 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 41 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 010 012(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1101 0100 1100 0000 0000 0011 0011 1110 1110 0000 1100 1111 0011 0(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1101 0100 1100 0000 0000 0011 0011 1110 1110 0000 1100 1111 0011 0(2) × 20 =

1.1010 1001 1000 0000 0000 0110 0111 1101 1100 0001 1001 1110 0110(2) × 2-41


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): -41

Mantissa (not normalized):
1.1010 1001 1000 0000 0000 0110 0111 1101 1100 0001 1001 1110 0110


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-41 + 2(11-1) - 1 =

(-41 + 1 023)(10) =

982(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;
  • 982 ÷ 2 = 491 + 0;
  • 491 ÷ 2 = 245 + 1;
  • 245 ÷ 2 = 122 + 1;
  • 122 ÷ 2 = 61 + 0;
  • 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) =

982(10) =

011 1101 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. 1010 1001 1000 0000 0000 0110 0111 1101 1100 0001 1001 1110 0110 =

1010 1001 1000 0000 0000 0110 0111 1101 1100 0001 1001 1110 0110


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 0110

Mantissa (52 bits) =
1010 1001 1000 0000 0000 0110 0111 1101 1100 0001 1001 1110 0110


The base ten decimal number 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 010 012 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 011 1101 0110 - 1010 1001 1000 0000 0000 0110 0111 1101 1100 0001 1001 1110 0110

More operations with decimal numbers converted to 64 bit double precision IEEE 754 binary floating point representation:

» Number 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 010 011 converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point representation = ?

» Number 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 010 013 converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point representation = ?

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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