0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 009 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 009(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
0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 009(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: 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 009.

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) 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 009 × 2 = 0 + 0.000 000 000 001 511 680 022 220 202 202 020 200 220 000 000 000 002 200 002 020 022 018;
  • 2) 0.000 000 000 001 511 680 022 220 202 202 020 200 220 000 000 000 002 200 002 020 022 018 × 2 = 0 + 0.000 000 000 003 023 360 044 440 404 404 040 400 440 000 000 000 004 400 004 040 044 036;
  • 3) 0.000 000 000 003 023 360 044 440 404 404 040 400 440 000 000 000 004 400 004 040 044 036 × 2 = 0 + 0.000 000 000 006 046 720 088 880 808 808 080 800 880 000 000 000 008 800 008 080 088 072;
  • 4) 0.000 000 000 006 046 720 088 880 808 808 080 800 880 000 000 000 008 800 008 080 088 072 × 2 = 0 + 0.000 000 000 012 093 440 177 761 617 616 161 601 760 000 000 000 017 600 016 160 176 144;
  • 5) 0.000 000 000 012 093 440 177 761 617 616 161 601 760 000 000 000 017 600 016 160 176 144 × 2 = 0 + 0.000 000 000 024 186 880 355 523 235 232 323 203 520 000 000 000 035 200 032 320 352 288;
  • 6) 0.000 000 000 024 186 880 355 523 235 232 323 203 520 000 000 000 035 200 032 320 352 288 × 2 = 0 + 0.000 000 000 048 373 760 711 046 470 464 646 407 040 000 000 000 070 400 064 640 704 576;
  • 7) 0.000 000 000 048 373 760 711 046 470 464 646 407 040 000 000 000 070 400 064 640 704 576 × 2 = 0 + 0.000 000 000 096 747 521 422 092 940 929 292 814 080 000 000 000 140 800 129 281 409 152;
  • 8) 0.000 000 000 096 747 521 422 092 940 929 292 814 080 000 000 000 140 800 129 281 409 152 × 2 = 0 + 0.000 000 000 193 495 042 844 185 881 858 585 628 160 000 000 000 281 600 258 562 818 304;
  • 9) 0.000 000 000 193 495 042 844 185 881 858 585 628 160 000 000 000 281 600 258 562 818 304 × 2 = 0 + 0.000 000 000 386 990 085 688 371 763 717 171 256 320 000 000 000 563 200 517 125 636 608;
  • 10) 0.000 000 000 386 990 085 688 371 763 717 171 256 320 000 000 000 563 200 517 125 636 608 × 2 = 0 + 0.000 000 000 773 980 171 376 743 527 434 342 512 640 000 000 001 126 401 034 251 273 216;
  • 11) 0.000 000 000 773 980 171 376 743 527 434 342 512 640 000 000 001 126 401 034 251 273 216 × 2 = 0 + 0.000 000 001 547 960 342 753 487 054 868 685 025 280 000 000 002 252 802 068 502 546 432;
  • 12) 0.000 000 001 547 960 342 753 487 054 868 685 025 280 000 000 002 252 802 068 502 546 432 × 2 = 0 + 0.000 000 003 095 920 685 506 974 109 737 370 050 560 000 000 004 505 604 137 005 092 864;
  • 13) 0.000 000 003 095 920 685 506 974 109 737 370 050 560 000 000 004 505 604 137 005 092 864 × 2 = 0 + 0.000 000 006 191 841 371 013 948 219 474 740 101 120 000 000 009 011 208 274 010 185 728;
  • 14) 0.000 000 006 191 841 371 013 948 219 474 740 101 120 000 000 009 011 208 274 010 185 728 × 2 = 0 + 0.000 000 012 383 682 742 027 896 438 949 480 202 240 000 000 018 022 416 548 020 371 456;
  • 15) 0.000 000 012 383 682 742 027 896 438 949 480 202 240 000 000 018 022 416 548 020 371 456 × 2 = 0 + 0.000 000 024 767 365 484 055 792 877 898 960 404 480 000 000 036 044 833 096 040 742 912;
  • 16) 0.000 000 024 767 365 484 055 792 877 898 960 404 480 000 000 036 044 833 096 040 742 912 × 2 = 0 + 0.000 000 049 534 730 968 111 585 755 797 920 808 960 000 000 072 089 666 192 081 485 824;
  • 17) 0.000 000 049 534 730 968 111 585 755 797 920 808 960 000 000 072 089 666 192 081 485 824 × 2 = 0 + 0.000 000 099 069 461 936 223 171 511 595 841 617 920 000 000 144 179 332 384 162 971 648;
  • 18) 0.000 000 099 069 461 936 223 171 511 595 841 617 920 000 000 144 179 332 384 162 971 648 × 2 = 0 + 0.000 000 198 138 923 872 446 343 023 191 683 235 840 000 000 288 358 664 768 325 943 296;
  • 19) 0.000 000 198 138 923 872 446 343 023 191 683 235 840 000 000 288 358 664 768 325 943 296 × 2 = 0 + 0.000 000 396 277 847 744 892 686 046 383 366 471 680 000 000 576 717 329 536 651 886 592;
  • 20) 0.000 000 396 277 847 744 892 686 046 383 366 471 680 000 000 576 717 329 536 651 886 592 × 2 = 0 + 0.000 000 792 555 695 489 785 372 092 766 732 943 360 000 001 153 434 659 073 303 773 184;
  • 21) 0.000 000 792 555 695 489 785 372 092 766 732 943 360 000 001 153 434 659 073 303 773 184 × 2 = 0 + 0.000 001 585 111 390 979 570 744 185 533 465 886 720 000 002 306 869 318 146 607 546 368;
  • 22) 0.000 001 585 111 390 979 570 744 185 533 465 886 720 000 002 306 869 318 146 607 546 368 × 2 = 0 + 0.000 003 170 222 781 959 141 488 371 066 931 773 440 000 004 613 738 636 293 215 092 736;
  • 23) 0.000 003 170 222 781 959 141 488 371 066 931 773 440 000 004 613 738 636 293 215 092 736 × 2 = 0 + 0.000 006 340 445 563 918 282 976 742 133 863 546 880 000 009 227 477 272 586 430 185 472;
  • 24) 0.000 006 340 445 563 918 282 976 742 133 863 546 880 000 009 227 477 272 586 430 185 472 × 2 = 0 + 0.000 012 680 891 127 836 565 953 484 267 727 093 760 000 018 454 954 545 172 860 370 944;
  • 25) 0.000 012 680 891 127 836 565 953 484 267 727 093 760 000 018 454 954 545 172 860 370 944 × 2 = 0 + 0.000 025 361 782 255 673 131 906 968 535 454 187 520 000 036 909 909 090 345 720 741 888;
  • 26) 0.000 025 361 782 255 673 131 906 968 535 454 187 520 000 036 909 909 090 345 720 741 888 × 2 = 0 + 0.000 050 723 564 511 346 263 813 937 070 908 375 040 000 073 819 818 180 691 441 483 776;
  • 27) 0.000 050 723 564 511 346 263 813 937 070 908 375 040 000 073 819 818 180 691 441 483 776 × 2 = 0 + 0.000 101 447 129 022 692 527 627 874 141 816 750 080 000 147 639 636 361 382 882 967 552;
  • 28) 0.000 101 447 129 022 692 527 627 874 141 816 750 080 000 147 639 636 361 382 882 967 552 × 2 = 0 + 0.000 202 894 258 045 385 055 255 748 283 633 500 160 000 295 279 272 722 765 765 935 104;
  • 29) 0.000 202 894 258 045 385 055 255 748 283 633 500 160 000 295 279 272 722 765 765 935 104 × 2 = 0 + 0.000 405 788 516 090 770 110 511 496 567 267 000 320 000 590 558 545 445 531 531 870 208;
  • 30) 0.000 405 788 516 090 770 110 511 496 567 267 000 320 000 590 558 545 445 531 531 870 208 × 2 = 0 + 0.000 811 577 032 181 540 221 022 993 134 534 000 640 001 181 117 090 891 063 063 740 416;
  • 31) 0.000 811 577 032 181 540 221 022 993 134 534 000 640 001 181 117 090 891 063 063 740 416 × 2 = 0 + 0.001 623 154 064 363 080 442 045 986 269 068 001 280 002 362 234 181 782 126 127 480 832;
  • 32) 0.001 623 154 064 363 080 442 045 986 269 068 001 280 002 362 234 181 782 126 127 480 832 × 2 = 0 + 0.003 246 308 128 726 160 884 091 972 538 136 002 560 004 724 468 363 564 252 254 961 664;
  • 33) 0.003 246 308 128 726 160 884 091 972 538 136 002 560 004 724 468 363 564 252 254 961 664 × 2 = 0 + 0.006 492 616 257 452 321 768 183 945 076 272 005 120 009 448 936 727 128 504 509 923 328;
  • 34) 0.006 492 616 257 452 321 768 183 945 076 272 005 120 009 448 936 727 128 504 509 923 328 × 2 = 0 + 0.012 985 232 514 904 643 536 367 890 152 544 010 240 018 897 873 454 257 009 019 846 656;
  • 35) 0.012 985 232 514 904 643 536 367 890 152 544 010 240 018 897 873 454 257 009 019 846 656 × 2 = 0 + 0.025 970 465 029 809 287 072 735 780 305 088 020 480 037 795 746 908 514 018 039 693 312;
  • 36) 0.025 970 465 029 809 287 072 735 780 305 088 020 480 037 795 746 908 514 018 039 693 312 × 2 = 0 + 0.051 940 930 059 618 574 145 471 560 610 176 040 960 075 591 493 817 028 036 079 386 624;
  • 37) 0.051 940 930 059 618 574 145 471 560 610 176 040 960 075 591 493 817 028 036 079 386 624 × 2 = 0 + 0.103 881 860 119 237 148 290 943 121 220 352 081 920 151 182 987 634 056 072 158 773 248;
  • 38) 0.103 881 860 119 237 148 290 943 121 220 352 081 920 151 182 987 634 056 072 158 773 248 × 2 = 0 + 0.207 763 720 238 474 296 581 886 242 440 704 163 840 302 365 975 268 112 144 317 546 496;
  • 39) 0.207 763 720 238 474 296 581 886 242 440 704 163 840 302 365 975 268 112 144 317 546 496 × 2 = 0 + 0.415 527 440 476 948 593 163 772 484 881 408 327 680 604 731 950 536 224 288 635 092 992;
  • 40) 0.415 527 440 476 948 593 163 772 484 881 408 327 680 604 731 950 536 224 288 635 092 992 × 2 = 0 + 0.831 054 880 953 897 186 327 544 969 762 816 655 361 209 463 901 072 448 577 270 185 984;
  • 41) 0.831 054 880 953 897 186 327 544 969 762 816 655 361 209 463 901 072 448 577 270 185 984 × 2 = 1 + 0.662 109 761 907 794 372 655 089 939 525 633 310 722 418 927 802 144 897 154 540 371 968;
  • 42) 0.662 109 761 907 794 372 655 089 939 525 633 310 722 418 927 802 144 897 154 540 371 968 × 2 = 1 + 0.324 219 523 815 588 745 310 179 879 051 266 621 444 837 855 604 289 794 309 080 743 936;
  • 43) 0.324 219 523 815 588 745 310 179 879 051 266 621 444 837 855 604 289 794 309 080 743 936 × 2 = 0 + 0.648 439 047 631 177 490 620 359 758 102 533 242 889 675 711 208 579 588 618 161 487 872;
  • 44) 0.648 439 047 631 177 490 620 359 758 102 533 242 889 675 711 208 579 588 618 161 487 872 × 2 = 1 + 0.296 878 095 262 354 981 240 719 516 205 066 485 779 351 422 417 159 177 236 322 975 744;
  • 45) 0.296 878 095 262 354 981 240 719 516 205 066 485 779 351 422 417 159 177 236 322 975 744 × 2 = 0 + 0.593 756 190 524 709 962 481 439 032 410 132 971 558 702 844 834 318 354 472 645 951 488;
  • 46) 0.593 756 190 524 709 962 481 439 032 410 132 971 558 702 844 834 318 354 472 645 951 488 × 2 = 1 + 0.187 512 381 049 419 924 962 878 064 820 265 943 117 405 689 668 636 708 945 291 902 976;
  • 47) 0.187 512 381 049 419 924 962 878 064 820 265 943 117 405 689 668 636 708 945 291 902 976 × 2 = 0 + 0.375 024 762 098 839 849 925 756 129 640 531 886 234 811 379 337 273 417 890 583 805 952;
  • 48) 0.375 024 762 098 839 849 925 756 129 640 531 886 234 811 379 337 273 417 890 583 805 952 × 2 = 0 + 0.750 049 524 197 679 699 851 512 259 281 063 772 469 622 758 674 546 835 781 167 611 904;
  • 49) 0.750 049 524 197 679 699 851 512 259 281 063 772 469 622 758 674 546 835 781 167 611 904 × 2 = 1 + 0.500 099 048 395 359 399 703 024 518 562 127 544 939 245 517 349 093 671 562 335 223 808;
  • 50) 0.500 099 048 395 359 399 703 024 518 562 127 544 939 245 517 349 093 671 562 335 223 808 × 2 = 1 + 0.000 198 096 790 718 799 406 049 037 124 255 089 878 491 034 698 187 343 124 670 447 616;
  • 51) 0.000 198 096 790 718 799 406 049 037 124 255 089 878 491 034 698 187 343 124 670 447 616 × 2 = 0 + 0.000 396 193 581 437 598 812 098 074 248 510 179 756 982 069 396 374 686 249 340 895 232;
  • 52) 0.000 396 193 581 437 598 812 098 074 248 510 179 756 982 069 396 374 686 249 340 895 232 × 2 = 0 + 0.000 792 387 162 875 197 624 196 148 497 020 359 513 964 138 792 749 372 498 681 790 464;
  • 53) 0.000 792 387 162 875 197 624 196 148 497 020 359 513 964 138 792 749 372 498 681 790 464 × 2 = 0 + 0.001 584 774 325 750 395 248 392 296 994 040 719 027 928 277 585 498 744 997 363 580 928;
  • 54) 0.001 584 774 325 750 395 248 392 296 994 040 719 027 928 277 585 498 744 997 363 580 928 × 2 = 0 + 0.003 169 548 651 500 790 496 784 593 988 081 438 055 856 555 170 997 489 994 727 161 856;
  • 55) 0.003 169 548 651 500 790 496 784 593 988 081 438 055 856 555 170 997 489 994 727 161 856 × 2 = 0 + 0.006 339 097 303 001 580 993 569 187 976 162 876 111 713 110 341 994 979 989 454 323 712;
  • 56) 0.006 339 097 303 001 580 993 569 187 976 162 876 111 713 110 341 994 979 989 454 323 712 × 2 = 0 + 0.012 678 194 606 003 161 987 138 375 952 325 752 223 426 220 683 989 959 978 908 647 424;
  • 57) 0.012 678 194 606 003 161 987 138 375 952 325 752 223 426 220 683 989 959 978 908 647 424 × 2 = 0 + 0.025 356 389 212 006 323 974 276 751 904 651 504 446 852 441 367 979 919 957 817 294 848;
  • 58) 0.025 356 389 212 006 323 974 276 751 904 651 504 446 852 441 367 979 919 957 817 294 848 × 2 = 0 + 0.050 712 778 424 012 647 948 553 503 809 303 008 893 704 882 735 959 839 915 634 589 696;
  • 59) 0.050 712 778 424 012 647 948 553 503 809 303 008 893 704 882 735 959 839 915 634 589 696 × 2 = 0 + 0.101 425 556 848 025 295 897 107 007 618 606 017 787 409 765 471 919 679 831 269 179 392;
  • 60) 0.101 425 556 848 025 295 897 107 007 618 606 017 787 409 765 471 919 679 831 269 179 392 × 2 = 0 + 0.202 851 113 696 050 591 794 214 015 237 212 035 574 819 530 943 839 359 662 538 358 784;
  • 61) 0.202 851 113 696 050 591 794 214 015 237 212 035 574 819 530 943 839 359 662 538 358 784 × 2 = 0 + 0.405 702 227 392 101 183 588 428 030 474 424 071 149 639 061 887 678 719 325 076 717 568;
  • 62) 0.405 702 227 392 101 183 588 428 030 474 424 071 149 639 061 887 678 719 325 076 717 568 × 2 = 0 + 0.811 404 454 784 202 367 176 856 060 948 848 142 299 278 123 775 357 438 650 153 435 136;
  • 63) 0.811 404 454 784 202 367 176 856 060 948 848 142 299 278 123 775 357 438 650 153 435 136 × 2 = 1 + 0.622 808 909 568 404 734 353 712 121 897 696 284 598 556 247 550 714 877 300 306 870 272;
  • 64) 0.622 808 909 568 404 734 353 712 121 897 696 284 598 556 247 550 714 877 300 306 870 272 × 2 = 1 + 0.245 617 819 136 809 468 707 424 243 795 392 569 197 112 495 101 429 754 600 613 740 544;
  • 65) 0.245 617 819 136 809 468 707 424 243 795 392 569 197 112 495 101 429 754 600 613 740 544 × 2 = 0 + 0.491 235 638 273 618 937 414 848 487 590 785 138 394 224 990 202 859 509 201 227 481 088;
  • 66) 0.491 235 638 273 618 937 414 848 487 590 785 138 394 224 990 202 859 509 201 227 481 088 × 2 = 0 + 0.982 471 276 547 237 874 829 696 975 181 570 276 788 449 980 405 719 018 402 454 962 176;
  • 67) 0.982 471 276 547 237 874 829 696 975 181 570 276 788 449 980 405 719 018 402 454 962 176 × 2 = 1 + 0.964 942 553 094 475 749 659 393 950 363 140 553 576 899 960 811 438 036 804 909 924 352;
  • 68) 0.964 942 553 094 475 749 659 393 950 363 140 553 576 899 960 811 438 036 804 909 924 352 × 2 = 1 + 0.929 885 106 188 951 499 318 787 900 726 281 107 153 799 921 622 876 073 609 819 848 704;
  • 69) 0.929 885 106 188 951 499 318 787 900 726 281 107 153 799 921 622 876 073 609 819 848 704 × 2 = 1 + 0.859 770 212 377 902 998 637 575 801 452 562 214 307 599 843 245 752 147 219 639 697 408;
  • 70) 0.859 770 212 377 902 998 637 575 801 452 562 214 307 599 843 245 752 147 219 639 697 408 × 2 = 1 + 0.719 540 424 755 805 997 275 151 602 905 124 428 615 199 686 491 504 294 439 279 394 816;
  • 71) 0.719 540 424 755 805 997 275 151 602 905 124 428 615 199 686 491 504 294 439 279 394 816 × 2 = 1 + 0.439 080 849 511 611 994 550 303 205 810 248 857 230 399 372 983 008 588 878 558 789 632;
  • 72) 0.439 080 849 511 611 994 550 303 205 810 248 857 230 399 372 983 008 588 878 558 789 632 × 2 = 0 + 0.878 161 699 023 223 989 100 606 411 620 497 714 460 798 745 966 017 177 757 117 579 264;
  • 73) 0.878 161 699 023 223 989 100 606 411 620 497 714 460 798 745 966 017 177 757 117 579 264 × 2 = 1 + 0.756 323 398 046 447 978 201 212 823 240 995 428 921 597 491 932 034 355 514 235 158 528;
  • 74) 0.756 323 398 046 447 978 201 212 823 240 995 428 921 597 491 932 034 355 514 235 158 528 × 2 = 1 + 0.512 646 796 092 895 956 402 425 646 481 990 857 843 194 983 864 068 711 028 470 317 056;
  • 75) 0.512 646 796 092 895 956 402 425 646 481 990 857 843 194 983 864 068 711 028 470 317 056 × 2 = 1 + 0.025 293 592 185 791 912 804 851 292 963 981 715 686 389 967 728 137 422 056 940 634 112;
  • 76) 0.025 293 592 185 791 912 804 851 292 963 981 715 686 389 967 728 137 422 056 940 634 112 × 2 = 0 + 0.050 587 184 371 583 825 609 702 585 927 963 431 372 779 935 456 274 844 113 881 268 224;
  • 77) 0.050 587 184 371 583 825 609 702 585 927 963 431 372 779 935 456 274 844 113 881 268 224 × 2 = 0 + 0.101 174 368 743 167 651 219 405 171 855 926 862 745 559 870 912 549 688 227 762 536 448;
  • 78) 0.101 174 368 743 167 651 219 405 171 855 926 862 745 559 870 912 549 688 227 762 536 448 × 2 = 0 + 0.202 348 737 486 335 302 438 810 343 711 853 725 491 119 741 825 099 376 455 525 072 896;
  • 79) 0.202 348 737 486 335 302 438 810 343 711 853 725 491 119 741 825 099 376 455 525 072 896 × 2 = 0 + 0.404 697 474 972 670 604 877 620 687 423 707 450 982 239 483 650 198 752 911 050 145 792;
  • 80) 0.404 697 474 972 670 604 877 620 687 423 707 450 982 239 483 650 198 752 911 050 145 792 × 2 = 0 + 0.809 394 949 945 341 209 755 241 374 847 414 901 964 478 967 300 397 505 822 100 291 584;
  • 81) 0.809 394 949 945 341 209 755 241 374 847 414 901 964 478 967 300 397 505 822 100 291 584 × 2 = 1 + 0.618 789 899 890 682 419 510 482 749 694 829 803 928 957 934 600 795 011 644 200 583 168;
  • 82) 0.618 789 899 890 682 419 510 482 749 694 829 803 928 957 934 600 795 011 644 200 583 168 × 2 = 1 + 0.237 579 799 781 364 839 020 965 499 389 659 607 857 915 869 201 590 023 288 401 166 336;
  • 83) 0.237 579 799 781 364 839 020 965 499 389 659 607 857 915 869 201 590 023 288 401 166 336 × 2 = 0 + 0.475 159 599 562 729 678 041 930 998 779 319 215 715 831 738 403 180 046 576 802 332 672;
  • 84) 0.475 159 599 562 729 678 041 930 998 779 319 215 715 831 738 403 180 046 576 802 332 672 × 2 = 0 + 0.950 319 199 125 459 356 083 861 997 558 638 431 431 663 476 806 360 093 153 604 665 344;
  • 85) 0.950 319 199 125 459 356 083 861 997 558 638 431 431 663 476 806 360 093 153 604 665 344 × 2 = 1 + 0.900 638 398 250 918 712 167 723 995 117 276 862 863 326 953 612 720 186 307 209 330 688;
  • 86) 0.900 638 398 250 918 712 167 723 995 117 276 862 863 326 953 612 720 186 307 209 330 688 × 2 = 1 + 0.801 276 796 501 837 424 335 447 990 234 553 725 726 653 907 225 440 372 614 418 661 376;
  • 87) 0.801 276 796 501 837 424 335 447 990 234 553 725 726 653 907 225 440 372 614 418 661 376 × 2 = 1 + 0.602 553 593 003 674 848 670 895 980 469 107 451 453 307 814 450 880 745 228 837 322 752;
  • 88) 0.602 553 593 003 674 848 670 895 980 469 107 451 453 307 814 450 880 745 228 837 322 752 × 2 = 1 + 0.205 107 186 007 349 697 341 791 960 938 214 902 906 615 628 901 761 490 457 674 645 504;
  • 89) 0.205 107 186 007 349 697 341 791 960 938 214 902 906 615 628 901 761 490 457 674 645 504 × 2 = 0 + 0.410 214 372 014 699 394 683 583 921 876 429 805 813 231 257 803 522 980 915 349 291 008;
  • 90) 0.410 214 372 014 699 394 683 583 921 876 429 805 813 231 257 803 522 980 915 349 291 008 × 2 = 0 + 0.820 428 744 029 398 789 367 167 843 752 859 611 626 462 515 607 045 961 830 698 582 016;
  • 91) 0.820 428 744 029 398 789 367 167 843 752 859 611 626 462 515 607 045 961 830 698 582 016 × 2 = 1 + 0.640 857 488 058 797 578 734 335 687 505 719 223 252 925 031 214 091 923 661 397 164 032;
  • 92) 0.640 857 488 058 797 578 734 335 687 505 719 223 252 925 031 214 091 923 661 397 164 032 × 2 = 1 + 0.281 714 976 117 595 157 468 671 375 011 438 446 505 850 062 428 183 847 322 794 328 064;
  • 93) 0.281 714 976 117 595 157 468 671 375 011 438 446 505 850 062 428 183 847 322 794 328 064 × 2 = 0 + 0.563 429 952 235 190 314 937 342 750 022 876 893 011 700 124 856 367 694 645 588 656 128;

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:


0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 009(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:

0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 009(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:


0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 009(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


Decimal number 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 100 001 010 011 009 converted to 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

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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