0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 027 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 027(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
0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 027(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: 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 027.

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) 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 027 × 2 = 0 + 0.000 000 000 001 511 680 022 220 202 202 020 200 220 000 000 000 002 054;
  • 2) 0.000 000 000 001 511 680 022 220 202 202 020 200 220 000 000 000 002 054 × 2 = 0 + 0.000 000 000 003 023 360 044 440 404 404 040 400 440 000 000 000 004 108;
  • 3) 0.000 000 000 003 023 360 044 440 404 404 040 400 440 000 000 000 004 108 × 2 = 0 + 0.000 000 000 006 046 720 088 880 808 808 080 800 880 000 000 000 008 216;
  • 4) 0.000 000 000 006 046 720 088 880 808 808 080 800 880 000 000 000 008 216 × 2 = 0 + 0.000 000 000 012 093 440 177 761 617 616 161 601 760 000 000 000 016 432;
  • 5) 0.000 000 000 012 093 440 177 761 617 616 161 601 760 000 000 000 016 432 × 2 = 0 + 0.000 000 000 024 186 880 355 523 235 232 323 203 520 000 000 000 032 864;
  • 6) 0.000 000 000 024 186 880 355 523 235 232 323 203 520 000 000 000 032 864 × 2 = 0 + 0.000 000 000 048 373 760 711 046 470 464 646 407 040 000 000 000 065 728;
  • 7) 0.000 000 000 048 373 760 711 046 470 464 646 407 040 000 000 000 065 728 × 2 = 0 + 0.000 000 000 096 747 521 422 092 940 929 292 814 080 000 000 000 131 456;
  • 8) 0.000 000 000 096 747 521 422 092 940 929 292 814 080 000 000 000 131 456 × 2 = 0 + 0.000 000 000 193 495 042 844 185 881 858 585 628 160 000 000 000 262 912;
  • 9) 0.000 000 000 193 495 042 844 185 881 858 585 628 160 000 000 000 262 912 × 2 = 0 + 0.000 000 000 386 990 085 688 371 763 717 171 256 320 000 000 000 525 824;
  • 10) 0.000 000 000 386 990 085 688 371 763 717 171 256 320 000 000 000 525 824 × 2 = 0 + 0.000 000 000 773 980 171 376 743 527 434 342 512 640 000 000 001 051 648;
  • 11) 0.000 000 000 773 980 171 376 743 527 434 342 512 640 000 000 001 051 648 × 2 = 0 + 0.000 000 001 547 960 342 753 487 054 868 685 025 280 000 000 002 103 296;
  • 12) 0.000 000 001 547 960 342 753 487 054 868 685 025 280 000 000 002 103 296 × 2 = 0 + 0.000 000 003 095 920 685 506 974 109 737 370 050 560 000 000 004 206 592;
  • 13) 0.000 000 003 095 920 685 506 974 109 737 370 050 560 000 000 004 206 592 × 2 = 0 + 0.000 000 006 191 841 371 013 948 219 474 740 101 120 000 000 008 413 184;
  • 14) 0.000 000 006 191 841 371 013 948 219 474 740 101 120 000 000 008 413 184 × 2 = 0 + 0.000 000 012 383 682 742 027 896 438 949 480 202 240 000 000 016 826 368;
  • 15) 0.000 000 012 383 682 742 027 896 438 949 480 202 240 000 000 016 826 368 × 2 = 0 + 0.000 000 024 767 365 484 055 792 877 898 960 404 480 000 000 033 652 736;
  • 16) 0.000 000 024 767 365 484 055 792 877 898 960 404 480 000 000 033 652 736 × 2 = 0 + 0.000 000 049 534 730 968 111 585 755 797 920 808 960 000 000 067 305 472;
  • 17) 0.000 000 049 534 730 968 111 585 755 797 920 808 960 000 000 067 305 472 × 2 = 0 + 0.000 000 099 069 461 936 223 171 511 595 841 617 920 000 000 134 610 944;
  • 18) 0.000 000 099 069 461 936 223 171 511 595 841 617 920 000 000 134 610 944 × 2 = 0 + 0.000 000 198 138 923 872 446 343 023 191 683 235 840 000 000 269 221 888;
  • 19) 0.000 000 198 138 923 872 446 343 023 191 683 235 840 000 000 269 221 888 × 2 = 0 + 0.000 000 396 277 847 744 892 686 046 383 366 471 680 000 000 538 443 776;
  • 20) 0.000 000 396 277 847 744 892 686 046 383 366 471 680 000 000 538 443 776 × 2 = 0 + 0.000 000 792 555 695 489 785 372 092 766 732 943 360 000 001 076 887 552;
  • 21) 0.000 000 792 555 695 489 785 372 092 766 732 943 360 000 001 076 887 552 × 2 = 0 + 0.000 001 585 111 390 979 570 744 185 533 465 886 720 000 002 153 775 104;
  • 22) 0.000 001 585 111 390 979 570 744 185 533 465 886 720 000 002 153 775 104 × 2 = 0 + 0.000 003 170 222 781 959 141 488 371 066 931 773 440 000 004 307 550 208;
  • 23) 0.000 003 170 222 781 959 141 488 371 066 931 773 440 000 004 307 550 208 × 2 = 0 + 0.000 006 340 445 563 918 282 976 742 133 863 546 880 000 008 615 100 416;
  • 24) 0.000 006 340 445 563 918 282 976 742 133 863 546 880 000 008 615 100 416 × 2 = 0 + 0.000 012 680 891 127 836 565 953 484 267 727 093 760 000 017 230 200 832;
  • 25) 0.000 012 680 891 127 836 565 953 484 267 727 093 760 000 017 230 200 832 × 2 = 0 + 0.000 025 361 782 255 673 131 906 968 535 454 187 520 000 034 460 401 664;
  • 26) 0.000 025 361 782 255 673 131 906 968 535 454 187 520 000 034 460 401 664 × 2 = 0 + 0.000 050 723 564 511 346 263 813 937 070 908 375 040 000 068 920 803 328;
  • 27) 0.000 050 723 564 511 346 263 813 937 070 908 375 040 000 068 920 803 328 × 2 = 0 + 0.000 101 447 129 022 692 527 627 874 141 816 750 080 000 137 841 606 656;
  • 28) 0.000 101 447 129 022 692 527 627 874 141 816 750 080 000 137 841 606 656 × 2 = 0 + 0.000 202 894 258 045 385 055 255 748 283 633 500 160 000 275 683 213 312;
  • 29) 0.000 202 894 258 045 385 055 255 748 283 633 500 160 000 275 683 213 312 × 2 = 0 + 0.000 405 788 516 090 770 110 511 496 567 267 000 320 000 551 366 426 624;
  • 30) 0.000 405 788 516 090 770 110 511 496 567 267 000 320 000 551 366 426 624 × 2 = 0 + 0.000 811 577 032 181 540 221 022 993 134 534 000 640 001 102 732 853 248;
  • 31) 0.000 811 577 032 181 540 221 022 993 134 534 000 640 001 102 732 853 248 × 2 = 0 + 0.001 623 154 064 363 080 442 045 986 269 068 001 280 002 205 465 706 496;
  • 32) 0.001 623 154 064 363 080 442 045 986 269 068 001 280 002 205 465 706 496 × 2 = 0 + 0.003 246 308 128 726 160 884 091 972 538 136 002 560 004 410 931 412 992;
  • 33) 0.003 246 308 128 726 160 884 091 972 538 136 002 560 004 410 931 412 992 × 2 = 0 + 0.006 492 616 257 452 321 768 183 945 076 272 005 120 008 821 862 825 984;
  • 34) 0.006 492 616 257 452 321 768 183 945 076 272 005 120 008 821 862 825 984 × 2 = 0 + 0.012 985 232 514 904 643 536 367 890 152 544 010 240 017 643 725 651 968;
  • 35) 0.012 985 232 514 904 643 536 367 890 152 544 010 240 017 643 725 651 968 × 2 = 0 + 0.025 970 465 029 809 287 072 735 780 305 088 020 480 035 287 451 303 936;
  • 36) 0.025 970 465 029 809 287 072 735 780 305 088 020 480 035 287 451 303 936 × 2 = 0 + 0.051 940 930 059 618 574 145 471 560 610 176 040 960 070 574 902 607 872;
  • 37) 0.051 940 930 059 618 574 145 471 560 610 176 040 960 070 574 902 607 872 × 2 = 0 + 0.103 881 860 119 237 148 290 943 121 220 352 081 920 141 149 805 215 744;
  • 38) 0.103 881 860 119 237 148 290 943 121 220 352 081 920 141 149 805 215 744 × 2 = 0 + 0.207 763 720 238 474 296 581 886 242 440 704 163 840 282 299 610 431 488;
  • 39) 0.207 763 720 238 474 296 581 886 242 440 704 163 840 282 299 610 431 488 × 2 = 0 + 0.415 527 440 476 948 593 163 772 484 881 408 327 680 564 599 220 862 976;
  • 40) 0.415 527 440 476 948 593 163 772 484 881 408 327 680 564 599 220 862 976 × 2 = 0 + 0.831 054 880 953 897 186 327 544 969 762 816 655 361 129 198 441 725 952;
  • 41) 0.831 054 880 953 897 186 327 544 969 762 816 655 361 129 198 441 725 952 × 2 = 1 + 0.662 109 761 907 794 372 655 089 939 525 633 310 722 258 396 883 451 904;
  • 42) 0.662 109 761 907 794 372 655 089 939 525 633 310 722 258 396 883 451 904 × 2 = 1 + 0.324 219 523 815 588 745 310 179 879 051 266 621 444 516 793 766 903 808;
  • 43) 0.324 219 523 815 588 745 310 179 879 051 266 621 444 516 793 766 903 808 × 2 = 0 + 0.648 439 047 631 177 490 620 359 758 102 533 242 889 033 587 533 807 616;
  • 44) 0.648 439 047 631 177 490 620 359 758 102 533 242 889 033 587 533 807 616 × 2 = 1 + 0.296 878 095 262 354 981 240 719 516 205 066 485 778 067 175 067 615 232;
  • 45) 0.296 878 095 262 354 981 240 719 516 205 066 485 778 067 175 067 615 232 × 2 = 0 + 0.593 756 190 524 709 962 481 439 032 410 132 971 556 134 350 135 230 464;
  • 46) 0.593 756 190 524 709 962 481 439 032 410 132 971 556 134 350 135 230 464 × 2 = 1 + 0.187 512 381 049 419 924 962 878 064 820 265 943 112 268 700 270 460 928;
  • 47) 0.187 512 381 049 419 924 962 878 064 820 265 943 112 268 700 270 460 928 × 2 = 0 + 0.375 024 762 098 839 849 925 756 129 640 531 886 224 537 400 540 921 856;
  • 48) 0.375 024 762 098 839 849 925 756 129 640 531 886 224 537 400 540 921 856 × 2 = 0 + 0.750 049 524 197 679 699 851 512 259 281 063 772 449 074 801 081 843 712;
  • 49) 0.750 049 524 197 679 699 851 512 259 281 063 772 449 074 801 081 843 712 × 2 = 1 + 0.500 099 048 395 359 399 703 024 518 562 127 544 898 149 602 163 687 424;
  • 50) 0.500 099 048 395 359 399 703 024 518 562 127 544 898 149 602 163 687 424 × 2 = 1 + 0.000 198 096 790 718 799 406 049 037 124 255 089 796 299 204 327 374 848;
  • 51) 0.000 198 096 790 718 799 406 049 037 124 255 089 796 299 204 327 374 848 × 2 = 0 + 0.000 396 193 581 437 598 812 098 074 248 510 179 592 598 408 654 749 696;
  • 52) 0.000 396 193 581 437 598 812 098 074 248 510 179 592 598 408 654 749 696 × 2 = 0 + 0.000 792 387 162 875 197 624 196 148 497 020 359 185 196 817 309 499 392;
  • 53) 0.000 792 387 162 875 197 624 196 148 497 020 359 185 196 817 309 499 392 × 2 = 0 + 0.001 584 774 325 750 395 248 392 296 994 040 718 370 393 634 618 998 784;
  • 54) 0.001 584 774 325 750 395 248 392 296 994 040 718 370 393 634 618 998 784 × 2 = 0 + 0.003 169 548 651 500 790 496 784 593 988 081 436 740 787 269 237 997 568;
  • 55) 0.003 169 548 651 500 790 496 784 593 988 081 436 740 787 269 237 997 568 × 2 = 0 + 0.006 339 097 303 001 580 993 569 187 976 162 873 481 574 538 475 995 136;
  • 56) 0.006 339 097 303 001 580 993 569 187 976 162 873 481 574 538 475 995 136 × 2 = 0 + 0.012 678 194 606 003 161 987 138 375 952 325 746 963 149 076 951 990 272;
  • 57) 0.012 678 194 606 003 161 987 138 375 952 325 746 963 149 076 951 990 272 × 2 = 0 + 0.025 356 389 212 006 323 974 276 751 904 651 493 926 298 153 903 980 544;
  • 58) 0.025 356 389 212 006 323 974 276 751 904 651 493 926 298 153 903 980 544 × 2 = 0 + 0.050 712 778 424 012 647 948 553 503 809 302 987 852 596 307 807 961 088;
  • 59) 0.050 712 778 424 012 647 948 553 503 809 302 987 852 596 307 807 961 088 × 2 = 0 + 0.101 425 556 848 025 295 897 107 007 618 605 975 705 192 615 615 922 176;
  • 60) 0.101 425 556 848 025 295 897 107 007 618 605 975 705 192 615 615 922 176 × 2 = 0 + 0.202 851 113 696 050 591 794 214 015 237 211 951 410 385 231 231 844 352;
  • 61) 0.202 851 113 696 050 591 794 214 015 237 211 951 410 385 231 231 844 352 × 2 = 0 + 0.405 702 227 392 101 183 588 428 030 474 423 902 820 770 462 463 688 704;
  • 62) 0.405 702 227 392 101 183 588 428 030 474 423 902 820 770 462 463 688 704 × 2 = 0 + 0.811 404 454 784 202 367 176 856 060 948 847 805 641 540 924 927 377 408;
  • 63) 0.811 404 454 784 202 367 176 856 060 948 847 805 641 540 924 927 377 408 × 2 = 1 + 0.622 808 909 568 404 734 353 712 121 897 695 611 283 081 849 854 754 816;
  • 64) 0.622 808 909 568 404 734 353 712 121 897 695 611 283 081 849 854 754 816 × 2 = 1 + 0.245 617 819 136 809 468 707 424 243 795 391 222 566 163 699 709 509 632;
  • 65) 0.245 617 819 136 809 468 707 424 243 795 391 222 566 163 699 709 509 632 × 2 = 0 + 0.491 235 638 273 618 937 414 848 487 590 782 445 132 327 399 419 019 264;
  • 66) 0.491 235 638 273 618 937 414 848 487 590 782 445 132 327 399 419 019 264 × 2 = 0 + 0.982 471 276 547 237 874 829 696 975 181 564 890 264 654 798 838 038 528;
  • 67) 0.982 471 276 547 237 874 829 696 975 181 564 890 264 654 798 838 038 528 × 2 = 1 + 0.964 942 553 094 475 749 659 393 950 363 129 780 529 309 597 676 077 056;
  • 68) 0.964 942 553 094 475 749 659 393 950 363 129 780 529 309 597 676 077 056 × 2 = 1 + 0.929 885 106 188 951 499 318 787 900 726 259 561 058 619 195 352 154 112;
  • 69) 0.929 885 106 188 951 499 318 787 900 726 259 561 058 619 195 352 154 112 × 2 = 1 + 0.859 770 212 377 902 998 637 575 801 452 519 122 117 238 390 704 308 224;
  • 70) 0.859 770 212 377 902 998 637 575 801 452 519 122 117 238 390 704 308 224 × 2 = 1 + 0.719 540 424 755 805 997 275 151 602 905 038 244 234 476 781 408 616 448;
  • 71) 0.719 540 424 755 805 997 275 151 602 905 038 244 234 476 781 408 616 448 × 2 = 1 + 0.439 080 849 511 611 994 550 303 205 810 076 488 468 953 562 817 232 896;
  • 72) 0.439 080 849 511 611 994 550 303 205 810 076 488 468 953 562 817 232 896 × 2 = 0 + 0.878 161 699 023 223 989 100 606 411 620 152 976 937 907 125 634 465 792;
  • 73) 0.878 161 699 023 223 989 100 606 411 620 152 976 937 907 125 634 465 792 × 2 = 1 + 0.756 323 398 046 447 978 201 212 823 240 305 953 875 814 251 268 931 584;
  • 74) 0.756 323 398 046 447 978 201 212 823 240 305 953 875 814 251 268 931 584 × 2 = 1 + 0.512 646 796 092 895 956 402 425 646 480 611 907 751 628 502 537 863 168;
  • 75) 0.512 646 796 092 895 956 402 425 646 480 611 907 751 628 502 537 863 168 × 2 = 1 + 0.025 293 592 185 791 912 804 851 292 961 223 815 503 257 005 075 726 336;
  • 76) 0.025 293 592 185 791 912 804 851 292 961 223 815 503 257 005 075 726 336 × 2 = 0 + 0.050 587 184 371 583 825 609 702 585 922 447 631 006 514 010 151 452 672;
  • 77) 0.050 587 184 371 583 825 609 702 585 922 447 631 006 514 010 151 452 672 × 2 = 0 + 0.101 174 368 743 167 651 219 405 171 844 895 262 013 028 020 302 905 344;
  • 78) 0.101 174 368 743 167 651 219 405 171 844 895 262 013 028 020 302 905 344 × 2 = 0 + 0.202 348 737 486 335 302 438 810 343 689 790 524 026 056 040 605 810 688;
  • 79) 0.202 348 737 486 335 302 438 810 343 689 790 524 026 056 040 605 810 688 × 2 = 0 + 0.404 697 474 972 670 604 877 620 687 379 581 048 052 112 081 211 621 376;
  • 80) 0.404 697 474 972 670 604 877 620 687 379 581 048 052 112 081 211 621 376 × 2 = 0 + 0.809 394 949 945 341 209 755 241 374 759 162 096 104 224 162 423 242 752;
  • 81) 0.809 394 949 945 341 209 755 241 374 759 162 096 104 224 162 423 242 752 × 2 = 1 + 0.618 789 899 890 682 419 510 482 749 518 324 192 208 448 324 846 485 504;
  • 82) 0.618 789 899 890 682 419 510 482 749 518 324 192 208 448 324 846 485 504 × 2 = 1 + 0.237 579 799 781 364 839 020 965 499 036 648 384 416 896 649 692 971 008;
  • 83) 0.237 579 799 781 364 839 020 965 499 036 648 384 416 896 649 692 971 008 × 2 = 0 + 0.475 159 599 562 729 678 041 930 998 073 296 768 833 793 299 385 942 016;
  • 84) 0.475 159 599 562 729 678 041 930 998 073 296 768 833 793 299 385 942 016 × 2 = 0 + 0.950 319 199 125 459 356 083 861 996 146 593 537 667 586 598 771 884 032;
  • 85) 0.950 319 199 125 459 356 083 861 996 146 593 537 667 586 598 771 884 032 × 2 = 1 + 0.900 638 398 250 918 712 167 723 992 293 187 075 335 173 197 543 768 064;
  • 86) 0.900 638 398 250 918 712 167 723 992 293 187 075 335 173 197 543 768 064 × 2 = 1 + 0.801 276 796 501 837 424 335 447 984 586 374 150 670 346 395 087 536 128;
  • 87) 0.801 276 796 501 837 424 335 447 984 586 374 150 670 346 395 087 536 128 × 2 = 1 + 0.602 553 593 003 674 848 670 895 969 172 748 301 340 692 790 175 072 256;
  • 88) 0.602 553 593 003 674 848 670 895 969 172 748 301 340 692 790 175 072 256 × 2 = 1 + 0.205 107 186 007 349 697 341 791 938 345 496 602 681 385 580 350 144 512;
  • 89) 0.205 107 186 007 349 697 341 791 938 345 496 602 681 385 580 350 144 512 × 2 = 0 + 0.410 214 372 014 699 394 683 583 876 690 993 205 362 771 160 700 289 024;
  • 90) 0.410 214 372 014 699 394 683 583 876 690 993 205 362 771 160 700 289 024 × 2 = 0 + 0.820 428 744 029 398 789 367 167 753 381 986 410 725 542 321 400 578 048;
  • 91) 0.820 428 744 029 398 789 367 167 753 381 986 410 725 542 321 400 578 048 × 2 = 1 + 0.640 857 488 058 797 578 734 335 506 763 972 821 451 084 642 801 156 096;
  • 92) 0.640 857 488 058 797 578 734 335 506 763 972 821 451 084 642 801 156 096 × 2 = 1 + 0.281 714 976 117 595 157 468 671 013 527 945 642 902 169 285 602 312 192;
  • 93) 0.281 714 976 117 595 157 468 671 013 527 945 642 902 169 285 602 312 192 × 2 = 0 + 0.563 429 952 235 190 314 937 342 027 055 891 285 804 338 571 204 624 384;

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:


0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 027(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:

0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 027(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:


0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 027(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 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 027 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

» Convert decimal 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 000 971 to 64 bit double precision IEEE 754 binary floating point representation standard

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» Month 10, 2026 [October]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: October - by our visitors

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