0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 183 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 183(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
0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 183(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: 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 183.

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) 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 183 × 2 = 0 + 0.000 000 000 001 511 680 022 220 202 202 020 200 220 000 000 000 002 366;
  • 2) 0.000 000 000 001 511 680 022 220 202 202 020 200 220 000 000 000 002 366 × 2 = 0 + 0.000 000 000 003 023 360 044 440 404 404 040 400 440 000 000 000 004 732;
  • 3) 0.000 000 000 003 023 360 044 440 404 404 040 400 440 000 000 000 004 732 × 2 = 0 + 0.000 000 000 006 046 720 088 880 808 808 080 800 880 000 000 000 009 464;
  • 4) 0.000 000 000 006 046 720 088 880 808 808 080 800 880 000 000 000 009 464 × 2 = 0 + 0.000 000 000 012 093 440 177 761 617 616 161 601 760 000 000 000 018 928;
  • 5) 0.000 000 000 012 093 440 177 761 617 616 161 601 760 000 000 000 018 928 × 2 = 0 + 0.000 000 000 024 186 880 355 523 235 232 323 203 520 000 000 000 037 856;
  • 6) 0.000 000 000 024 186 880 355 523 235 232 323 203 520 000 000 000 037 856 × 2 = 0 + 0.000 000 000 048 373 760 711 046 470 464 646 407 040 000 000 000 075 712;
  • 7) 0.000 000 000 048 373 760 711 046 470 464 646 407 040 000 000 000 075 712 × 2 = 0 + 0.000 000 000 096 747 521 422 092 940 929 292 814 080 000 000 000 151 424;
  • 8) 0.000 000 000 096 747 521 422 092 940 929 292 814 080 000 000 000 151 424 × 2 = 0 + 0.000 000 000 193 495 042 844 185 881 858 585 628 160 000 000 000 302 848;
  • 9) 0.000 000 000 193 495 042 844 185 881 858 585 628 160 000 000 000 302 848 × 2 = 0 + 0.000 000 000 386 990 085 688 371 763 717 171 256 320 000 000 000 605 696;
  • 10) 0.000 000 000 386 990 085 688 371 763 717 171 256 320 000 000 000 605 696 × 2 = 0 + 0.000 000 000 773 980 171 376 743 527 434 342 512 640 000 000 001 211 392;
  • 11) 0.000 000 000 773 980 171 376 743 527 434 342 512 640 000 000 001 211 392 × 2 = 0 + 0.000 000 001 547 960 342 753 487 054 868 685 025 280 000 000 002 422 784;
  • 12) 0.000 000 001 547 960 342 753 487 054 868 685 025 280 000 000 002 422 784 × 2 = 0 + 0.000 000 003 095 920 685 506 974 109 737 370 050 560 000 000 004 845 568;
  • 13) 0.000 000 003 095 920 685 506 974 109 737 370 050 560 000 000 004 845 568 × 2 = 0 + 0.000 000 006 191 841 371 013 948 219 474 740 101 120 000 000 009 691 136;
  • 14) 0.000 000 006 191 841 371 013 948 219 474 740 101 120 000 000 009 691 136 × 2 = 0 + 0.000 000 012 383 682 742 027 896 438 949 480 202 240 000 000 019 382 272;
  • 15) 0.000 000 012 383 682 742 027 896 438 949 480 202 240 000 000 019 382 272 × 2 = 0 + 0.000 000 024 767 365 484 055 792 877 898 960 404 480 000 000 038 764 544;
  • 16) 0.000 000 024 767 365 484 055 792 877 898 960 404 480 000 000 038 764 544 × 2 = 0 + 0.000 000 049 534 730 968 111 585 755 797 920 808 960 000 000 077 529 088;
  • 17) 0.000 000 049 534 730 968 111 585 755 797 920 808 960 000 000 077 529 088 × 2 = 0 + 0.000 000 099 069 461 936 223 171 511 595 841 617 920 000 000 155 058 176;
  • 18) 0.000 000 099 069 461 936 223 171 511 595 841 617 920 000 000 155 058 176 × 2 = 0 + 0.000 000 198 138 923 872 446 343 023 191 683 235 840 000 000 310 116 352;
  • 19) 0.000 000 198 138 923 872 446 343 023 191 683 235 840 000 000 310 116 352 × 2 = 0 + 0.000 000 396 277 847 744 892 686 046 383 366 471 680 000 000 620 232 704;
  • 20) 0.000 000 396 277 847 744 892 686 046 383 366 471 680 000 000 620 232 704 × 2 = 0 + 0.000 000 792 555 695 489 785 372 092 766 732 943 360 000 001 240 465 408;
  • 21) 0.000 000 792 555 695 489 785 372 092 766 732 943 360 000 001 240 465 408 × 2 = 0 + 0.000 001 585 111 390 979 570 744 185 533 465 886 720 000 002 480 930 816;
  • 22) 0.000 001 585 111 390 979 570 744 185 533 465 886 720 000 002 480 930 816 × 2 = 0 + 0.000 003 170 222 781 959 141 488 371 066 931 773 440 000 004 961 861 632;
  • 23) 0.000 003 170 222 781 959 141 488 371 066 931 773 440 000 004 961 861 632 × 2 = 0 + 0.000 006 340 445 563 918 282 976 742 133 863 546 880 000 009 923 723 264;
  • 24) 0.000 006 340 445 563 918 282 976 742 133 863 546 880 000 009 923 723 264 × 2 = 0 + 0.000 012 680 891 127 836 565 953 484 267 727 093 760 000 019 847 446 528;
  • 25) 0.000 012 680 891 127 836 565 953 484 267 727 093 760 000 019 847 446 528 × 2 = 0 + 0.000 025 361 782 255 673 131 906 968 535 454 187 520 000 039 694 893 056;
  • 26) 0.000 025 361 782 255 673 131 906 968 535 454 187 520 000 039 694 893 056 × 2 = 0 + 0.000 050 723 564 511 346 263 813 937 070 908 375 040 000 079 389 786 112;
  • 27) 0.000 050 723 564 511 346 263 813 937 070 908 375 040 000 079 389 786 112 × 2 = 0 + 0.000 101 447 129 022 692 527 627 874 141 816 750 080 000 158 779 572 224;
  • 28) 0.000 101 447 129 022 692 527 627 874 141 816 750 080 000 158 779 572 224 × 2 = 0 + 0.000 202 894 258 045 385 055 255 748 283 633 500 160 000 317 559 144 448;
  • 29) 0.000 202 894 258 045 385 055 255 748 283 633 500 160 000 317 559 144 448 × 2 = 0 + 0.000 405 788 516 090 770 110 511 496 567 267 000 320 000 635 118 288 896;
  • 30) 0.000 405 788 516 090 770 110 511 496 567 267 000 320 000 635 118 288 896 × 2 = 0 + 0.000 811 577 032 181 540 221 022 993 134 534 000 640 001 270 236 577 792;
  • 31) 0.000 811 577 032 181 540 221 022 993 134 534 000 640 001 270 236 577 792 × 2 = 0 + 0.001 623 154 064 363 080 442 045 986 269 068 001 280 002 540 473 155 584;
  • 32) 0.001 623 154 064 363 080 442 045 986 269 068 001 280 002 540 473 155 584 × 2 = 0 + 0.003 246 308 128 726 160 884 091 972 538 136 002 560 005 080 946 311 168;
  • 33) 0.003 246 308 128 726 160 884 091 972 538 136 002 560 005 080 946 311 168 × 2 = 0 + 0.006 492 616 257 452 321 768 183 945 076 272 005 120 010 161 892 622 336;
  • 34) 0.006 492 616 257 452 321 768 183 945 076 272 005 120 010 161 892 622 336 × 2 = 0 + 0.012 985 232 514 904 643 536 367 890 152 544 010 240 020 323 785 244 672;
  • 35) 0.012 985 232 514 904 643 536 367 890 152 544 010 240 020 323 785 244 672 × 2 = 0 + 0.025 970 465 029 809 287 072 735 780 305 088 020 480 040 647 570 489 344;
  • 36) 0.025 970 465 029 809 287 072 735 780 305 088 020 480 040 647 570 489 344 × 2 = 0 + 0.051 940 930 059 618 574 145 471 560 610 176 040 960 081 295 140 978 688;
  • 37) 0.051 940 930 059 618 574 145 471 560 610 176 040 960 081 295 140 978 688 × 2 = 0 + 0.103 881 860 119 237 148 290 943 121 220 352 081 920 162 590 281 957 376;
  • 38) 0.103 881 860 119 237 148 290 943 121 220 352 081 920 162 590 281 957 376 × 2 = 0 + 0.207 763 720 238 474 296 581 886 242 440 704 163 840 325 180 563 914 752;
  • 39) 0.207 763 720 238 474 296 581 886 242 440 704 163 840 325 180 563 914 752 × 2 = 0 + 0.415 527 440 476 948 593 163 772 484 881 408 327 680 650 361 127 829 504;
  • 40) 0.415 527 440 476 948 593 163 772 484 881 408 327 680 650 361 127 829 504 × 2 = 0 + 0.831 054 880 953 897 186 327 544 969 762 816 655 361 300 722 255 659 008;
  • 41) 0.831 054 880 953 897 186 327 544 969 762 816 655 361 300 722 255 659 008 × 2 = 1 + 0.662 109 761 907 794 372 655 089 939 525 633 310 722 601 444 511 318 016;
  • 42) 0.662 109 761 907 794 372 655 089 939 525 633 310 722 601 444 511 318 016 × 2 = 1 + 0.324 219 523 815 588 745 310 179 879 051 266 621 445 202 889 022 636 032;
  • 43) 0.324 219 523 815 588 745 310 179 879 051 266 621 445 202 889 022 636 032 × 2 = 0 + 0.648 439 047 631 177 490 620 359 758 102 533 242 890 405 778 045 272 064;
  • 44) 0.648 439 047 631 177 490 620 359 758 102 533 242 890 405 778 045 272 064 × 2 = 1 + 0.296 878 095 262 354 981 240 719 516 205 066 485 780 811 556 090 544 128;
  • 45) 0.296 878 095 262 354 981 240 719 516 205 066 485 780 811 556 090 544 128 × 2 = 0 + 0.593 756 190 524 709 962 481 439 032 410 132 971 561 623 112 181 088 256;
  • 46) 0.593 756 190 524 709 962 481 439 032 410 132 971 561 623 112 181 088 256 × 2 = 1 + 0.187 512 381 049 419 924 962 878 064 820 265 943 123 246 224 362 176 512;
  • 47) 0.187 512 381 049 419 924 962 878 064 820 265 943 123 246 224 362 176 512 × 2 = 0 + 0.375 024 762 098 839 849 925 756 129 640 531 886 246 492 448 724 353 024;
  • 48) 0.375 024 762 098 839 849 925 756 129 640 531 886 246 492 448 724 353 024 × 2 = 0 + 0.750 049 524 197 679 699 851 512 259 281 063 772 492 984 897 448 706 048;
  • 49) 0.750 049 524 197 679 699 851 512 259 281 063 772 492 984 897 448 706 048 × 2 = 1 + 0.500 099 048 395 359 399 703 024 518 562 127 544 985 969 794 897 412 096;
  • 50) 0.500 099 048 395 359 399 703 024 518 562 127 544 985 969 794 897 412 096 × 2 = 1 + 0.000 198 096 790 718 799 406 049 037 124 255 089 971 939 589 794 824 192;
  • 51) 0.000 198 096 790 718 799 406 049 037 124 255 089 971 939 589 794 824 192 × 2 = 0 + 0.000 396 193 581 437 598 812 098 074 248 510 179 943 879 179 589 648 384;
  • 52) 0.000 396 193 581 437 598 812 098 074 248 510 179 943 879 179 589 648 384 × 2 = 0 + 0.000 792 387 162 875 197 624 196 148 497 020 359 887 758 359 179 296 768;
  • 53) 0.000 792 387 162 875 197 624 196 148 497 020 359 887 758 359 179 296 768 × 2 = 0 + 0.001 584 774 325 750 395 248 392 296 994 040 719 775 516 718 358 593 536;
  • 54) 0.001 584 774 325 750 395 248 392 296 994 040 719 775 516 718 358 593 536 × 2 = 0 + 0.003 169 548 651 500 790 496 784 593 988 081 439 551 033 436 717 187 072;
  • 55) 0.003 169 548 651 500 790 496 784 593 988 081 439 551 033 436 717 187 072 × 2 = 0 + 0.006 339 097 303 001 580 993 569 187 976 162 879 102 066 873 434 374 144;
  • 56) 0.006 339 097 303 001 580 993 569 187 976 162 879 102 066 873 434 374 144 × 2 = 0 + 0.012 678 194 606 003 161 987 138 375 952 325 758 204 133 746 868 748 288;
  • 57) 0.012 678 194 606 003 161 987 138 375 952 325 758 204 133 746 868 748 288 × 2 = 0 + 0.025 356 389 212 006 323 974 276 751 904 651 516 408 267 493 737 496 576;
  • 58) 0.025 356 389 212 006 323 974 276 751 904 651 516 408 267 493 737 496 576 × 2 = 0 + 0.050 712 778 424 012 647 948 553 503 809 303 032 816 534 987 474 993 152;
  • 59) 0.050 712 778 424 012 647 948 553 503 809 303 032 816 534 987 474 993 152 × 2 = 0 + 0.101 425 556 848 025 295 897 107 007 618 606 065 633 069 974 949 986 304;
  • 60) 0.101 425 556 848 025 295 897 107 007 618 606 065 633 069 974 949 986 304 × 2 = 0 + 0.202 851 113 696 050 591 794 214 015 237 212 131 266 139 949 899 972 608;
  • 61) 0.202 851 113 696 050 591 794 214 015 237 212 131 266 139 949 899 972 608 × 2 = 0 + 0.405 702 227 392 101 183 588 428 030 474 424 262 532 279 899 799 945 216;
  • 62) 0.405 702 227 392 101 183 588 428 030 474 424 262 532 279 899 799 945 216 × 2 = 0 + 0.811 404 454 784 202 367 176 856 060 948 848 525 064 559 799 599 890 432;
  • 63) 0.811 404 454 784 202 367 176 856 060 948 848 525 064 559 799 599 890 432 × 2 = 1 + 0.622 808 909 568 404 734 353 712 121 897 697 050 129 119 599 199 780 864;
  • 64) 0.622 808 909 568 404 734 353 712 121 897 697 050 129 119 599 199 780 864 × 2 = 1 + 0.245 617 819 136 809 468 707 424 243 795 394 100 258 239 198 399 561 728;
  • 65) 0.245 617 819 136 809 468 707 424 243 795 394 100 258 239 198 399 561 728 × 2 = 0 + 0.491 235 638 273 618 937 414 848 487 590 788 200 516 478 396 799 123 456;
  • 66) 0.491 235 638 273 618 937 414 848 487 590 788 200 516 478 396 799 123 456 × 2 = 0 + 0.982 471 276 547 237 874 829 696 975 181 576 401 032 956 793 598 246 912;
  • 67) 0.982 471 276 547 237 874 829 696 975 181 576 401 032 956 793 598 246 912 × 2 = 1 + 0.964 942 553 094 475 749 659 393 950 363 152 802 065 913 587 196 493 824;
  • 68) 0.964 942 553 094 475 749 659 393 950 363 152 802 065 913 587 196 493 824 × 2 = 1 + 0.929 885 106 188 951 499 318 787 900 726 305 604 131 827 174 392 987 648;
  • 69) 0.929 885 106 188 951 499 318 787 900 726 305 604 131 827 174 392 987 648 × 2 = 1 + 0.859 770 212 377 902 998 637 575 801 452 611 208 263 654 348 785 975 296;
  • 70) 0.859 770 212 377 902 998 637 575 801 452 611 208 263 654 348 785 975 296 × 2 = 1 + 0.719 540 424 755 805 997 275 151 602 905 222 416 527 308 697 571 950 592;
  • 71) 0.719 540 424 755 805 997 275 151 602 905 222 416 527 308 697 571 950 592 × 2 = 1 + 0.439 080 849 511 611 994 550 303 205 810 444 833 054 617 395 143 901 184;
  • 72) 0.439 080 849 511 611 994 550 303 205 810 444 833 054 617 395 143 901 184 × 2 = 0 + 0.878 161 699 023 223 989 100 606 411 620 889 666 109 234 790 287 802 368;
  • 73) 0.878 161 699 023 223 989 100 606 411 620 889 666 109 234 790 287 802 368 × 2 = 1 + 0.756 323 398 046 447 978 201 212 823 241 779 332 218 469 580 575 604 736;
  • 74) 0.756 323 398 046 447 978 201 212 823 241 779 332 218 469 580 575 604 736 × 2 = 1 + 0.512 646 796 092 895 956 402 425 646 483 558 664 436 939 161 151 209 472;
  • 75) 0.512 646 796 092 895 956 402 425 646 483 558 664 436 939 161 151 209 472 × 2 = 1 + 0.025 293 592 185 791 912 804 851 292 967 117 328 873 878 322 302 418 944;
  • 76) 0.025 293 592 185 791 912 804 851 292 967 117 328 873 878 322 302 418 944 × 2 = 0 + 0.050 587 184 371 583 825 609 702 585 934 234 657 747 756 644 604 837 888;
  • 77) 0.050 587 184 371 583 825 609 702 585 934 234 657 747 756 644 604 837 888 × 2 = 0 + 0.101 174 368 743 167 651 219 405 171 868 469 315 495 513 289 209 675 776;
  • 78) 0.101 174 368 743 167 651 219 405 171 868 469 315 495 513 289 209 675 776 × 2 = 0 + 0.202 348 737 486 335 302 438 810 343 736 938 630 991 026 578 419 351 552;
  • 79) 0.202 348 737 486 335 302 438 810 343 736 938 630 991 026 578 419 351 552 × 2 = 0 + 0.404 697 474 972 670 604 877 620 687 473 877 261 982 053 156 838 703 104;
  • 80) 0.404 697 474 972 670 604 877 620 687 473 877 261 982 053 156 838 703 104 × 2 = 0 + 0.809 394 949 945 341 209 755 241 374 947 754 523 964 106 313 677 406 208;
  • 81) 0.809 394 949 945 341 209 755 241 374 947 754 523 964 106 313 677 406 208 × 2 = 1 + 0.618 789 899 890 682 419 510 482 749 895 509 047 928 212 627 354 812 416;
  • 82) 0.618 789 899 890 682 419 510 482 749 895 509 047 928 212 627 354 812 416 × 2 = 1 + 0.237 579 799 781 364 839 020 965 499 791 018 095 856 425 254 709 624 832;
  • 83) 0.237 579 799 781 364 839 020 965 499 791 018 095 856 425 254 709 624 832 × 2 = 0 + 0.475 159 599 562 729 678 041 930 999 582 036 191 712 850 509 419 249 664;
  • 84) 0.475 159 599 562 729 678 041 930 999 582 036 191 712 850 509 419 249 664 × 2 = 0 + 0.950 319 199 125 459 356 083 861 999 164 072 383 425 701 018 838 499 328;
  • 85) 0.950 319 199 125 459 356 083 861 999 164 072 383 425 701 018 838 499 328 × 2 = 1 + 0.900 638 398 250 918 712 167 723 998 328 144 766 851 402 037 676 998 656;
  • 86) 0.900 638 398 250 918 712 167 723 998 328 144 766 851 402 037 676 998 656 × 2 = 1 + 0.801 276 796 501 837 424 335 447 996 656 289 533 702 804 075 353 997 312;
  • 87) 0.801 276 796 501 837 424 335 447 996 656 289 533 702 804 075 353 997 312 × 2 = 1 + 0.602 553 593 003 674 848 670 895 993 312 579 067 405 608 150 707 994 624;
  • 88) 0.602 553 593 003 674 848 670 895 993 312 579 067 405 608 150 707 994 624 × 2 = 1 + 0.205 107 186 007 349 697 341 791 986 625 158 134 811 216 301 415 989 248;
  • 89) 0.205 107 186 007 349 697 341 791 986 625 158 134 811 216 301 415 989 248 × 2 = 0 + 0.410 214 372 014 699 394 683 583 973 250 316 269 622 432 602 831 978 496;
  • 90) 0.410 214 372 014 699 394 683 583 973 250 316 269 622 432 602 831 978 496 × 2 = 0 + 0.820 428 744 029 398 789 367 167 946 500 632 539 244 865 205 663 956 992;
  • 91) 0.820 428 744 029 398 789 367 167 946 500 632 539 244 865 205 663 956 992 × 2 = 1 + 0.640 857 488 058 797 578 734 335 893 001 265 078 489 730 411 327 913 984;
  • 92) 0.640 857 488 058 797 578 734 335 893 001 265 078 489 730 411 327 913 984 × 2 = 1 + 0.281 714 976 117 595 157 468 671 786 002 530 156 979 460 822 655 827 968;
  • 93) 0.281 714 976 117 595 157 468 671 786 002 530 156 979 460 822 655 827 968 × 2 = 0 + 0.563 429 952 235 190 314 937 343 572 005 060 313 958 921 645 311 655 936;

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:


0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 183(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:

0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 183(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:


0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 183(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 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 183 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 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 214 to 64 bit double precision IEEE 754 binary floating point representation standard

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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