0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 024 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 024(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
0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 024(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: 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 024.

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) 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 024 × 2 = 0 + 0.000 000 000 001 511 680 022 220 202 202 020 200 220 000 000 000 002 048;
  • 2) 0.000 000 000 001 511 680 022 220 202 202 020 200 220 000 000 000 002 048 × 2 = 0 + 0.000 000 000 003 023 360 044 440 404 404 040 400 440 000 000 000 004 096;
  • 3) 0.000 000 000 003 023 360 044 440 404 404 040 400 440 000 000 000 004 096 × 2 = 0 + 0.000 000 000 006 046 720 088 880 808 808 080 800 880 000 000 000 008 192;
  • 4) 0.000 000 000 006 046 720 088 880 808 808 080 800 880 000 000 000 008 192 × 2 = 0 + 0.000 000 000 012 093 440 177 761 617 616 161 601 760 000 000 000 016 384;
  • 5) 0.000 000 000 012 093 440 177 761 617 616 161 601 760 000 000 000 016 384 × 2 = 0 + 0.000 000 000 024 186 880 355 523 235 232 323 203 520 000 000 000 032 768;
  • 6) 0.000 000 000 024 186 880 355 523 235 232 323 203 520 000 000 000 032 768 × 2 = 0 + 0.000 000 000 048 373 760 711 046 470 464 646 407 040 000 000 000 065 536;
  • 7) 0.000 000 000 048 373 760 711 046 470 464 646 407 040 000 000 000 065 536 × 2 = 0 + 0.000 000 000 096 747 521 422 092 940 929 292 814 080 000 000 000 131 072;
  • 8) 0.000 000 000 096 747 521 422 092 940 929 292 814 080 000 000 000 131 072 × 2 = 0 + 0.000 000 000 193 495 042 844 185 881 858 585 628 160 000 000 000 262 144;
  • 9) 0.000 000 000 193 495 042 844 185 881 858 585 628 160 000 000 000 262 144 × 2 = 0 + 0.000 000 000 386 990 085 688 371 763 717 171 256 320 000 000 000 524 288;
  • 10) 0.000 000 000 386 990 085 688 371 763 717 171 256 320 000 000 000 524 288 × 2 = 0 + 0.000 000 000 773 980 171 376 743 527 434 342 512 640 000 000 001 048 576;
  • 11) 0.000 000 000 773 980 171 376 743 527 434 342 512 640 000 000 001 048 576 × 2 = 0 + 0.000 000 001 547 960 342 753 487 054 868 685 025 280 000 000 002 097 152;
  • 12) 0.000 000 001 547 960 342 753 487 054 868 685 025 280 000 000 002 097 152 × 2 = 0 + 0.000 000 003 095 920 685 506 974 109 737 370 050 560 000 000 004 194 304;
  • 13) 0.000 000 003 095 920 685 506 974 109 737 370 050 560 000 000 004 194 304 × 2 = 0 + 0.000 000 006 191 841 371 013 948 219 474 740 101 120 000 000 008 388 608;
  • 14) 0.000 000 006 191 841 371 013 948 219 474 740 101 120 000 000 008 388 608 × 2 = 0 + 0.000 000 012 383 682 742 027 896 438 949 480 202 240 000 000 016 777 216;
  • 15) 0.000 000 012 383 682 742 027 896 438 949 480 202 240 000 000 016 777 216 × 2 = 0 + 0.000 000 024 767 365 484 055 792 877 898 960 404 480 000 000 033 554 432;
  • 16) 0.000 000 024 767 365 484 055 792 877 898 960 404 480 000 000 033 554 432 × 2 = 0 + 0.000 000 049 534 730 968 111 585 755 797 920 808 960 000 000 067 108 864;
  • 17) 0.000 000 049 534 730 968 111 585 755 797 920 808 960 000 000 067 108 864 × 2 = 0 + 0.000 000 099 069 461 936 223 171 511 595 841 617 920 000 000 134 217 728;
  • 18) 0.000 000 099 069 461 936 223 171 511 595 841 617 920 000 000 134 217 728 × 2 = 0 + 0.000 000 198 138 923 872 446 343 023 191 683 235 840 000 000 268 435 456;
  • 19) 0.000 000 198 138 923 872 446 343 023 191 683 235 840 000 000 268 435 456 × 2 = 0 + 0.000 000 396 277 847 744 892 686 046 383 366 471 680 000 000 536 870 912;
  • 20) 0.000 000 396 277 847 744 892 686 046 383 366 471 680 000 000 536 870 912 × 2 = 0 + 0.000 000 792 555 695 489 785 372 092 766 732 943 360 000 001 073 741 824;
  • 21) 0.000 000 792 555 695 489 785 372 092 766 732 943 360 000 001 073 741 824 × 2 = 0 + 0.000 001 585 111 390 979 570 744 185 533 465 886 720 000 002 147 483 648;
  • 22) 0.000 001 585 111 390 979 570 744 185 533 465 886 720 000 002 147 483 648 × 2 = 0 + 0.000 003 170 222 781 959 141 488 371 066 931 773 440 000 004 294 967 296;
  • 23) 0.000 003 170 222 781 959 141 488 371 066 931 773 440 000 004 294 967 296 × 2 = 0 + 0.000 006 340 445 563 918 282 976 742 133 863 546 880 000 008 589 934 592;
  • 24) 0.000 006 340 445 563 918 282 976 742 133 863 546 880 000 008 589 934 592 × 2 = 0 + 0.000 012 680 891 127 836 565 953 484 267 727 093 760 000 017 179 869 184;
  • 25) 0.000 012 680 891 127 836 565 953 484 267 727 093 760 000 017 179 869 184 × 2 = 0 + 0.000 025 361 782 255 673 131 906 968 535 454 187 520 000 034 359 738 368;
  • 26) 0.000 025 361 782 255 673 131 906 968 535 454 187 520 000 034 359 738 368 × 2 = 0 + 0.000 050 723 564 511 346 263 813 937 070 908 375 040 000 068 719 476 736;
  • 27) 0.000 050 723 564 511 346 263 813 937 070 908 375 040 000 068 719 476 736 × 2 = 0 + 0.000 101 447 129 022 692 527 627 874 141 816 750 080 000 137 438 953 472;
  • 28) 0.000 101 447 129 022 692 527 627 874 141 816 750 080 000 137 438 953 472 × 2 = 0 + 0.000 202 894 258 045 385 055 255 748 283 633 500 160 000 274 877 906 944;
  • 29) 0.000 202 894 258 045 385 055 255 748 283 633 500 160 000 274 877 906 944 × 2 = 0 + 0.000 405 788 516 090 770 110 511 496 567 267 000 320 000 549 755 813 888;
  • 30) 0.000 405 788 516 090 770 110 511 496 567 267 000 320 000 549 755 813 888 × 2 = 0 + 0.000 811 577 032 181 540 221 022 993 134 534 000 640 001 099 511 627 776;
  • 31) 0.000 811 577 032 181 540 221 022 993 134 534 000 640 001 099 511 627 776 × 2 = 0 + 0.001 623 154 064 363 080 442 045 986 269 068 001 280 002 199 023 255 552;
  • 32) 0.001 623 154 064 363 080 442 045 986 269 068 001 280 002 199 023 255 552 × 2 = 0 + 0.003 246 308 128 726 160 884 091 972 538 136 002 560 004 398 046 511 104;
  • 33) 0.003 246 308 128 726 160 884 091 972 538 136 002 560 004 398 046 511 104 × 2 = 0 + 0.006 492 616 257 452 321 768 183 945 076 272 005 120 008 796 093 022 208;
  • 34) 0.006 492 616 257 452 321 768 183 945 076 272 005 120 008 796 093 022 208 × 2 = 0 + 0.012 985 232 514 904 643 536 367 890 152 544 010 240 017 592 186 044 416;
  • 35) 0.012 985 232 514 904 643 536 367 890 152 544 010 240 017 592 186 044 416 × 2 = 0 + 0.025 970 465 029 809 287 072 735 780 305 088 020 480 035 184 372 088 832;
  • 36) 0.025 970 465 029 809 287 072 735 780 305 088 020 480 035 184 372 088 832 × 2 = 0 + 0.051 940 930 059 618 574 145 471 560 610 176 040 960 070 368 744 177 664;
  • 37) 0.051 940 930 059 618 574 145 471 560 610 176 040 960 070 368 744 177 664 × 2 = 0 + 0.103 881 860 119 237 148 290 943 121 220 352 081 920 140 737 488 355 328;
  • 38) 0.103 881 860 119 237 148 290 943 121 220 352 081 920 140 737 488 355 328 × 2 = 0 + 0.207 763 720 238 474 296 581 886 242 440 704 163 840 281 474 976 710 656;
  • 39) 0.207 763 720 238 474 296 581 886 242 440 704 163 840 281 474 976 710 656 × 2 = 0 + 0.415 527 440 476 948 593 163 772 484 881 408 327 680 562 949 953 421 312;
  • 40) 0.415 527 440 476 948 593 163 772 484 881 408 327 680 562 949 953 421 312 × 2 = 0 + 0.831 054 880 953 897 186 327 544 969 762 816 655 361 125 899 906 842 624;
  • 41) 0.831 054 880 953 897 186 327 544 969 762 816 655 361 125 899 906 842 624 × 2 = 1 + 0.662 109 761 907 794 372 655 089 939 525 633 310 722 251 799 813 685 248;
  • 42) 0.662 109 761 907 794 372 655 089 939 525 633 310 722 251 799 813 685 248 × 2 = 1 + 0.324 219 523 815 588 745 310 179 879 051 266 621 444 503 599 627 370 496;
  • 43) 0.324 219 523 815 588 745 310 179 879 051 266 621 444 503 599 627 370 496 × 2 = 0 + 0.648 439 047 631 177 490 620 359 758 102 533 242 889 007 199 254 740 992;
  • 44) 0.648 439 047 631 177 490 620 359 758 102 533 242 889 007 199 254 740 992 × 2 = 1 + 0.296 878 095 262 354 981 240 719 516 205 066 485 778 014 398 509 481 984;
  • 45) 0.296 878 095 262 354 981 240 719 516 205 066 485 778 014 398 509 481 984 × 2 = 0 + 0.593 756 190 524 709 962 481 439 032 410 132 971 556 028 797 018 963 968;
  • 46) 0.593 756 190 524 709 962 481 439 032 410 132 971 556 028 797 018 963 968 × 2 = 1 + 0.187 512 381 049 419 924 962 878 064 820 265 943 112 057 594 037 927 936;
  • 47) 0.187 512 381 049 419 924 962 878 064 820 265 943 112 057 594 037 927 936 × 2 = 0 + 0.375 024 762 098 839 849 925 756 129 640 531 886 224 115 188 075 855 872;
  • 48) 0.375 024 762 098 839 849 925 756 129 640 531 886 224 115 188 075 855 872 × 2 = 0 + 0.750 049 524 197 679 699 851 512 259 281 063 772 448 230 376 151 711 744;
  • 49) 0.750 049 524 197 679 699 851 512 259 281 063 772 448 230 376 151 711 744 × 2 = 1 + 0.500 099 048 395 359 399 703 024 518 562 127 544 896 460 752 303 423 488;
  • 50) 0.500 099 048 395 359 399 703 024 518 562 127 544 896 460 752 303 423 488 × 2 = 1 + 0.000 198 096 790 718 799 406 049 037 124 255 089 792 921 504 606 846 976;
  • 51) 0.000 198 096 790 718 799 406 049 037 124 255 089 792 921 504 606 846 976 × 2 = 0 + 0.000 396 193 581 437 598 812 098 074 248 510 179 585 843 009 213 693 952;
  • 52) 0.000 396 193 581 437 598 812 098 074 248 510 179 585 843 009 213 693 952 × 2 = 0 + 0.000 792 387 162 875 197 624 196 148 497 020 359 171 686 018 427 387 904;
  • 53) 0.000 792 387 162 875 197 624 196 148 497 020 359 171 686 018 427 387 904 × 2 = 0 + 0.001 584 774 325 750 395 248 392 296 994 040 718 343 372 036 854 775 808;
  • 54) 0.001 584 774 325 750 395 248 392 296 994 040 718 343 372 036 854 775 808 × 2 = 0 + 0.003 169 548 651 500 790 496 784 593 988 081 436 686 744 073 709 551 616;
  • 55) 0.003 169 548 651 500 790 496 784 593 988 081 436 686 744 073 709 551 616 × 2 = 0 + 0.006 339 097 303 001 580 993 569 187 976 162 873 373 488 147 419 103 232;
  • 56) 0.006 339 097 303 001 580 993 569 187 976 162 873 373 488 147 419 103 232 × 2 = 0 + 0.012 678 194 606 003 161 987 138 375 952 325 746 746 976 294 838 206 464;
  • 57) 0.012 678 194 606 003 161 987 138 375 952 325 746 746 976 294 838 206 464 × 2 = 0 + 0.025 356 389 212 006 323 974 276 751 904 651 493 493 952 589 676 412 928;
  • 58) 0.025 356 389 212 006 323 974 276 751 904 651 493 493 952 589 676 412 928 × 2 = 0 + 0.050 712 778 424 012 647 948 553 503 809 302 986 987 905 179 352 825 856;
  • 59) 0.050 712 778 424 012 647 948 553 503 809 302 986 987 905 179 352 825 856 × 2 = 0 + 0.101 425 556 848 025 295 897 107 007 618 605 973 975 810 358 705 651 712;
  • 60) 0.101 425 556 848 025 295 897 107 007 618 605 973 975 810 358 705 651 712 × 2 = 0 + 0.202 851 113 696 050 591 794 214 015 237 211 947 951 620 717 411 303 424;
  • 61) 0.202 851 113 696 050 591 794 214 015 237 211 947 951 620 717 411 303 424 × 2 = 0 + 0.405 702 227 392 101 183 588 428 030 474 423 895 903 241 434 822 606 848;
  • 62) 0.405 702 227 392 101 183 588 428 030 474 423 895 903 241 434 822 606 848 × 2 = 0 + 0.811 404 454 784 202 367 176 856 060 948 847 791 806 482 869 645 213 696;
  • 63) 0.811 404 454 784 202 367 176 856 060 948 847 791 806 482 869 645 213 696 × 2 = 1 + 0.622 808 909 568 404 734 353 712 121 897 695 583 612 965 739 290 427 392;
  • 64) 0.622 808 909 568 404 734 353 712 121 897 695 583 612 965 739 290 427 392 × 2 = 1 + 0.245 617 819 136 809 468 707 424 243 795 391 167 225 931 478 580 854 784;
  • 65) 0.245 617 819 136 809 468 707 424 243 795 391 167 225 931 478 580 854 784 × 2 = 0 + 0.491 235 638 273 618 937 414 848 487 590 782 334 451 862 957 161 709 568;
  • 66) 0.491 235 638 273 618 937 414 848 487 590 782 334 451 862 957 161 709 568 × 2 = 0 + 0.982 471 276 547 237 874 829 696 975 181 564 668 903 725 914 323 419 136;
  • 67) 0.982 471 276 547 237 874 829 696 975 181 564 668 903 725 914 323 419 136 × 2 = 1 + 0.964 942 553 094 475 749 659 393 950 363 129 337 807 451 828 646 838 272;
  • 68) 0.964 942 553 094 475 749 659 393 950 363 129 337 807 451 828 646 838 272 × 2 = 1 + 0.929 885 106 188 951 499 318 787 900 726 258 675 614 903 657 293 676 544;
  • 69) 0.929 885 106 188 951 499 318 787 900 726 258 675 614 903 657 293 676 544 × 2 = 1 + 0.859 770 212 377 902 998 637 575 801 452 517 351 229 807 314 587 353 088;
  • 70) 0.859 770 212 377 902 998 637 575 801 452 517 351 229 807 314 587 353 088 × 2 = 1 + 0.719 540 424 755 805 997 275 151 602 905 034 702 459 614 629 174 706 176;
  • 71) 0.719 540 424 755 805 997 275 151 602 905 034 702 459 614 629 174 706 176 × 2 = 1 + 0.439 080 849 511 611 994 550 303 205 810 069 404 919 229 258 349 412 352;
  • 72) 0.439 080 849 511 611 994 550 303 205 810 069 404 919 229 258 349 412 352 × 2 = 0 + 0.878 161 699 023 223 989 100 606 411 620 138 809 838 458 516 698 824 704;
  • 73) 0.878 161 699 023 223 989 100 606 411 620 138 809 838 458 516 698 824 704 × 2 = 1 + 0.756 323 398 046 447 978 201 212 823 240 277 619 676 917 033 397 649 408;
  • 74) 0.756 323 398 046 447 978 201 212 823 240 277 619 676 917 033 397 649 408 × 2 = 1 + 0.512 646 796 092 895 956 402 425 646 480 555 239 353 834 066 795 298 816;
  • 75) 0.512 646 796 092 895 956 402 425 646 480 555 239 353 834 066 795 298 816 × 2 = 1 + 0.025 293 592 185 791 912 804 851 292 961 110 478 707 668 133 590 597 632;
  • 76) 0.025 293 592 185 791 912 804 851 292 961 110 478 707 668 133 590 597 632 × 2 = 0 + 0.050 587 184 371 583 825 609 702 585 922 220 957 415 336 267 181 195 264;
  • 77) 0.050 587 184 371 583 825 609 702 585 922 220 957 415 336 267 181 195 264 × 2 = 0 + 0.101 174 368 743 167 651 219 405 171 844 441 914 830 672 534 362 390 528;
  • 78) 0.101 174 368 743 167 651 219 405 171 844 441 914 830 672 534 362 390 528 × 2 = 0 + 0.202 348 737 486 335 302 438 810 343 688 883 829 661 345 068 724 781 056;
  • 79) 0.202 348 737 486 335 302 438 810 343 688 883 829 661 345 068 724 781 056 × 2 = 0 + 0.404 697 474 972 670 604 877 620 687 377 767 659 322 690 137 449 562 112;
  • 80) 0.404 697 474 972 670 604 877 620 687 377 767 659 322 690 137 449 562 112 × 2 = 0 + 0.809 394 949 945 341 209 755 241 374 755 535 318 645 380 274 899 124 224;
  • 81) 0.809 394 949 945 341 209 755 241 374 755 535 318 645 380 274 899 124 224 × 2 = 1 + 0.618 789 899 890 682 419 510 482 749 511 070 637 290 760 549 798 248 448;
  • 82) 0.618 789 899 890 682 419 510 482 749 511 070 637 290 760 549 798 248 448 × 2 = 1 + 0.237 579 799 781 364 839 020 965 499 022 141 274 581 521 099 596 496 896;
  • 83) 0.237 579 799 781 364 839 020 965 499 022 141 274 581 521 099 596 496 896 × 2 = 0 + 0.475 159 599 562 729 678 041 930 998 044 282 549 163 042 199 192 993 792;
  • 84) 0.475 159 599 562 729 678 041 930 998 044 282 549 163 042 199 192 993 792 × 2 = 0 + 0.950 319 199 125 459 356 083 861 996 088 565 098 326 084 398 385 987 584;
  • 85) 0.950 319 199 125 459 356 083 861 996 088 565 098 326 084 398 385 987 584 × 2 = 1 + 0.900 638 398 250 918 712 167 723 992 177 130 196 652 168 796 771 975 168;
  • 86) 0.900 638 398 250 918 712 167 723 992 177 130 196 652 168 796 771 975 168 × 2 = 1 + 0.801 276 796 501 837 424 335 447 984 354 260 393 304 337 593 543 950 336;
  • 87) 0.801 276 796 501 837 424 335 447 984 354 260 393 304 337 593 543 950 336 × 2 = 1 + 0.602 553 593 003 674 848 670 895 968 708 520 786 608 675 187 087 900 672;
  • 88) 0.602 553 593 003 674 848 670 895 968 708 520 786 608 675 187 087 900 672 × 2 = 1 + 0.205 107 186 007 349 697 341 791 937 417 041 573 217 350 374 175 801 344;
  • 89) 0.205 107 186 007 349 697 341 791 937 417 041 573 217 350 374 175 801 344 × 2 = 0 + 0.410 214 372 014 699 394 683 583 874 834 083 146 434 700 748 351 602 688;
  • 90) 0.410 214 372 014 699 394 683 583 874 834 083 146 434 700 748 351 602 688 × 2 = 0 + 0.820 428 744 029 398 789 367 167 749 668 166 292 869 401 496 703 205 376;
  • 91) 0.820 428 744 029 398 789 367 167 749 668 166 292 869 401 496 703 205 376 × 2 = 1 + 0.640 857 488 058 797 578 734 335 499 336 332 585 738 802 993 406 410 752;
  • 92) 0.640 857 488 058 797 578 734 335 499 336 332 585 738 802 993 406 410 752 × 2 = 1 + 0.281 714 976 117 595 157 468 670 998 672 665 171 477 605 986 812 821 504;
  • 93) 0.281 714 976 117 595 157 468 670 998 672 665 171 477 605 986 812 821 504 × 2 = 0 + 0.563 429 952 235 190 314 937 341 997 345 330 342 955 211 973 625 643 008;

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:


0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 024(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:

0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 024(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:


0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 024(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 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 024 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 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 045 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