0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 000 971 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 000 971(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
0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 000 971(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: 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 000 971.

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) 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 000 971 × 2 = 0 + 0.000 000 000 001 511 680 022 220 202 202 020 200 220 000 000 000 001 942;
  • 2) 0.000 000 000 001 511 680 022 220 202 202 020 200 220 000 000 000 001 942 × 2 = 0 + 0.000 000 000 003 023 360 044 440 404 404 040 400 440 000 000 000 003 884;
  • 3) 0.000 000 000 003 023 360 044 440 404 404 040 400 440 000 000 000 003 884 × 2 = 0 + 0.000 000 000 006 046 720 088 880 808 808 080 800 880 000 000 000 007 768;
  • 4) 0.000 000 000 006 046 720 088 880 808 808 080 800 880 000 000 000 007 768 × 2 = 0 + 0.000 000 000 012 093 440 177 761 617 616 161 601 760 000 000 000 015 536;
  • 5) 0.000 000 000 012 093 440 177 761 617 616 161 601 760 000 000 000 015 536 × 2 = 0 + 0.000 000 000 024 186 880 355 523 235 232 323 203 520 000 000 000 031 072;
  • 6) 0.000 000 000 024 186 880 355 523 235 232 323 203 520 000 000 000 031 072 × 2 = 0 + 0.000 000 000 048 373 760 711 046 470 464 646 407 040 000 000 000 062 144;
  • 7) 0.000 000 000 048 373 760 711 046 470 464 646 407 040 000 000 000 062 144 × 2 = 0 + 0.000 000 000 096 747 521 422 092 940 929 292 814 080 000 000 000 124 288;
  • 8) 0.000 000 000 096 747 521 422 092 940 929 292 814 080 000 000 000 124 288 × 2 = 0 + 0.000 000 000 193 495 042 844 185 881 858 585 628 160 000 000 000 248 576;
  • 9) 0.000 000 000 193 495 042 844 185 881 858 585 628 160 000 000 000 248 576 × 2 = 0 + 0.000 000 000 386 990 085 688 371 763 717 171 256 320 000 000 000 497 152;
  • 10) 0.000 000 000 386 990 085 688 371 763 717 171 256 320 000 000 000 497 152 × 2 = 0 + 0.000 000 000 773 980 171 376 743 527 434 342 512 640 000 000 000 994 304;
  • 11) 0.000 000 000 773 980 171 376 743 527 434 342 512 640 000 000 000 994 304 × 2 = 0 + 0.000 000 001 547 960 342 753 487 054 868 685 025 280 000 000 001 988 608;
  • 12) 0.000 000 001 547 960 342 753 487 054 868 685 025 280 000 000 001 988 608 × 2 = 0 + 0.000 000 003 095 920 685 506 974 109 737 370 050 560 000 000 003 977 216;
  • 13) 0.000 000 003 095 920 685 506 974 109 737 370 050 560 000 000 003 977 216 × 2 = 0 + 0.000 000 006 191 841 371 013 948 219 474 740 101 120 000 000 007 954 432;
  • 14) 0.000 000 006 191 841 371 013 948 219 474 740 101 120 000 000 007 954 432 × 2 = 0 + 0.000 000 012 383 682 742 027 896 438 949 480 202 240 000 000 015 908 864;
  • 15) 0.000 000 012 383 682 742 027 896 438 949 480 202 240 000 000 015 908 864 × 2 = 0 + 0.000 000 024 767 365 484 055 792 877 898 960 404 480 000 000 031 817 728;
  • 16) 0.000 000 024 767 365 484 055 792 877 898 960 404 480 000 000 031 817 728 × 2 = 0 + 0.000 000 049 534 730 968 111 585 755 797 920 808 960 000 000 063 635 456;
  • 17) 0.000 000 049 534 730 968 111 585 755 797 920 808 960 000 000 063 635 456 × 2 = 0 + 0.000 000 099 069 461 936 223 171 511 595 841 617 920 000 000 127 270 912;
  • 18) 0.000 000 099 069 461 936 223 171 511 595 841 617 920 000 000 127 270 912 × 2 = 0 + 0.000 000 198 138 923 872 446 343 023 191 683 235 840 000 000 254 541 824;
  • 19) 0.000 000 198 138 923 872 446 343 023 191 683 235 840 000 000 254 541 824 × 2 = 0 + 0.000 000 396 277 847 744 892 686 046 383 366 471 680 000 000 509 083 648;
  • 20) 0.000 000 396 277 847 744 892 686 046 383 366 471 680 000 000 509 083 648 × 2 = 0 + 0.000 000 792 555 695 489 785 372 092 766 732 943 360 000 001 018 167 296;
  • 21) 0.000 000 792 555 695 489 785 372 092 766 732 943 360 000 001 018 167 296 × 2 = 0 + 0.000 001 585 111 390 979 570 744 185 533 465 886 720 000 002 036 334 592;
  • 22) 0.000 001 585 111 390 979 570 744 185 533 465 886 720 000 002 036 334 592 × 2 = 0 + 0.000 003 170 222 781 959 141 488 371 066 931 773 440 000 004 072 669 184;
  • 23) 0.000 003 170 222 781 959 141 488 371 066 931 773 440 000 004 072 669 184 × 2 = 0 + 0.000 006 340 445 563 918 282 976 742 133 863 546 880 000 008 145 338 368;
  • 24) 0.000 006 340 445 563 918 282 976 742 133 863 546 880 000 008 145 338 368 × 2 = 0 + 0.000 012 680 891 127 836 565 953 484 267 727 093 760 000 016 290 676 736;
  • 25) 0.000 012 680 891 127 836 565 953 484 267 727 093 760 000 016 290 676 736 × 2 = 0 + 0.000 025 361 782 255 673 131 906 968 535 454 187 520 000 032 581 353 472;
  • 26) 0.000 025 361 782 255 673 131 906 968 535 454 187 520 000 032 581 353 472 × 2 = 0 + 0.000 050 723 564 511 346 263 813 937 070 908 375 040 000 065 162 706 944;
  • 27) 0.000 050 723 564 511 346 263 813 937 070 908 375 040 000 065 162 706 944 × 2 = 0 + 0.000 101 447 129 022 692 527 627 874 141 816 750 080 000 130 325 413 888;
  • 28) 0.000 101 447 129 022 692 527 627 874 141 816 750 080 000 130 325 413 888 × 2 = 0 + 0.000 202 894 258 045 385 055 255 748 283 633 500 160 000 260 650 827 776;
  • 29) 0.000 202 894 258 045 385 055 255 748 283 633 500 160 000 260 650 827 776 × 2 = 0 + 0.000 405 788 516 090 770 110 511 496 567 267 000 320 000 521 301 655 552;
  • 30) 0.000 405 788 516 090 770 110 511 496 567 267 000 320 000 521 301 655 552 × 2 = 0 + 0.000 811 577 032 181 540 221 022 993 134 534 000 640 001 042 603 311 104;
  • 31) 0.000 811 577 032 181 540 221 022 993 134 534 000 640 001 042 603 311 104 × 2 = 0 + 0.001 623 154 064 363 080 442 045 986 269 068 001 280 002 085 206 622 208;
  • 32) 0.001 623 154 064 363 080 442 045 986 269 068 001 280 002 085 206 622 208 × 2 = 0 + 0.003 246 308 128 726 160 884 091 972 538 136 002 560 004 170 413 244 416;
  • 33) 0.003 246 308 128 726 160 884 091 972 538 136 002 560 004 170 413 244 416 × 2 = 0 + 0.006 492 616 257 452 321 768 183 945 076 272 005 120 008 340 826 488 832;
  • 34) 0.006 492 616 257 452 321 768 183 945 076 272 005 120 008 340 826 488 832 × 2 = 0 + 0.012 985 232 514 904 643 536 367 890 152 544 010 240 016 681 652 977 664;
  • 35) 0.012 985 232 514 904 643 536 367 890 152 544 010 240 016 681 652 977 664 × 2 = 0 + 0.025 970 465 029 809 287 072 735 780 305 088 020 480 033 363 305 955 328;
  • 36) 0.025 970 465 029 809 287 072 735 780 305 088 020 480 033 363 305 955 328 × 2 = 0 + 0.051 940 930 059 618 574 145 471 560 610 176 040 960 066 726 611 910 656;
  • 37) 0.051 940 930 059 618 574 145 471 560 610 176 040 960 066 726 611 910 656 × 2 = 0 + 0.103 881 860 119 237 148 290 943 121 220 352 081 920 133 453 223 821 312;
  • 38) 0.103 881 860 119 237 148 290 943 121 220 352 081 920 133 453 223 821 312 × 2 = 0 + 0.207 763 720 238 474 296 581 886 242 440 704 163 840 266 906 447 642 624;
  • 39) 0.207 763 720 238 474 296 581 886 242 440 704 163 840 266 906 447 642 624 × 2 = 0 + 0.415 527 440 476 948 593 163 772 484 881 408 327 680 533 812 895 285 248;
  • 40) 0.415 527 440 476 948 593 163 772 484 881 408 327 680 533 812 895 285 248 × 2 = 0 + 0.831 054 880 953 897 186 327 544 969 762 816 655 361 067 625 790 570 496;
  • 41) 0.831 054 880 953 897 186 327 544 969 762 816 655 361 067 625 790 570 496 × 2 = 1 + 0.662 109 761 907 794 372 655 089 939 525 633 310 722 135 251 581 140 992;
  • 42) 0.662 109 761 907 794 372 655 089 939 525 633 310 722 135 251 581 140 992 × 2 = 1 + 0.324 219 523 815 588 745 310 179 879 051 266 621 444 270 503 162 281 984;
  • 43) 0.324 219 523 815 588 745 310 179 879 051 266 621 444 270 503 162 281 984 × 2 = 0 + 0.648 439 047 631 177 490 620 359 758 102 533 242 888 541 006 324 563 968;
  • 44) 0.648 439 047 631 177 490 620 359 758 102 533 242 888 541 006 324 563 968 × 2 = 1 + 0.296 878 095 262 354 981 240 719 516 205 066 485 777 082 012 649 127 936;
  • 45) 0.296 878 095 262 354 981 240 719 516 205 066 485 777 082 012 649 127 936 × 2 = 0 + 0.593 756 190 524 709 962 481 439 032 410 132 971 554 164 025 298 255 872;
  • 46) 0.593 756 190 524 709 962 481 439 032 410 132 971 554 164 025 298 255 872 × 2 = 1 + 0.187 512 381 049 419 924 962 878 064 820 265 943 108 328 050 596 511 744;
  • 47) 0.187 512 381 049 419 924 962 878 064 820 265 943 108 328 050 596 511 744 × 2 = 0 + 0.375 024 762 098 839 849 925 756 129 640 531 886 216 656 101 193 023 488;
  • 48) 0.375 024 762 098 839 849 925 756 129 640 531 886 216 656 101 193 023 488 × 2 = 0 + 0.750 049 524 197 679 699 851 512 259 281 063 772 433 312 202 386 046 976;
  • 49) 0.750 049 524 197 679 699 851 512 259 281 063 772 433 312 202 386 046 976 × 2 = 1 + 0.500 099 048 395 359 399 703 024 518 562 127 544 866 624 404 772 093 952;
  • 50) 0.500 099 048 395 359 399 703 024 518 562 127 544 866 624 404 772 093 952 × 2 = 1 + 0.000 198 096 790 718 799 406 049 037 124 255 089 733 248 809 544 187 904;
  • 51) 0.000 198 096 790 718 799 406 049 037 124 255 089 733 248 809 544 187 904 × 2 = 0 + 0.000 396 193 581 437 598 812 098 074 248 510 179 466 497 619 088 375 808;
  • 52) 0.000 396 193 581 437 598 812 098 074 248 510 179 466 497 619 088 375 808 × 2 = 0 + 0.000 792 387 162 875 197 624 196 148 497 020 358 932 995 238 176 751 616;
  • 53) 0.000 792 387 162 875 197 624 196 148 497 020 358 932 995 238 176 751 616 × 2 = 0 + 0.001 584 774 325 750 395 248 392 296 994 040 717 865 990 476 353 503 232;
  • 54) 0.001 584 774 325 750 395 248 392 296 994 040 717 865 990 476 353 503 232 × 2 = 0 + 0.003 169 548 651 500 790 496 784 593 988 081 435 731 980 952 707 006 464;
  • 55) 0.003 169 548 651 500 790 496 784 593 988 081 435 731 980 952 707 006 464 × 2 = 0 + 0.006 339 097 303 001 580 993 569 187 976 162 871 463 961 905 414 012 928;
  • 56) 0.006 339 097 303 001 580 993 569 187 976 162 871 463 961 905 414 012 928 × 2 = 0 + 0.012 678 194 606 003 161 987 138 375 952 325 742 927 923 810 828 025 856;
  • 57) 0.012 678 194 606 003 161 987 138 375 952 325 742 927 923 810 828 025 856 × 2 = 0 + 0.025 356 389 212 006 323 974 276 751 904 651 485 855 847 621 656 051 712;
  • 58) 0.025 356 389 212 006 323 974 276 751 904 651 485 855 847 621 656 051 712 × 2 = 0 + 0.050 712 778 424 012 647 948 553 503 809 302 971 711 695 243 312 103 424;
  • 59) 0.050 712 778 424 012 647 948 553 503 809 302 971 711 695 243 312 103 424 × 2 = 0 + 0.101 425 556 848 025 295 897 107 007 618 605 943 423 390 486 624 206 848;
  • 60) 0.101 425 556 848 025 295 897 107 007 618 605 943 423 390 486 624 206 848 × 2 = 0 + 0.202 851 113 696 050 591 794 214 015 237 211 886 846 780 973 248 413 696;
  • 61) 0.202 851 113 696 050 591 794 214 015 237 211 886 846 780 973 248 413 696 × 2 = 0 + 0.405 702 227 392 101 183 588 428 030 474 423 773 693 561 946 496 827 392;
  • 62) 0.405 702 227 392 101 183 588 428 030 474 423 773 693 561 946 496 827 392 × 2 = 0 + 0.811 404 454 784 202 367 176 856 060 948 847 547 387 123 892 993 654 784;
  • 63) 0.811 404 454 784 202 367 176 856 060 948 847 547 387 123 892 993 654 784 × 2 = 1 + 0.622 808 909 568 404 734 353 712 121 897 695 094 774 247 785 987 309 568;
  • 64) 0.622 808 909 568 404 734 353 712 121 897 695 094 774 247 785 987 309 568 × 2 = 1 + 0.245 617 819 136 809 468 707 424 243 795 390 189 548 495 571 974 619 136;
  • 65) 0.245 617 819 136 809 468 707 424 243 795 390 189 548 495 571 974 619 136 × 2 = 0 + 0.491 235 638 273 618 937 414 848 487 590 780 379 096 991 143 949 238 272;
  • 66) 0.491 235 638 273 618 937 414 848 487 590 780 379 096 991 143 949 238 272 × 2 = 0 + 0.982 471 276 547 237 874 829 696 975 181 560 758 193 982 287 898 476 544;
  • 67) 0.982 471 276 547 237 874 829 696 975 181 560 758 193 982 287 898 476 544 × 2 = 1 + 0.964 942 553 094 475 749 659 393 950 363 121 516 387 964 575 796 953 088;
  • 68) 0.964 942 553 094 475 749 659 393 950 363 121 516 387 964 575 796 953 088 × 2 = 1 + 0.929 885 106 188 951 499 318 787 900 726 243 032 775 929 151 593 906 176;
  • 69) 0.929 885 106 188 951 499 318 787 900 726 243 032 775 929 151 593 906 176 × 2 = 1 + 0.859 770 212 377 902 998 637 575 801 452 486 065 551 858 303 187 812 352;
  • 70) 0.859 770 212 377 902 998 637 575 801 452 486 065 551 858 303 187 812 352 × 2 = 1 + 0.719 540 424 755 805 997 275 151 602 904 972 131 103 716 606 375 624 704;
  • 71) 0.719 540 424 755 805 997 275 151 602 904 972 131 103 716 606 375 624 704 × 2 = 1 + 0.439 080 849 511 611 994 550 303 205 809 944 262 207 433 212 751 249 408;
  • 72) 0.439 080 849 511 611 994 550 303 205 809 944 262 207 433 212 751 249 408 × 2 = 0 + 0.878 161 699 023 223 989 100 606 411 619 888 524 414 866 425 502 498 816;
  • 73) 0.878 161 699 023 223 989 100 606 411 619 888 524 414 866 425 502 498 816 × 2 = 1 + 0.756 323 398 046 447 978 201 212 823 239 777 048 829 732 851 004 997 632;
  • 74) 0.756 323 398 046 447 978 201 212 823 239 777 048 829 732 851 004 997 632 × 2 = 1 + 0.512 646 796 092 895 956 402 425 646 479 554 097 659 465 702 009 995 264;
  • 75) 0.512 646 796 092 895 956 402 425 646 479 554 097 659 465 702 009 995 264 × 2 = 1 + 0.025 293 592 185 791 912 804 851 292 959 108 195 318 931 404 019 990 528;
  • 76) 0.025 293 592 185 791 912 804 851 292 959 108 195 318 931 404 019 990 528 × 2 = 0 + 0.050 587 184 371 583 825 609 702 585 918 216 390 637 862 808 039 981 056;
  • 77) 0.050 587 184 371 583 825 609 702 585 918 216 390 637 862 808 039 981 056 × 2 = 0 + 0.101 174 368 743 167 651 219 405 171 836 432 781 275 725 616 079 962 112;
  • 78) 0.101 174 368 743 167 651 219 405 171 836 432 781 275 725 616 079 962 112 × 2 = 0 + 0.202 348 737 486 335 302 438 810 343 672 865 562 551 451 232 159 924 224;
  • 79) 0.202 348 737 486 335 302 438 810 343 672 865 562 551 451 232 159 924 224 × 2 = 0 + 0.404 697 474 972 670 604 877 620 687 345 731 125 102 902 464 319 848 448;
  • 80) 0.404 697 474 972 670 604 877 620 687 345 731 125 102 902 464 319 848 448 × 2 = 0 + 0.809 394 949 945 341 209 755 241 374 691 462 250 205 804 928 639 696 896;
  • 81) 0.809 394 949 945 341 209 755 241 374 691 462 250 205 804 928 639 696 896 × 2 = 1 + 0.618 789 899 890 682 419 510 482 749 382 924 500 411 609 857 279 393 792;
  • 82) 0.618 789 899 890 682 419 510 482 749 382 924 500 411 609 857 279 393 792 × 2 = 1 + 0.237 579 799 781 364 839 020 965 498 765 849 000 823 219 714 558 787 584;
  • 83) 0.237 579 799 781 364 839 020 965 498 765 849 000 823 219 714 558 787 584 × 2 = 0 + 0.475 159 599 562 729 678 041 930 997 531 698 001 646 439 429 117 575 168;
  • 84) 0.475 159 599 562 729 678 041 930 997 531 698 001 646 439 429 117 575 168 × 2 = 0 + 0.950 319 199 125 459 356 083 861 995 063 396 003 292 878 858 235 150 336;
  • 85) 0.950 319 199 125 459 356 083 861 995 063 396 003 292 878 858 235 150 336 × 2 = 1 + 0.900 638 398 250 918 712 167 723 990 126 792 006 585 757 716 470 300 672;
  • 86) 0.900 638 398 250 918 712 167 723 990 126 792 006 585 757 716 470 300 672 × 2 = 1 + 0.801 276 796 501 837 424 335 447 980 253 584 013 171 515 432 940 601 344;
  • 87) 0.801 276 796 501 837 424 335 447 980 253 584 013 171 515 432 940 601 344 × 2 = 1 + 0.602 553 593 003 674 848 670 895 960 507 168 026 343 030 865 881 202 688;
  • 88) 0.602 553 593 003 674 848 670 895 960 507 168 026 343 030 865 881 202 688 × 2 = 1 + 0.205 107 186 007 349 697 341 791 921 014 336 052 686 061 731 762 405 376;
  • 89) 0.205 107 186 007 349 697 341 791 921 014 336 052 686 061 731 762 405 376 × 2 = 0 + 0.410 214 372 014 699 394 683 583 842 028 672 105 372 123 463 524 810 752;
  • 90) 0.410 214 372 014 699 394 683 583 842 028 672 105 372 123 463 524 810 752 × 2 = 0 + 0.820 428 744 029 398 789 367 167 684 057 344 210 744 246 927 049 621 504;
  • 91) 0.820 428 744 029 398 789 367 167 684 057 344 210 744 246 927 049 621 504 × 2 = 1 + 0.640 857 488 058 797 578 734 335 368 114 688 421 488 493 854 099 243 008;
  • 92) 0.640 857 488 058 797 578 734 335 368 114 688 421 488 493 854 099 243 008 × 2 = 1 + 0.281 714 976 117 595 157 468 670 736 229 376 842 976 987 708 198 486 016;
  • 93) 0.281 714 976 117 595 157 468 670 736 229 376 842 976 987 708 198 486 016 × 2 = 0 + 0.563 429 952 235 190 314 937 341 472 458 753 685 953 975 416 396 972 032;

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:


0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 000 971(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:

0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 000 971(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:


0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 000 971(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 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 000 971 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 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 000 894 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