0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 045 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 045(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
0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 045(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: 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 045.

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) 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 045 × 2 = 0 + 0.000 000 000 001 511 680 022 220 202 202 020 200 220 000 000 000 002 09;
  • 2) 0.000 000 000 001 511 680 022 220 202 202 020 200 220 000 000 000 002 09 × 2 = 0 + 0.000 000 000 003 023 360 044 440 404 404 040 400 440 000 000 000 004 18;
  • 3) 0.000 000 000 003 023 360 044 440 404 404 040 400 440 000 000 000 004 18 × 2 = 0 + 0.000 000 000 006 046 720 088 880 808 808 080 800 880 000 000 000 008 36;
  • 4) 0.000 000 000 006 046 720 088 880 808 808 080 800 880 000 000 000 008 36 × 2 = 0 + 0.000 000 000 012 093 440 177 761 617 616 161 601 760 000 000 000 016 72;
  • 5) 0.000 000 000 012 093 440 177 761 617 616 161 601 760 000 000 000 016 72 × 2 = 0 + 0.000 000 000 024 186 880 355 523 235 232 323 203 520 000 000 000 033 44;
  • 6) 0.000 000 000 024 186 880 355 523 235 232 323 203 520 000 000 000 033 44 × 2 = 0 + 0.000 000 000 048 373 760 711 046 470 464 646 407 040 000 000 000 066 88;
  • 7) 0.000 000 000 048 373 760 711 046 470 464 646 407 040 000 000 000 066 88 × 2 = 0 + 0.000 000 000 096 747 521 422 092 940 929 292 814 080 000 000 000 133 76;
  • 8) 0.000 000 000 096 747 521 422 092 940 929 292 814 080 000 000 000 133 76 × 2 = 0 + 0.000 000 000 193 495 042 844 185 881 858 585 628 160 000 000 000 267 52;
  • 9) 0.000 000 000 193 495 042 844 185 881 858 585 628 160 000 000 000 267 52 × 2 = 0 + 0.000 000 000 386 990 085 688 371 763 717 171 256 320 000 000 000 535 04;
  • 10) 0.000 000 000 386 990 085 688 371 763 717 171 256 320 000 000 000 535 04 × 2 = 0 + 0.000 000 000 773 980 171 376 743 527 434 342 512 640 000 000 001 070 08;
  • 11) 0.000 000 000 773 980 171 376 743 527 434 342 512 640 000 000 001 070 08 × 2 = 0 + 0.000 000 001 547 960 342 753 487 054 868 685 025 280 000 000 002 140 16;
  • 12) 0.000 000 001 547 960 342 753 487 054 868 685 025 280 000 000 002 140 16 × 2 = 0 + 0.000 000 003 095 920 685 506 974 109 737 370 050 560 000 000 004 280 32;
  • 13) 0.000 000 003 095 920 685 506 974 109 737 370 050 560 000 000 004 280 32 × 2 = 0 + 0.000 000 006 191 841 371 013 948 219 474 740 101 120 000 000 008 560 64;
  • 14) 0.000 000 006 191 841 371 013 948 219 474 740 101 120 000 000 008 560 64 × 2 = 0 + 0.000 000 012 383 682 742 027 896 438 949 480 202 240 000 000 017 121 28;
  • 15) 0.000 000 012 383 682 742 027 896 438 949 480 202 240 000 000 017 121 28 × 2 = 0 + 0.000 000 024 767 365 484 055 792 877 898 960 404 480 000 000 034 242 56;
  • 16) 0.000 000 024 767 365 484 055 792 877 898 960 404 480 000 000 034 242 56 × 2 = 0 + 0.000 000 049 534 730 968 111 585 755 797 920 808 960 000 000 068 485 12;
  • 17) 0.000 000 049 534 730 968 111 585 755 797 920 808 960 000 000 068 485 12 × 2 = 0 + 0.000 000 099 069 461 936 223 171 511 595 841 617 920 000 000 136 970 24;
  • 18) 0.000 000 099 069 461 936 223 171 511 595 841 617 920 000 000 136 970 24 × 2 = 0 + 0.000 000 198 138 923 872 446 343 023 191 683 235 840 000 000 273 940 48;
  • 19) 0.000 000 198 138 923 872 446 343 023 191 683 235 840 000 000 273 940 48 × 2 = 0 + 0.000 000 396 277 847 744 892 686 046 383 366 471 680 000 000 547 880 96;
  • 20) 0.000 000 396 277 847 744 892 686 046 383 366 471 680 000 000 547 880 96 × 2 = 0 + 0.000 000 792 555 695 489 785 372 092 766 732 943 360 000 001 095 761 92;
  • 21) 0.000 000 792 555 695 489 785 372 092 766 732 943 360 000 001 095 761 92 × 2 = 0 + 0.000 001 585 111 390 979 570 744 185 533 465 886 720 000 002 191 523 84;
  • 22) 0.000 001 585 111 390 979 570 744 185 533 465 886 720 000 002 191 523 84 × 2 = 0 + 0.000 003 170 222 781 959 141 488 371 066 931 773 440 000 004 383 047 68;
  • 23) 0.000 003 170 222 781 959 141 488 371 066 931 773 440 000 004 383 047 68 × 2 = 0 + 0.000 006 340 445 563 918 282 976 742 133 863 546 880 000 008 766 095 36;
  • 24) 0.000 006 340 445 563 918 282 976 742 133 863 546 880 000 008 766 095 36 × 2 = 0 + 0.000 012 680 891 127 836 565 953 484 267 727 093 760 000 017 532 190 72;
  • 25) 0.000 012 680 891 127 836 565 953 484 267 727 093 760 000 017 532 190 72 × 2 = 0 + 0.000 025 361 782 255 673 131 906 968 535 454 187 520 000 035 064 381 44;
  • 26) 0.000 025 361 782 255 673 131 906 968 535 454 187 520 000 035 064 381 44 × 2 = 0 + 0.000 050 723 564 511 346 263 813 937 070 908 375 040 000 070 128 762 88;
  • 27) 0.000 050 723 564 511 346 263 813 937 070 908 375 040 000 070 128 762 88 × 2 = 0 + 0.000 101 447 129 022 692 527 627 874 141 816 750 080 000 140 257 525 76;
  • 28) 0.000 101 447 129 022 692 527 627 874 141 816 750 080 000 140 257 525 76 × 2 = 0 + 0.000 202 894 258 045 385 055 255 748 283 633 500 160 000 280 515 051 52;
  • 29) 0.000 202 894 258 045 385 055 255 748 283 633 500 160 000 280 515 051 52 × 2 = 0 + 0.000 405 788 516 090 770 110 511 496 567 267 000 320 000 561 030 103 04;
  • 30) 0.000 405 788 516 090 770 110 511 496 567 267 000 320 000 561 030 103 04 × 2 = 0 + 0.000 811 577 032 181 540 221 022 993 134 534 000 640 001 122 060 206 08;
  • 31) 0.000 811 577 032 181 540 221 022 993 134 534 000 640 001 122 060 206 08 × 2 = 0 + 0.001 623 154 064 363 080 442 045 986 269 068 001 280 002 244 120 412 16;
  • 32) 0.001 623 154 064 363 080 442 045 986 269 068 001 280 002 244 120 412 16 × 2 = 0 + 0.003 246 308 128 726 160 884 091 972 538 136 002 560 004 488 240 824 32;
  • 33) 0.003 246 308 128 726 160 884 091 972 538 136 002 560 004 488 240 824 32 × 2 = 0 + 0.006 492 616 257 452 321 768 183 945 076 272 005 120 008 976 481 648 64;
  • 34) 0.006 492 616 257 452 321 768 183 945 076 272 005 120 008 976 481 648 64 × 2 = 0 + 0.012 985 232 514 904 643 536 367 890 152 544 010 240 017 952 963 297 28;
  • 35) 0.012 985 232 514 904 643 536 367 890 152 544 010 240 017 952 963 297 28 × 2 = 0 + 0.025 970 465 029 809 287 072 735 780 305 088 020 480 035 905 926 594 56;
  • 36) 0.025 970 465 029 809 287 072 735 780 305 088 020 480 035 905 926 594 56 × 2 = 0 + 0.051 940 930 059 618 574 145 471 560 610 176 040 960 071 811 853 189 12;
  • 37) 0.051 940 930 059 618 574 145 471 560 610 176 040 960 071 811 853 189 12 × 2 = 0 + 0.103 881 860 119 237 148 290 943 121 220 352 081 920 143 623 706 378 24;
  • 38) 0.103 881 860 119 237 148 290 943 121 220 352 081 920 143 623 706 378 24 × 2 = 0 + 0.207 763 720 238 474 296 581 886 242 440 704 163 840 287 247 412 756 48;
  • 39) 0.207 763 720 238 474 296 581 886 242 440 704 163 840 287 247 412 756 48 × 2 = 0 + 0.415 527 440 476 948 593 163 772 484 881 408 327 680 574 494 825 512 96;
  • 40) 0.415 527 440 476 948 593 163 772 484 881 408 327 680 574 494 825 512 96 × 2 = 0 + 0.831 054 880 953 897 186 327 544 969 762 816 655 361 148 989 651 025 92;
  • 41) 0.831 054 880 953 897 186 327 544 969 762 816 655 361 148 989 651 025 92 × 2 = 1 + 0.662 109 761 907 794 372 655 089 939 525 633 310 722 297 979 302 051 84;
  • 42) 0.662 109 761 907 794 372 655 089 939 525 633 310 722 297 979 302 051 84 × 2 = 1 + 0.324 219 523 815 588 745 310 179 879 051 266 621 444 595 958 604 103 68;
  • 43) 0.324 219 523 815 588 745 310 179 879 051 266 621 444 595 958 604 103 68 × 2 = 0 + 0.648 439 047 631 177 490 620 359 758 102 533 242 889 191 917 208 207 36;
  • 44) 0.648 439 047 631 177 490 620 359 758 102 533 242 889 191 917 208 207 36 × 2 = 1 + 0.296 878 095 262 354 981 240 719 516 205 066 485 778 383 834 416 414 72;
  • 45) 0.296 878 095 262 354 981 240 719 516 205 066 485 778 383 834 416 414 72 × 2 = 0 + 0.593 756 190 524 709 962 481 439 032 410 132 971 556 767 668 832 829 44;
  • 46) 0.593 756 190 524 709 962 481 439 032 410 132 971 556 767 668 832 829 44 × 2 = 1 + 0.187 512 381 049 419 924 962 878 064 820 265 943 113 535 337 665 658 88;
  • 47) 0.187 512 381 049 419 924 962 878 064 820 265 943 113 535 337 665 658 88 × 2 = 0 + 0.375 024 762 098 839 849 925 756 129 640 531 886 227 070 675 331 317 76;
  • 48) 0.375 024 762 098 839 849 925 756 129 640 531 886 227 070 675 331 317 76 × 2 = 0 + 0.750 049 524 197 679 699 851 512 259 281 063 772 454 141 350 662 635 52;
  • 49) 0.750 049 524 197 679 699 851 512 259 281 063 772 454 141 350 662 635 52 × 2 = 1 + 0.500 099 048 395 359 399 703 024 518 562 127 544 908 282 701 325 271 04;
  • 50) 0.500 099 048 395 359 399 703 024 518 562 127 544 908 282 701 325 271 04 × 2 = 1 + 0.000 198 096 790 718 799 406 049 037 124 255 089 816 565 402 650 542 08;
  • 51) 0.000 198 096 790 718 799 406 049 037 124 255 089 816 565 402 650 542 08 × 2 = 0 + 0.000 396 193 581 437 598 812 098 074 248 510 179 633 130 805 301 084 16;
  • 52) 0.000 396 193 581 437 598 812 098 074 248 510 179 633 130 805 301 084 16 × 2 = 0 + 0.000 792 387 162 875 197 624 196 148 497 020 359 266 261 610 602 168 32;
  • 53) 0.000 792 387 162 875 197 624 196 148 497 020 359 266 261 610 602 168 32 × 2 = 0 + 0.001 584 774 325 750 395 248 392 296 994 040 718 532 523 221 204 336 64;
  • 54) 0.001 584 774 325 750 395 248 392 296 994 040 718 532 523 221 204 336 64 × 2 = 0 + 0.003 169 548 651 500 790 496 784 593 988 081 437 065 046 442 408 673 28;
  • 55) 0.003 169 548 651 500 790 496 784 593 988 081 437 065 046 442 408 673 28 × 2 = 0 + 0.006 339 097 303 001 580 993 569 187 976 162 874 130 092 884 817 346 56;
  • 56) 0.006 339 097 303 001 580 993 569 187 976 162 874 130 092 884 817 346 56 × 2 = 0 + 0.012 678 194 606 003 161 987 138 375 952 325 748 260 185 769 634 693 12;
  • 57) 0.012 678 194 606 003 161 987 138 375 952 325 748 260 185 769 634 693 12 × 2 = 0 + 0.025 356 389 212 006 323 974 276 751 904 651 496 520 371 539 269 386 24;
  • 58) 0.025 356 389 212 006 323 974 276 751 904 651 496 520 371 539 269 386 24 × 2 = 0 + 0.050 712 778 424 012 647 948 553 503 809 302 993 040 743 078 538 772 48;
  • 59) 0.050 712 778 424 012 647 948 553 503 809 302 993 040 743 078 538 772 48 × 2 = 0 + 0.101 425 556 848 025 295 897 107 007 618 605 986 081 486 157 077 544 96;
  • 60) 0.101 425 556 848 025 295 897 107 007 618 605 986 081 486 157 077 544 96 × 2 = 0 + 0.202 851 113 696 050 591 794 214 015 237 211 972 162 972 314 155 089 92;
  • 61) 0.202 851 113 696 050 591 794 214 015 237 211 972 162 972 314 155 089 92 × 2 = 0 + 0.405 702 227 392 101 183 588 428 030 474 423 944 325 944 628 310 179 84;
  • 62) 0.405 702 227 392 101 183 588 428 030 474 423 944 325 944 628 310 179 84 × 2 = 0 + 0.811 404 454 784 202 367 176 856 060 948 847 888 651 889 256 620 359 68;
  • 63) 0.811 404 454 784 202 367 176 856 060 948 847 888 651 889 256 620 359 68 × 2 = 1 + 0.622 808 909 568 404 734 353 712 121 897 695 777 303 778 513 240 719 36;
  • 64) 0.622 808 909 568 404 734 353 712 121 897 695 777 303 778 513 240 719 36 × 2 = 1 + 0.245 617 819 136 809 468 707 424 243 795 391 554 607 557 026 481 438 72;
  • 65) 0.245 617 819 136 809 468 707 424 243 795 391 554 607 557 026 481 438 72 × 2 = 0 + 0.491 235 638 273 618 937 414 848 487 590 783 109 215 114 052 962 877 44;
  • 66) 0.491 235 638 273 618 937 414 848 487 590 783 109 215 114 052 962 877 44 × 2 = 0 + 0.982 471 276 547 237 874 829 696 975 181 566 218 430 228 105 925 754 88;
  • 67) 0.982 471 276 547 237 874 829 696 975 181 566 218 430 228 105 925 754 88 × 2 = 1 + 0.964 942 553 094 475 749 659 393 950 363 132 436 860 456 211 851 509 76;
  • 68) 0.964 942 553 094 475 749 659 393 950 363 132 436 860 456 211 851 509 76 × 2 = 1 + 0.929 885 106 188 951 499 318 787 900 726 264 873 720 912 423 703 019 52;
  • 69) 0.929 885 106 188 951 499 318 787 900 726 264 873 720 912 423 703 019 52 × 2 = 1 + 0.859 770 212 377 902 998 637 575 801 452 529 747 441 824 847 406 039 04;
  • 70) 0.859 770 212 377 902 998 637 575 801 452 529 747 441 824 847 406 039 04 × 2 = 1 + 0.719 540 424 755 805 997 275 151 602 905 059 494 883 649 694 812 078 08;
  • 71) 0.719 540 424 755 805 997 275 151 602 905 059 494 883 649 694 812 078 08 × 2 = 1 + 0.439 080 849 511 611 994 550 303 205 810 118 989 767 299 389 624 156 16;
  • 72) 0.439 080 849 511 611 994 550 303 205 810 118 989 767 299 389 624 156 16 × 2 = 0 + 0.878 161 699 023 223 989 100 606 411 620 237 979 534 598 779 248 312 32;
  • 73) 0.878 161 699 023 223 989 100 606 411 620 237 979 534 598 779 248 312 32 × 2 = 1 + 0.756 323 398 046 447 978 201 212 823 240 475 959 069 197 558 496 624 64;
  • 74) 0.756 323 398 046 447 978 201 212 823 240 475 959 069 197 558 496 624 64 × 2 = 1 + 0.512 646 796 092 895 956 402 425 646 480 951 918 138 395 116 993 249 28;
  • 75) 0.512 646 796 092 895 956 402 425 646 480 951 918 138 395 116 993 249 28 × 2 = 1 + 0.025 293 592 185 791 912 804 851 292 961 903 836 276 790 233 986 498 56;
  • 76) 0.025 293 592 185 791 912 804 851 292 961 903 836 276 790 233 986 498 56 × 2 = 0 + 0.050 587 184 371 583 825 609 702 585 923 807 672 553 580 467 972 997 12;
  • 77) 0.050 587 184 371 583 825 609 702 585 923 807 672 553 580 467 972 997 12 × 2 = 0 + 0.101 174 368 743 167 651 219 405 171 847 615 345 107 160 935 945 994 24;
  • 78) 0.101 174 368 743 167 651 219 405 171 847 615 345 107 160 935 945 994 24 × 2 = 0 + 0.202 348 737 486 335 302 438 810 343 695 230 690 214 321 871 891 988 48;
  • 79) 0.202 348 737 486 335 302 438 810 343 695 230 690 214 321 871 891 988 48 × 2 = 0 + 0.404 697 474 972 670 604 877 620 687 390 461 380 428 643 743 783 976 96;
  • 80) 0.404 697 474 972 670 604 877 620 687 390 461 380 428 643 743 783 976 96 × 2 = 0 + 0.809 394 949 945 341 209 755 241 374 780 922 760 857 287 487 567 953 92;
  • 81) 0.809 394 949 945 341 209 755 241 374 780 922 760 857 287 487 567 953 92 × 2 = 1 + 0.618 789 899 890 682 419 510 482 749 561 845 521 714 574 975 135 907 84;
  • 82) 0.618 789 899 890 682 419 510 482 749 561 845 521 714 574 975 135 907 84 × 2 = 1 + 0.237 579 799 781 364 839 020 965 499 123 691 043 429 149 950 271 815 68;
  • 83) 0.237 579 799 781 364 839 020 965 499 123 691 043 429 149 950 271 815 68 × 2 = 0 + 0.475 159 599 562 729 678 041 930 998 247 382 086 858 299 900 543 631 36;
  • 84) 0.475 159 599 562 729 678 041 930 998 247 382 086 858 299 900 543 631 36 × 2 = 0 + 0.950 319 199 125 459 356 083 861 996 494 764 173 716 599 801 087 262 72;
  • 85) 0.950 319 199 125 459 356 083 861 996 494 764 173 716 599 801 087 262 72 × 2 = 1 + 0.900 638 398 250 918 712 167 723 992 989 528 347 433 199 602 174 525 44;
  • 86) 0.900 638 398 250 918 712 167 723 992 989 528 347 433 199 602 174 525 44 × 2 = 1 + 0.801 276 796 501 837 424 335 447 985 979 056 694 866 399 204 349 050 88;
  • 87) 0.801 276 796 501 837 424 335 447 985 979 056 694 866 399 204 349 050 88 × 2 = 1 + 0.602 553 593 003 674 848 670 895 971 958 113 389 732 798 408 698 101 76;
  • 88) 0.602 553 593 003 674 848 670 895 971 958 113 389 732 798 408 698 101 76 × 2 = 1 + 0.205 107 186 007 349 697 341 791 943 916 226 779 465 596 817 396 203 52;
  • 89) 0.205 107 186 007 349 697 341 791 943 916 226 779 465 596 817 396 203 52 × 2 = 0 + 0.410 214 372 014 699 394 683 583 887 832 453 558 931 193 634 792 407 04;
  • 90) 0.410 214 372 014 699 394 683 583 887 832 453 558 931 193 634 792 407 04 × 2 = 0 + 0.820 428 744 029 398 789 367 167 775 664 907 117 862 387 269 584 814 08;
  • 91) 0.820 428 744 029 398 789 367 167 775 664 907 117 862 387 269 584 814 08 × 2 = 1 + 0.640 857 488 058 797 578 734 335 551 329 814 235 724 774 539 169 628 16;
  • 92) 0.640 857 488 058 797 578 734 335 551 329 814 235 724 774 539 169 628 16 × 2 = 1 + 0.281 714 976 117 595 157 468 671 102 659 628 471 449 549 078 339 256 32;
  • 93) 0.281 714 976 117 595 157 468 671 102 659 628 471 449 549 078 339 256 32 × 2 = 0 + 0.563 429 952 235 190 314 937 342 205 319 256 942 899 098 156 678 512 64;

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:


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

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


0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 045(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 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 001 045 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 0.000 000 000 000 755 840 011 110 101 101 010 100 110 000 000 000 000 986 to 64 bit double precision IEEE 754 binary floating point representation standard

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» Month 11, 2026 [November]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: November - 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