0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 361 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 361 2(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
0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 361 2(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: 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 361 2.

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) 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 361 2 × 2 = 0 + 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 722 4;
  • 2) 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 722 4 × 2 = 0 + 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 444 8;
  • 3) 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 444 8 × 2 = 0 + 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 889 6;
  • 4) 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 889 6 × 2 = 0 + 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 779 2;
  • 5) 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 779 2 × 2 = 0 + 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 558 4;
  • 6) 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 558 4 × 2 = 0 + 0.000 000 000 000 000 790 124 025 252 133 724 078 653 463 116 8;
  • 7) 0.000 000 000 000 000 790 124 025 252 133 724 078 653 463 116 8 × 2 = 0 + 0.000 000 000 000 001 580 248 050 504 267 448 157 306 926 233 6;
  • 8) 0.000 000 000 000 001 580 248 050 504 267 448 157 306 926 233 6 × 2 = 0 + 0.000 000 000 000 003 160 496 101 008 534 896 314 613 852 467 2;
  • 9) 0.000 000 000 000 003 160 496 101 008 534 896 314 613 852 467 2 × 2 = 0 + 0.000 000 000 000 006 320 992 202 017 069 792 629 227 704 934 4;
  • 10) 0.000 000 000 000 006 320 992 202 017 069 792 629 227 704 934 4 × 2 = 0 + 0.000 000 000 000 012 641 984 404 034 139 585 258 455 409 868 8;
  • 11) 0.000 000 000 000 012 641 984 404 034 139 585 258 455 409 868 8 × 2 = 0 + 0.000 000 000 000 025 283 968 808 068 279 170 516 910 819 737 6;
  • 12) 0.000 000 000 000 025 283 968 808 068 279 170 516 910 819 737 6 × 2 = 0 + 0.000 000 000 000 050 567 937 616 136 558 341 033 821 639 475 2;
  • 13) 0.000 000 000 000 050 567 937 616 136 558 341 033 821 639 475 2 × 2 = 0 + 0.000 000 000 000 101 135 875 232 273 116 682 067 643 278 950 4;
  • 14) 0.000 000 000 000 101 135 875 232 273 116 682 067 643 278 950 4 × 2 = 0 + 0.000 000 000 000 202 271 750 464 546 233 364 135 286 557 900 8;
  • 15) 0.000 000 000 000 202 271 750 464 546 233 364 135 286 557 900 8 × 2 = 0 + 0.000 000 000 000 404 543 500 929 092 466 728 270 573 115 801 6;
  • 16) 0.000 000 000 000 404 543 500 929 092 466 728 270 573 115 801 6 × 2 = 0 + 0.000 000 000 000 809 087 001 858 184 933 456 541 146 231 603 2;
  • 17) 0.000 000 000 000 809 087 001 858 184 933 456 541 146 231 603 2 × 2 = 0 + 0.000 000 000 001 618 174 003 716 369 866 913 082 292 463 206 4;
  • 18) 0.000 000 000 001 618 174 003 716 369 866 913 082 292 463 206 4 × 2 = 0 + 0.000 000 000 003 236 348 007 432 739 733 826 164 584 926 412 8;
  • 19) 0.000 000 000 003 236 348 007 432 739 733 826 164 584 926 412 8 × 2 = 0 + 0.000 000 000 006 472 696 014 865 479 467 652 329 169 852 825 6;
  • 20) 0.000 000 000 006 472 696 014 865 479 467 652 329 169 852 825 6 × 2 = 0 + 0.000 000 000 012 945 392 029 730 958 935 304 658 339 705 651 2;
  • 21) 0.000 000 000 012 945 392 029 730 958 935 304 658 339 705 651 2 × 2 = 0 + 0.000 000 000 025 890 784 059 461 917 870 609 316 679 411 302 4;
  • 22) 0.000 000 000 025 890 784 059 461 917 870 609 316 679 411 302 4 × 2 = 0 + 0.000 000 000 051 781 568 118 923 835 741 218 633 358 822 604 8;
  • 23) 0.000 000 000 051 781 568 118 923 835 741 218 633 358 822 604 8 × 2 = 0 + 0.000 000 000 103 563 136 237 847 671 482 437 266 717 645 209 6;
  • 24) 0.000 000 000 103 563 136 237 847 671 482 437 266 717 645 209 6 × 2 = 0 + 0.000 000 000 207 126 272 475 695 342 964 874 533 435 290 419 2;
  • 25) 0.000 000 000 207 126 272 475 695 342 964 874 533 435 290 419 2 × 2 = 0 + 0.000 000 000 414 252 544 951 390 685 929 749 066 870 580 838 4;
  • 26) 0.000 000 000 414 252 544 951 390 685 929 749 066 870 580 838 4 × 2 = 0 + 0.000 000 000 828 505 089 902 781 371 859 498 133 741 161 676 8;
  • 27) 0.000 000 000 828 505 089 902 781 371 859 498 133 741 161 676 8 × 2 = 0 + 0.000 000 001 657 010 179 805 562 743 718 996 267 482 323 353 6;
  • 28) 0.000 000 001 657 010 179 805 562 743 718 996 267 482 323 353 6 × 2 = 0 + 0.000 000 003 314 020 359 611 125 487 437 992 534 964 646 707 2;
  • 29) 0.000 000 003 314 020 359 611 125 487 437 992 534 964 646 707 2 × 2 = 0 + 0.000 000 006 628 040 719 222 250 974 875 985 069 929 293 414 4;
  • 30) 0.000 000 006 628 040 719 222 250 974 875 985 069 929 293 414 4 × 2 = 0 + 0.000 000 013 256 081 438 444 501 949 751 970 139 858 586 828 8;
  • 31) 0.000 000 013 256 081 438 444 501 949 751 970 139 858 586 828 8 × 2 = 0 + 0.000 000 026 512 162 876 889 003 899 503 940 279 717 173 657 6;
  • 32) 0.000 000 026 512 162 876 889 003 899 503 940 279 717 173 657 6 × 2 = 0 + 0.000 000 053 024 325 753 778 007 799 007 880 559 434 347 315 2;
  • 33) 0.000 000 053 024 325 753 778 007 799 007 880 559 434 347 315 2 × 2 = 0 + 0.000 000 106 048 651 507 556 015 598 015 761 118 868 694 630 4;
  • 34) 0.000 000 106 048 651 507 556 015 598 015 761 118 868 694 630 4 × 2 = 0 + 0.000 000 212 097 303 015 112 031 196 031 522 237 737 389 260 8;
  • 35) 0.000 000 212 097 303 015 112 031 196 031 522 237 737 389 260 8 × 2 = 0 + 0.000 000 424 194 606 030 224 062 392 063 044 475 474 778 521 6;
  • 36) 0.000 000 424 194 606 030 224 062 392 063 044 475 474 778 521 6 × 2 = 0 + 0.000 000 848 389 212 060 448 124 784 126 088 950 949 557 043 2;
  • 37) 0.000 000 848 389 212 060 448 124 784 126 088 950 949 557 043 2 × 2 = 0 + 0.000 001 696 778 424 120 896 249 568 252 177 901 899 114 086 4;
  • 38) 0.000 001 696 778 424 120 896 249 568 252 177 901 899 114 086 4 × 2 = 0 + 0.000 003 393 556 848 241 792 499 136 504 355 803 798 228 172 8;
  • 39) 0.000 003 393 556 848 241 792 499 136 504 355 803 798 228 172 8 × 2 = 0 + 0.000 006 787 113 696 483 584 998 273 008 711 607 596 456 345 6;
  • 40) 0.000 006 787 113 696 483 584 998 273 008 711 607 596 456 345 6 × 2 = 0 + 0.000 013 574 227 392 967 169 996 546 017 423 215 192 912 691 2;
  • 41) 0.000 013 574 227 392 967 169 996 546 017 423 215 192 912 691 2 × 2 = 0 + 0.000 027 148 454 785 934 339 993 092 034 846 430 385 825 382 4;
  • 42) 0.000 027 148 454 785 934 339 993 092 034 846 430 385 825 382 4 × 2 = 0 + 0.000 054 296 909 571 868 679 986 184 069 692 860 771 650 764 8;
  • 43) 0.000 054 296 909 571 868 679 986 184 069 692 860 771 650 764 8 × 2 = 0 + 0.000 108 593 819 143 737 359 972 368 139 385 721 543 301 529 6;
  • 44) 0.000 108 593 819 143 737 359 972 368 139 385 721 543 301 529 6 × 2 = 0 + 0.000 217 187 638 287 474 719 944 736 278 771 443 086 603 059 2;
  • 45) 0.000 217 187 638 287 474 719 944 736 278 771 443 086 603 059 2 × 2 = 0 + 0.000 434 375 276 574 949 439 889 472 557 542 886 173 206 118 4;
  • 46) 0.000 434 375 276 574 949 439 889 472 557 542 886 173 206 118 4 × 2 = 0 + 0.000 868 750 553 149 898 879 778 945 115 085 772 346 412 236 8;
  • 47) 0.000 868 750 553 149 898 879 778 945 115 085 772 346 412 236 8 × 2 = 0 + 0.001 737 501 106 299 797 759 557 890 230 171 544 692 824 473 6;
  • 48) 0.001 737 501 106 299 797 759 557 890 230 171 544 692 824 473 6 × 2 = 0 + 0.003 475 002 212 599 595 519 115 780 460 343 089 385 648 947 2;
  • 49) 0.003 475 002 212 599 595 519 115 780 460 343 089 385 648 947 2 × 2 = 0 + 0.006 950 004 425 199 191 038 231 560 920 686 178 771 297 894 4;
  • 50) 0.006 950 004 425 199 191 038 231 560 920 686 178 771 297 894 4 × 2 = 0 + 0.013 900 008 850 398 382 076 463 121 841 372 357 542 595 788 8;
  • 51) 0.013 900 008 850 398 382 076 463 121 841 372 357 542 595 788 8 × 2 = 0 + 0.027 800 017 700 796 764 152 926 243 682 744 715 085 191 577 6;
  • 52) 0.027 800 017 700 796 764 152 926 243 682 744 715 085 191 577 6 × 2 = 0 + 0.055 600 035 401 593 528 305 852 487 365 489 430 170 383 155 2;
  • 53) 0.055 600 035 401 593 528 305 852 487 365 489 430 170 383 155 2 × 2 = 0 + 0.111 200 070 803 187 056 611 704 974 730 978 860 340 766 310 4;
  • 54) 0.111 200 070 803 187 056 611 704 974 730 978 860 340 766 310 4 × 2 = 0 + 0.222 400 141 606 374 113 223 409 949 461 957 720 681 532 620 8;
  • 55) 0.222 400 141 606 374 113 223 409 949 461 957 720 681 532 620 8 × 2 = 0 + 0.444 800 283 212 748 226 446 819 898 923 915 441 363 065 241 6;
  • 56) 0.444 800 283 212 748 226 446 819 898 923 915 441 363 065 241 6 × 2 = 0 + 0.889 600 566 425 496 452 893 639 797 847 830 882 726 130 483 2;
  • 57) 0.889 600 566 425 496 452 893 639 797 847 830 882 726 130 483 2 × 2 = 1 + 0.779 201 132 850 992 905 787 279 595 695 661 765 452 260 966 4;
  • 58) 0.779 201 132 850 992 905 787 279 595 695 661 765 452 260 966 4 × 2 = 1 + 0.558 402 265 701 985 811 574 559 191 391 323 530 904 521 932 8;
  • 59) 0.558 402 265 701 985 811 574 559 191 391 323 530 904 521 932 8 × 2 = 1 + 0.116 804 531 403 971 623 149 118 382 782 647 061 809 043 865 6;
  • 60) 0.116 804 531 403 971 623 149 118 382 782 647 061 809 043 865 6 × 2 = 0 + 0.233 609 062 807 943 246 298 236 765 565 294 123 618 087 731 2;
  • 61) 0.233 609 062 807 943 246 298 236 765 565 294 123 618 087 731 2 × 2 = 0 + 0.467 218 125 615 886 492 596 473 531 130 588 247 236 175 462 4;
  • 62) 0.467 218 125 615 886 492 596 473 531 130 588 247 236 175 462 4 × 2 = 0 + 0.934 436 251 231 772 985 192 947 062 261 176 494 472 350 924 8;
  • 63) 0.934 436 251 231 772 985 192 947 062 261 176 494 472 350 924 8 × 2 = 1 + 0.868 872 502 463 545 970 385 894 124 522 352 988 944 701 849 6;
  • 64) 0.868 872 502 463 545 970 385 894 124 522 352 988 944 701 849 6 × 2 = 1 + 0.737 745 004 927 091 940 771 788 249 044 705 977 889 403 699 2;
  • 65) 0.737 745 004 927 091 940 771 788 249 044 705 977 889 403 699 2 × 2 = 1 + 0.475 490 009 854 183 881 543 576 498 089 411 955 778 807 398 4;
  • 66) 0.475 490 009 854 183 881 543 576 498 089 411 955 778 807 398 4 × 2 = 0 + 0.950 980 019 708 367 763 087 152 996 178 823 911 557 614 796 8;
  • 67) 0.950 980 019 708 367 763 087 152 996 178 823 911 557 614 796 8 × 2 = 1 + 0.901 960 039 416 735 526 174 305 992 357 647 823 115 229 593 6;
  • 68) 0.901 960 039 416 735 526 174 305 992 357 647 823 115 229 593 6 × 2 = 1 + 0.803 920 078 833 471 052 348 611 984 715 295 646 230 459 187 2;
  • 69) 0.803 920 078 833 471 052 348 611 984 715 295 646 230 459 187 2 × 2 = 1 + 0.607 840 157 666 942 104 697 223 969 430 591 292 460 918 374 4;
  • 70) 0.607 840 157 666 942 104 697 223 969 430 591 292 460 918 374 4 × 2 = 1 + 0.215 680 315 333 884 209 394 447 938 861 182 584 921 836 748 8;
  • 71) 0.215 680 315 333 884 209 394 447 938 861 182 584 921 836 748 8 × 2 = 0 + 0.431 360 630 667 768 418 788 895 877 722 365 169 843 673 497 6;
  • 72) 0.431 360 630 667 768 418 788 895 877 722 365 169 843 673 497 6 × 2 = 0 + 0.862 721 261 335 536 837 577 791 755 444 730 339 687 346 995 2;
  • 73) 0.862 721 261 335 536 837 577 791 755 444 730 339 687 346 995 2 × 2 = 1 + 0.725 442 522 671 073 675 155 583 510 889 460 679 374 693 990 4;
  • 74) 0.725 442 522 671 073 675 155 583 510 889 460 679 374 693 990 4 × 2 = 1 + 0.450 885 045 342 147 350 311 167 021 778 921 358 749 387 980 8;
  • 75) 0.450 885 045 342 147 350 311 167 021 778 921 358 749 387 980 8 × 2 = 0 + 0.901 770 090 684 294 700 622 334 043 557 842 717 498 775 961 6;
  • 76) 0.901 770 090 684 294 700 622 334 043 557 842 717 498 775 961 6 × 2 = 1 + 0.803 540 181 368 589 401 244 668 087 115 685 434 997 551 923 2;
  • 77) 0.803 540 181 368 589 401 244 668 087 115 685 434 997 551 923 2 × 2 = 1 + 0.607 080 362 737 178 802 489 336 174 231 370 869 995 103 846 4;
  • 78) 0.607 080 362 737 178 802 489 336 174 231 370 869 995 103 846 4 × 2 = 1 + 0.214 160 725 474 357 604 978 672 348 462 741 739 990 207 692 8;
  • 79) 0.214 160 725 474 357 604 978 672 348 462 741 739 990 207 692 8 × 2 = 0 + 0.428 321 450 948 715 209 957 344 696 925 483 479 980 415 385 6;
  • 80) 0.428 321 450 948 715 209 957 344 696 925 483 479 980 415 385 6 × 2 = 0 + 0.856 642 901 897 430 419 914 689 393 850 966 959 960 830 771 2;
  • 81) 0.856 642 901 897 430 419 914 689 393 850 966 959 960 830 771 2 × 2 = 1 + 0.713 285 803 794 860 839 829 378 787 701 933 919 921 661 542 4;
  • 82) 0.713 285 803 794 860 839 829 378 787 701 933 919 921 661 542 4 × 2 = 1 + 0.426 571 607 589 721 679 658 757 575 403 867 839 843 323 084 8;
  • 83) 0.426 571 607 589 721 679 658 757 575 403 867 839 843 323 084 8 × 2 = 0 + 0.853 143 215 179 443 359 317 515 150 807 735 679 686 646 169 6;
  • 84) 0.853 143 215 179 443 359 317 515 150 807 735 679 686 646 169 6 × 2 = 1 + 0.706 286 430 358 886 718 635 030 301 615 471 359 373 292 339 2;
  • 85) 0.706 286 430 358 886 718 635 030 301 615 471 359 373 292 339 2 × 2 = 1 + 0.412 572 860 717 773 437 270 060 603 230 942 718 746 584 678 4;
  • 86) 0.412 572 860 717 773 437 270 060 603 230 942 718 746 584 678 4 × 2 = 0 + 0.825 145 721 435 546 874 540 121 206 461 885 437 493 169 356 8;
  • 87) 0.825 145 721 435 546 874 540 121 206 461 885 437 493 169 356 8 × 2 = 1 + 0.650 291 442 871 093 749 080 242 412 923 770 874 986 338 713 6;
  • 88) 0.650 291 442 871 093 749 080 242 412 923 770 874 986 338 713 6 × 2 = 1 + 0.300 582 885 742 187 498 160 484 825 847 541 749 972 677 427 2;
  • 89) 0.300 582 885 742 187 498 160 484 825 847 541 749 972 677 427 2 × 2 = 0 + 0.601 165 771 484 374 996 320 969 651 695 083 499 945 354 854 4;
  • 90) 0.601 165 771 484 374 996 320 969 651 695 083 499 945 354 854 4 × 2 = 1 + 0.202 331 542 968 749 992 641 939 303 390 166 999 890 709 708 8;
  • 91) 0.202 331 542 968 749 992 641 939 303 390 166 999 890 709 708 8 × 2 = 0 + 0.404 663 085 937 499 985 283 878 606 780 333 999 781 419 417 6;
  • 92) 0.404 663 085 937 499 985 283 878 606 780 333 999 781 419 417 6 × 2 = 0 + 0.809 326 171 874 999 970 567 757 213 560 667 999 562 838 835 2;
  • 93) 0.809 326 171 874 999 970 567 757 213 560 667 999 562 838 835 2 × 2 = 1 + 0.618 652 343 749 999 941 135 514 427 121 335 999 125 677 670 4;
  • 94) 0.618 652 343 749 999 941 135 514 427 121 335 999 125 677 670 4 × 2 = 1 + 0.237 304 687 499 999 882 271 028 854 242 671 998 251 355 340 8;
  • 95) 0.237 304 687 499 999 882 271 028 854 242 671 998 251 355 340 8 × 2 = 0 + 0.474 609 374 999 999 764 542 057 708 485 343 996 502 710 681 6;
  • 96) 0.474 609 374 999 999 764 542 057 708 485 343 996 502 710 681 6 × 2 = 0 + 0.949 218 749 999 999 529 084 115 416 970 687 993 005 421 363 2;
  • 97) 0.949 218 749 999 999 529 084 115 416 970 687 993 005 421 363 2 × 2 = 1 + 0.898 437 499 999 999 058 168 230 833 941 375 986 010 842 726 4;
  • 98) 0.898 437 499 999 999 058 168 230 833 941 375 986 010 842 726 4 × 2 = 1 + 0.796 874 999 999 998 116 336 461 667 882 751 972 021 685 452 8;
  • 99) 0.796 874 999 999 998 116 336 461 667 882 751 972 021 685 452 8 × 2 = 1 + 0.593 749 999 999 996 232 672 923 335 765 503 944 043 370 905 6;
  • 100) 0.593 749 999 999 996 232 672 923 335 765 503 944 043 370 905 6 × 2 = 1 + 0.187 499 999 999 992 465 345 846 671 531 007 888 086 741 811 2;
  • 101) 0.187 499 999 999 992 465 345 846 671 531 007 888 086 741 811 2 × 2 = 0 + 0.374 999 999 999 984 930 691 693 343 062 015 776 173 483 622 4;
  • 102) 0.374 999 999 999 984 930 691 693 343 062 015 776 173 483 622 4 × 2 = 0 + 0.749 999 999 999 969 861 383 386 686 124 031 552 346 967 244 8;
  • 103) 0.749 999 999 999 969 861 383 386 686 124 031 552 346 967 244 8 × 2 = 1 + 0.499 999 999 999 939 722 766 773 372 248 063 104 693 934 489 6;
  • 104) 0.499 999 999 999 939 722 766 773 372 248 063 104 693 934 489 6 × 2 = 0 + 0.999 999 999 999 879 445 533 546 744 496 126 209 387 868 979 2;
  • 105) 0.999 999 999 999 879 445 533 546 744 496 126 209 387 868 979 2 × 2 = 1 + 0.999 999 999 999 758 891 067 093 488 992 252 418 775 737 958 4;
  • 106) 0.999 999 999 999 758 891 067 093 488 992 252 418 775 737 958 4 × 2 = 1 + 0.999 999 999 999 517 782 134 186 977 984 504 837 551 475 916 8;
  • 107) 0.999 999 999 999 517 782 134 186 977 984 504 837 551 475 916 8 × 2 = 1 + 0.999 999 999 999 035 564 268 373 955 969 009 675 102 951 833 6;
  • 108) 0.999 999 999 999 035 564 268 373 955 969 009 675 102 951 833 6 × 2 = 1 + 0.999 999 999 998 071 128 536 747 911 938 019 350 205 903 667 2;
  • 109) 0.999 999 999 998 071 128 536 747 911 938 019 350 205 903 667 2 × 2 = 1 + 0.999 999 999 996 142 257 073 495 823 876 038 700 411 807 334 4;

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:


0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 361 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110 0011 1011 1100 1101 1100 1101 1011 0100 1100 1111 0010 1111 1(2)

5. Positive number before normalization:

0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 361 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110 0011 1011 1100 1101 1100 1101 1011 0100 1100 1111 0010 1111 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 57 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 361 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110 0011 1011 1100 1101 1100 1101 1011 0100 1100 1111 0010 1111 1(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110 0011 1011 1100 1101 1100 1101 1011 0100 1100 1111 0010 1111 1(2) × 20 =

1.1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 1111(2) × 2-57


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): -57

Mantissa (not normalized):
1.1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 1111


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-57 + 2(11-1) - 1 =

(-57 + 1 023)(10) =

966(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;
  • 966 ÷ 2 = 483 + 0;
  • 483 ÷ 2 = 241 + 1;
  • 241 ÷ 2 = 120 + 1;
  • 120 ÷ 2 = 60 + 0;
  • 60 ÷ 2 = 30 + 0;
  • 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) =

966(10) =

011 1100 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. 1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 1111 =

1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 1111


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 1100 0110

Mantissa (52 bits) =
1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 1111


Decimal number 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 361 2 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1100 0110 - 1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 1111

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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