-0.000 000 045 419 381 005 632 302 850 869 008 985 917 885 112 826 297 699 939 459 562 301 635 729 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 045 419 381 005 632 302 850 869 008 985 917 885 112 826 297 699 939 459 562 301 635 729 5(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
-0.000 000 045 419 381 005 632 302 850 869 008 985 917 885 112 826 297 699 939 459 562 301 635 729 5(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. Start with the positive version of the number:

|-0.000 000 045 419 381 005 632 302 850 869 008 985 917 885 112 826 297 699 939 459 562 301 635 729 5| = 0.000 000 045 419 381 005 632 302 850 869 008 985 917 885 112 826 297 699 939 459 562 301 635 729 5


2. 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;

3. 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)


4. Convert to binary (base 2) the fractional part: 0.000 000 045 419 381 005 632 302 850 869 008 985 917 885 112 826 297 699 939 459 562 301 635 729 5.

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) 0.000 000 045 419 381 005 632 302 850 869 008 985 917 885 112 826 297 699 939 459 562 301 635 729 5 × 2 = 0 + 0.000 000 090 838 762 011 264 605 701 738 017 971 835 770 225 652 595 399 878 919 124 603 271 459;
  • 2) 0.000 000 090 838 762 011 264 605 701 738 017 971 835 770 225 652 595 399 878 919 124 603 271 459 × 2 = 0 + 0.000 000 181 677 524 022 529 211 403 476 035 943 671 540 451 305 190 799 757 838 249 206 542 918;
  • 3) 0.000 000 181 677 524 022 529 211 403 476 035 943 671 540 451 305 190 799 757 838 249 206 542 918 × 2 = 0 + 0.000 000 363 355 048 045 058 422 806 952 071 887 343 080 902 610 381 599 515 676 498 413 085 836;
  • 4) 0.000 000 363 355 048 045 058 422 806 952 071 887 343 080 902 610 381 599 515 676 498 413 085 836 × 2 = 0 + 0.000 000 726 710 096 090 116 845 613 904 143 774 686 161 805 220 763 199 031 352 996 826 171 672;
  • 5) 0.000 000 726 710 096 090 116 845 613 904 143 774 686 161 805 220 763 199 031 352 996 826 171 672 × 2 = 0 + 0.000 001 453 420 192 180 233 691 227 808 287 549 372 323 610 441 526 398 062 705 993 652 343 344;
  • 6) 0.000 001 453 420 192 180 233 691 227 808 287 549 372 323 610 441 526 398 062 705 993 652 343 344 × 2 = 0 + 0.000 002 906 840 384 360 467 382 455 616 575 098 744 647 220 883 052 796 125 411 987 304 686 688;
  • 7) 0.000 002 906 840 384 360 467 382 455 616 575 098 744 647 220 883 052 796 125 411 987 304 686 688 × 2 = 0 + 0.000 005 813 680 768 720 934 764 911 233 150 197 489 294 441 766 105 592 250 823 974 609 373 376;
  • 8) 0.000 005 813 680 768 720 934 764 911 233 150 197 489 294 441 766 105 592 250 823 974 609 373 376 × 2 = 0 + 0.000 011 627 361 537 441 869 529 822 466 300 394 978 588 883 532 211 184 501 647 949 218 746 752;
  • 9) 0.000 011 627 361 537 441 869 529 822 466 300 394 978 588 883 532 211 184 501 647 949 218 746 752 × 2 = 0 + 0.000 023 254 723 074 883 739 059 644 932 600 789 957 177 767 064 422 369 003 295 898 437 493 504;
  • 10) 0.000 023 254 723 074 883 739 059 644 932 600 789 957 177 767 064 422 369 003 295 898 437 493 504 × 2 = 0 + 0.000 046 509 446 149 767 478 119 289 865 201 579 914 355 534 128 844 738 006 591 796 874 987 008;
  • 11) 0.000 046 509 446 149 767 478 119 289 865 201 579 914 355 534 128 844 738 006 591 796 874 987 008 × 2 = 0 + 0.000 093 018 892 299 534 956 238 579 730 403 159 828 711 068 257 689 476 013 183 593 749 974 016;
  • 12) 0.000 093 018 892 299 534 956 238 579 730 403 159 828 711 068 257 689 476 013 183 593 749 974 016 × 2 = 0 + 0.000 186 037 784 599 069 912 477 159 460 806 319 657 422 136 515 378 952 026 367 187 499 948 032;
  • 13) 0.000 186 037 784 599 069 912 477 159 460 806 319 657 422 136 515 378 952 026 367 187 499 948 032 × 2 = 0 + 0.000 372 075 569 198 139 824 954 318 921 612 639 314 844 273 030 757 904 052 734 374 999 896 064;
  • 14) 0.000 372 075 569 198 139 824 954 318 921 612 639 314 844 273 030 757 904 052 734 374 999 896 064 × 2 = 0 + 0.000 744 151 138 396 279 649 908 637 843 225 278 629 688 546 061 515 808 105 468 749 999 792 128;
  • 15) 0.000 744 151 138 396 279 649 908 637 843 225 278 629 688 546 061 515 808 105 468 749 999 792 128 × 2 = 0 + 0.001 488 302 276 792 559 299 817 275 686 450 557 259 377 092 123 031 616 210 937 499 999 584 256;
  • 16) 0.001 488 302 276 792 559 299 817 275 686 450 557 259 377 092 123 031 616 210 937 499 999 584 256 × 2 = 0 + 0.002 976 604 553 585 118 599 634 551 372 901 114 518 754 184 246 063 232 421 874 999 999 168 512;
  • 17) 0.002 976 604 553 585 118 599 634 551 372 901 114 518 754 184 246 063 232 421 874 999 999 168 512 × 2 = 0 + 0.005 953 209 107 170 237 199 269 102 745 802 229 037 508 368 492 126 464 843 749 999 998 337 024;
  • 18) 0.005 953 209 107 170 237 199 269 102 745 802 229 037 508 368 492 126 464 843 749 999 998 337 024 × 2 = 0 + 0.011 906 418 214 340 474 398 538 205 491 604 458 075 016 736 984 252 929 687 499 999 996 674 048;
  • 19) 0.011 906 418 214 340 474 398 538 205 491 604 458 075 016 736 984 252 929 687 499 999 996 674 048 × 2 = 0 + 0.023 812 836 428 680 948 797 076 410 983 208 916 150 033 473 968 505 859 374 999 999 993 348 096;
  • 20) 0.023 812 836 428 680 948 797 076 410 983 208 916 150 033 473 968 505 859 374 999 999 993 348 096 × 2 = 0 + 0.047 625 672 857 361 897 594 152 821 966 417 832 300 066 947 937 011 718 749 999 999 986 696 192;
  • 21) 0.047 625 672 857 361 897 594 152 821 966 417 832 300 066 947 937 011 718 749 999 999 986 696 192 × 2 = 0 + 0.095 251 345 714 723 795 188 305 643 932 835 664 600 133 895 874 023 437 499 999 999 973 392 384;
  • 22) 0.095 251 345 714 723 795 188 305 643 932 835 664 600 133 895 874 023 437 499 999 999 973 392 384 × 2 = 0 + 0.190 502 691 429 447 590 376 611 287 865 671 329 200 267 791 748 046 874 999 999 999 946 784 768;
  • 23) 0.190 502 691 429 447 590 376 611 287 865 671 329 200 267 791 748 046 874 999 999 999 946 784 768 × 2 = 0 + 0.381 005 382 858 895 180 753 222 575 731 342 658 400 535 583 496 093 749 999 999 999 893 569 536;
  • 24) 0.381 005 382 858 895 180 753 222 575 731 342 658 400 535 583 496 093 749 999 999 999 893 569 536 × 2 = 0 + 0.762 010 765 717 790 361 506 445 151 462 685 316 801 071 166 992 187 499 999 999 999 787 139 072;
  • 25) 0.762 010 765 717 790 361 506 445 151 462 685 316 801 071 166 992 187 499 999 999 999 787 139 072 × 2 = 1 + 0.524 021 531 435 580 723 012 890 302 925 370 633 602 142 333 984 374 999 999 999 999 574 278 144;
  • 26) 0.524 021 531 435 580 723 012 890 302 925 370 633 602 142 333 984 374 999 999 999 999 574 278 144 × 2 = 1 + 0.048 043 062 871 161 446 025 780 605 850 741 267 204 284 667 968 749 999 999 999 999 148 556 288;
  • 27) 0.048 043 062 871 161 446 025 780 605 850 741 267 204 284 667 968 749 999 999 999 999 148 556 288 × 2 = 0 + 0.096 086 125 742 322 892 051 561 211 701 482 534 408 569 335 937 499 999 999 999 998 297 112 576;
  • 28) 0.096 086 125 742 322 892 051 561 211 701 482 534 408 569 335 937 499 999 999 999 998 297 112 576 × 2 = 0 + 0.192 172 251 484 645 784 103 122 423 402 965 068 817 138 671 874 999 999 999 999 996 594 225 152;
  • 29) 0.192 172 251 484 645 784 103 122 423 402 965 068 817 138 671 874 999 999 999 999 996 594 225 152 × 2 = 0 + 0.384 344 502 969 291 568 206 244 846 805 930 137 634 277 343 749 999 999 999 999 993 188 450 304;
  • 30) 0.384 344 502 969 291 568 206 244 846 805 930 137 634 277 343 749 999 999 999 999 993 188 450 304 × 2 = 0 + 0.768 689 005 938 583 136 412 489 693 611 860 275 268 554 687 499 999 999 999 999 986 376 900 608;
  • 31) 0.768 689 005 938 583 136 412 489 693 611 860 275 268 554 687 499 999 999 999 999 986 376 900 608 × 2 = 1 + 0.537 378 011 877 166 272 824 979 387 223 720 550 537 109 374 999 999 999 999 999 972 753 801 216;
  • 32) 0.537 378 011 877 166 272 824 979 387 223 720 550 537 109 374 999 999 999 999 999 972 753 801 216 × 2 = 1 + 0.074 756 023 754 332 545 649 958 774 447 441 101 074 218 749 999 999 999 999 999 945 507 602 432;
  • 33) 0.074 756 023 754 332 545 649 958 774 447 441 101 074 218 749 999 999 999 999 999 945 507 602 432 × 2 = 0 + 0.149 512 047 508 665 091 299 917 548 894 882 202 148 437 499 999 999 999 999 999 891 015 204 864;
  • 34) 0.149 512 047 508 665 091 299 917 548 894 882 202 148 437 499 999 999 999 999 999 891 015 204 864 × 2 = 0 + 0.299 024 095 017 330 182 599 835 097 789 764 404 296 874 999 999 999 999 999 999 782 030 409 728;
  • 35) 0.299 024 095 017 330 182 599 835 097 789 764 404 296 874 999 999 999 999 999 999 782 030 409 728 × 2 = 0 + 0.598 048 190 034 660 365 199 670 195 579 528 808 593 749 999 999 999 999 999 999 564 060 819 456;
  • 36) 0.598 048 190 034 660 365 199 670 195 579 528 808 593 749 999 999 999 999 999 999 564 060 819 456 × 2 = 1 + 0.196 096 380 069 320 730 399 340 391 159 057 617 187 499 999 999 999 999 999 999 128 121 638 912;
  • 37) 0.196 096 380 069 320 730 399 340 391 159 057 617 187 499 999 999 999 999 999 999 128 121 638 912 × 2 = 0 + 0.392 192 760 138 641 460 798 680 782 318 115 234 374 999 999 999 999 999 999 998 256 243 277 824;
  • 38) 0.392 192 760 138 641 460 798 680 782 318 115 234 374 999 999 999 999 999 999 998 256 243 277 824 × 2 = 0 + 0.784 385 520 277 282 921 597 361 564 636 230 468 749 999 999 999 999 999 999 996 512 486 555 648;
  • 39) 0.784 385 520 277 282 921 597 361 564 636 230 468 749 999 999 999 999 999 999 996 512 486 555 648 × 2 = 1 + 0.568 771 040 554 565 843 194 723 129 272 460 937 499 999 999 999 999 999 999 993 024 973 111 296;
  • 40) 0.568 771 040 554 565 843 194 723 129 272 460 937 499 999 999 999 999 999 999 993 024 973 111 296 × 2 = 1 + 0.137 542 081 109 131 686 389 446 258 544 921 874 999 999 999 999 999 999 999 986 049 946 222 592;
  • 41) 0.137 542 081 109 131 686 389 446 258 544 921 874 999 999 999 999 999 999 999 986 049 946 222 592 × 2 = 0 + 0.275 084 162 218 263 372 778 892 517 089 843 749 999 999 999 999 999 999 999 972 099 892 445 184;
  • 42) 0.275 084 162 218 263 372 778 892 517 089 843 749 999 999 999 999 999 999 999 972 099 892 445 184 × 2 = 0 + 0.550 168 324 436 526 745 557 785 034 179 687 499 999 999 999 999 999 999 999 944 199 784 890 368;
  • 43) 0.550 168 324 436 526 745 557 785 034 179 687 499 999 999 999 999 999 999 999 944 199 784 890 368 × 2 = 1 + 0.100 336 648 873 053 491 115 570 068 359 374 999 999 999 999 999 999 999 999 888 399 569 780 736;
  • 44) 0.100 336 648 873 053 491 115 570 068 359 374 999 999 999 999 999 999 999 999 888 399 569 780 736 × 2 = 0 + 0.200 673 297 746 106 982 231 140 136 718 749 999 999 999 999 999 999 999 999 776 799 139 561 472;
  • 45) 0.200 673 297 746 106 982 231 140 136 718 749 999 999 999 999 999 999 999 999 776 799 139 561 472 × 2 = 0 + 0.401 346 595 492 213 964 462 280 273 437 499 999 999 999 999 999 999 999 999 553 598 279 122 944;
  • 46) 0.401 346 595 492 213 964 462 280 273 437 499 999 999 999 999 999 999 999 999 553 598 279 122 944 × 2 = 0 + 0.802 693 190 984 427 928 924 560 546 874 999 999 999 999 999 999 999 999 999 107 196 558 245 888;
  • 47) 0.802 693 190 984 427 928 924 560 546 874 999 999 999 999 999 999 999 999 999 107 196 558 245 888 × 2 = 1 + 0.605 386 381 968 855 857 849 121 093 749 999 999 999 999 999 999 999 999 998 214 393 116 491 776;
  • 48) 0.605 386 381 968 855 857 849 121 093 749 999 999 999 999 999 999 999 999 998 214 393 116 491 776 × 2 = 1 + 0.210 772 763 937 711 715 698 242 187 499 999 999 999 999 999 999 999 999 996 428 786 232 983 552;
  • 49) 0.210 772 763 937 711 715 698 242 187 499 999 999 999 999 999 999 999 999 996 428 786 232 983 552 × 2 = 0 + 0.421 545 527 875 423 431 396 484 374 999 999 999 999 999 999 999 999 999 992 857 572 465 967 104;
  • 50) 0.421 545 527 875 423 431 396 484 374 999 999 999 999 999 999 999 999 999 992 857 572 465 967 104 × 2 = 0 + 0.843 091 055 750 846 862 792 968 749 999 999 999 999 999 999 999 999 999 985 715 144 931 934 208;
  • 51) 0.843 091 055 750 846 862 792 968 749 999 999 999 999 999 999 999 999 999 985 715 144 931 934 208 × 2 = 1 + 0.686 182 111 501 693 725 585 937 499 999 999 999 999 999 999 999 999 999 971 430 289 863 868 416;
  • 52) 0.686 182 111 501 693 725 585 937 499 999 999 999 999 999 999 999 999 999 971 430 289 863 868 416 × 2 = 1 + 0.372 364 223 003 387 451 171 874 999 999 999 999 999 999 999 999 999 999 942 860 579 727 736 832;
  • 53) 0.372 364 223 003 387 451 171 874 999 999 999 999 999 999 999 999 999 999 942 860 579 727 736 832 × 2 = 0 + 0.744 728 446 006 774 902 343 749 999 999 999 999 999 999 999 999 999 999 885 721 159 455 473 664;
  • 54) 0.744 728 446 006 774 902 343 749 999 999 999 999 999 999 999 999 999 999 885 721 159 455 473 664 × 2 = 1 + 0.489 456 892 013 549 804 687 499 999 999 999 999 999 999 999 999 999 999 771 442 318 910 947 328;
  • 55) 0.489 456 892 013 549 804 687 499 999 999 999 999 999 999 999 999 999 999 771 442 318 910 947 328 × 2 = 0 + 0.978 913 784 027 099 609 374 999 999 999 999 999 999 999 999 999 999 999 542 884 637 821 894 656;
  • 56) 0.978 913 784 027 099 609 374 999 999 999 999 999 999 999 999 999 999 999 542 884 637 821 894 656 × 2 = 1 + 0.957 827 568 054 199 218 749 999 999 999 999 999 999 999 999 999 999 999 085 769 275 643 789 312;
  • 57) 0.957 827 568 054 199 218 749 999 999 999 999 999 999 999 999 999 999 999 085 769 275 643 789 312 × 2 = 1 + 0.915 655 136 108 398 437 499 999 999 999 999 999 999 999 999 999 999 998 171 538 551 287 578 624;
  • 58) 0.915 655 136 108 398 437 499 999 999 999 999 999 999 999 999 999 999 998 171 538 551 287 578 624 × 2 = 1 + 0.831 310 272 216 796 874 999 999 999 999 999 999 999 999 999 999 999 996 343 077 102 575 157 248;
  • 59) 0.831 310 272 216 796 874 999 999 999 999 999 999 999 999 999 999 999 996 343 077 102 575 157 248 × 2 = 1 + 0.662 620 544 433 593 749 999 999 999 999 999 999 999 999 999 999 999 992 686 154 205 150 314 496;
  • 60) 0.662 620 544 433 593 749 999 999 999 999 999 999 999 999 999 999 999 992 686 154 205 150 314 496 × 2 = 1 + 0.325 241 088 867 187 499 999 999 999 999 999 999 999 999 999 999 999 985 372 308 410 300 628 992;
  • 61) 0.325 241 088 867 187 499 999 999 999 999 999 999 999 999 999 999 999 985 372 308 410 300 628 992 × 2 = 0 + 0.650 482 177 734 374 999 999 999 999 999 999 999 999 999 999 999 999 970 744 616 820 601 257 984;
  • 62) 0.650 482 177 734 374 999 999 999 999 999 999 999 999 999 999 999 999 970 744 616 820 601 257 984 × 2 = 1 + 0.300 964 355 468 749 999 999 999 999 999 999 999 999 999 999 999 999 941 489 233 641 202 515 968;
  • 63) 0.300 964 355 468 749 999 999 999 999 999 999 999 999 999 999 999 999 941 489 233 641 202 515 968 × 2 = 0 + 0.601 928 710 937 499 999 999 999 999 999 999 999 999 999 999 999 999 882 978 467 282 405 031 936;
  • 64) 0.601 928 710 937 499 999 999 999 999 999 999 999 999 999 999 999 999 882 978 467 282 405 031 936 × 2 = 1 + 0.203 857 421 874 999 999 999 999 999 999 999 999 999 999 999 999 999 765 956 934 564 810 063 872;
  • 65) 0.203 857 421 874 999 999 999 999 999 999 999 999 999 999 999 999 999 765 956 934 564 810 063 872 × 2 = 0 + 0.407 714 843 749 999 999 999 999 999 999 999 999 999 999 999 999 999 531 913 869 129 620 127 744;
  • 66) 0.407 714 843 749 999 999 999 999 999 999 999 999 999 999 999 999 999 531 913 869 129 620 127 744 × 2 = 0 + 0.815 429 687 499 999 999 999 999 999 999 999 999 999 999 999 999 999 063 827 738 259 240 255 488;
  • 67) 0.815 429 687 499 999 999 999 999 999 999 999 999 999 999 999 999 999 063 827 738 259 240 255 488 × 2 = 1 + 0.630 859 374 999 999 999 999 999 999 999 999 999 999 999 999 999 998 127 655 476 518 480 510 976;
  • 68) 0.630 859 374 999 999 999 999 999 999 999 999 999 999 999 999 999 998 127 655 476 518 480 510 976 × 2 = 1 + 0.261 718 749 999 999 999 999 999 999 999 999 999 999 999 999 999 996 255 310 953 036 961 021 952;
  • 69) 0.261 718 749 999 999 999 999 999 999 999 999 999 999 999 999 999 996 255 310 953 036 961 021 952 × 2 = 0 + 0.523 437 499 999 999 999 999 999 999 999 999 999 999 999 999 999 992 510 621 906 073 922 043 904;
  • 70) 0.523 437 499 999 999 999 999 999 999 999 999 999 999 999 999 999 992 510 621 906 073 922 043 904 × 2 = 1 + 0.046 874 999 999 999 999 999 999 999 999 999 999 999 999 999 999 985 021 243 812 147 844 087 808;
  • 71) 0.046 874 999 999 999 999 999 999 999 999 999 999 999 999 999 999 985 021 243 812 147 844 087 808 × 2 = 0 + 0.093 749 999 999 999 999 999 999 999 999 999 999 999 999 999 999 970 042 487 624 295 688 175 616;
  • 72) 0.093 749 999 999 999 999 999 999 999 999 999 999 999 999 999 999 970 042 487 624 295 688 175 616 × 2 = 0 + 0.187 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 940 084 975 248 591 376 351 232;
  • 73) 0.187 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 940 084 975 248 591 376 351 232 × 2 = 0 + 0.374 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 880 169 950 497 182 752 702 464;
  • 74) 0.374 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 880 169 950 497 182 752 702 464 × 2 = 0 + 0.749 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 760 339 900 994 365 505 404 928;
  • 75) 0.749 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 760 339 900 994 365 505 404 928 × 2 = 1 + 0.499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 520 679 801 988 731 010 809 856;
  • 76) 0.499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 520 679 801 988 731 010 809 856 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 041 359 603 977 462 021 619 712;
  • 77) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 041 359 603 977 462 021 619 712 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 082 719 207 954 924 043 239 424;

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).


5. 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:


0.000 000 045 419 381 005 632 302 850 869 008 985 917 885 112 826 297 699 939 459 562 301 635 729 5(10) =

0.0000 0000 0000 0000 0000 0000 1100 0011 0001 0011 0010 0011 0011 0101 1111 0101 0011 0100 0010 1(2)

6. Positive number before normalization:

0.000 000 045 419 381 005 632 302 850 869 008 985 917 885 112 826 297 699 939 459 562 301 635 729 5(10) =

0.0000 0000 0000 0000 0000 0000 1100 0011 0001 0011 0010 0011 0011 0101 1111 0101 0011 0100 0010 1(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 25 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 045 419 381 005 632 302 850 869 008 985 917 885 112 826 297 699 939 459 562 301 635 729 5(10) =

0.0000 0000 0000 0000 0000 0000 1100 0011 0001 0011 0010 0011 0011 0101 1111 0101 0011 0100 0010 1(2) =

0.0000 0000 0000 0000 0000 0000 1100 0011 0001 0011 0010 0011 0011 0101 1111 0101 0011 0100 0010 1(2) × 20 =

1.1000 0110 0010 0110 0100 0110 0110 1011 1110 1010 0110 1000 0101(2) × 2-25


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): -25

Mantissa (not normalized):
1.1000 0110 0010 0110 0100 0110 0110 1011 1110 1010 0110 1000 0101


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-25 + 2(11-1) - 1 =

(-25 + 1 023)(10) =

998(10)


10. 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;
  • 998 ÷ 2 = 499 + 0;
  • 499 ÷ 2 = 249 + 1;
  • 249 ÷ 2 = 124 + 1;
  • 124 ÷ 2 = 62 + 0;
  • 62 ÷ 2 = 31 + 0;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

11. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

998(10) =

011 1110 0110(2)


12. 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. 1000 0110 0010 0110 0100 0110 0110 1011 1110 1010 0110 1000 0101 =

1000 0110 0010 0110 0100 0110 0110 1011 1110 1010 0110 1000 0101


13. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
1 (a negative number)

Exponent (11 bits) =
011 1110 0110

Mantissa (52 bits) =
1000 0110 0010 0110 0100 0110 0110 1011 1110 1010 0110 1000 0101


Decimal number -0.000 000 045 419 381 005 632 302 850 869 008 985 917 885 112 826 297 699 939 459 562 301 635 729 5 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1110 0110 - 1000 0110 0010 0110 0100 0110 0110 1011 1110 1010 0110 1000 0101

» Convert decimal -0.000 000 045 419 381 005 632 302 850 869 008 985 917 885 112 826 297 699 939 459 562 301 635 737 5 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