0.000 000 000 000 000 012 345 687 894 564 589 438 731 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 012 345 687 894 564 589 438 731 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
0.000 000 000 000 000 012 345 687 894 564 589 438 731 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. 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: 0.000 000 000 000 000 012 345 687 894 564 589 438 731 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) 0.000 000 000 000 000 012 345 687 894 564 589 438 731 5 × 2 = 0 + 0.000 000 000 000 000 024 691 375 789 129 178 877 463;
  • 2) 0.000 000 000 000 000 024 691 375 789 129 178 877 463 × 2 = 0 + 0.000 000 000 000 000 049 382 751 578 258 357 754 926;
  • 3) 0.000 000 000 000 000 049 382 751 578 258 357 754 926 × 2 = 0 + 0.000 000 000 000 000 098 765 503 156 516 715 509 852;
  • 4) 0.000 000 000 000 000 098 765 503 156 516 715 509 852 × 2 = 0 + 0.000 000 000 000 000 197 531 006 313 033 431 019 704;
  • 5) 0.000 000 000 000 000 197 531 006 313 033 431 019 704 × 2 = 0 + 0.000 000 000 000 000 395 062 012 626 066 862 039 408;
  • 6) 0.000 000 000 000 000 395 062 012 626 066 862 039 408 × 2 = 0 + 0.000 000 000 000 000 790 124 025 252 133 724 078 816;
  • 7) 0.000 000 000 000 000 790 124 025 252 133 724 078 816 × 2 = 0 + 0.000 000 000 000 001 580 248 050 504 267 448 157 632;
  • 8) 0.000 000 000 000 001 580 248 050 504 267 448 157 632 × 2 = 0 + 0.000 000 000 000 003 160 496 101 008 534 896 315 264;
  • 9) 0.000 000 000 000 003 160 496 101 008 534 896 315 264 × 2 = 0 + 0.000 000 000 000 006 320 992 202 017 069 792 630 528;
  • 10) 0.000 000 000 000 006 320 992 202 017 069 792 630 528 × 2 = 0 + 0.000 000 000 000 012 641 984 404 034 139 585 261 056;
  • 11) 0.000 000 000 000 012 641 984 404 034 139 585 261 056 × 2 = 0 + 0.000 000 000 000 025 283 968 808 068 279 170 522 112;
  • 12) 0.000 000 000 000 025 283 968 808 068 279 170 522 112 × 2 = 0 + 0.000 000 000 000 050 567 937 616 136 558 341 044 224;
  • 13) 0.000 000 000 000 050 567 937 616 136 558 341 044 224 × 2 = 0 + 0.000 000 000 000 101 135 875 232 273 116 682 088 448;
  • 14) 0.000 000 000 000 101 135 875 232 273 116 682 088 448 × 2 = 0 + 0.000 000 000 000 202 271 750 464 546 233 364 176 896;
  • 15) 0.000 000 000 000 202 271 750 464 546 233 364 176 896 × 2 = 0 + 0.000 000 000 000 404 543 500 929 092 466 728 353 792;
  • 16) 0.000 000 000 000 404 543 500 929 092 466 728 353 792 × 2 = 0 + 0.000 000 000 000 809 087 001 858 184 933 456 707 584;
  • 17) 0.000 000 000 000 809 087 001 858 184 933 456 707 584 × 2 = 0 + 0.000 000 000 001 618 174 003 716 369 866 913 415 168;
  • 18) 0.000 000 000 001 618 174 003 716 369 866 913 415 168 × 2 = 0 + 0.000 000 000 003 236 348 007 432 739 733 826 830 336;
  • 19) 0.000 000 000 003 236 348 007 432 739 733 826 830 336 × 2 = 0 + 0.000 000 000 006 472 696 014 865 479 467 653 660 672;
  • 20) 0.000 000 000 006 472 696 014 865 479 467 653 660 672 × 2 = 0 + 0.000 000 000 012 945 392 029 730 958 935 307 321 344;
  • 21) 0.000 000 000 012 945 392 029 730 958 935 307 321 344 × 2 = 0 + 0.000 000 000 025 890 784 059 461 917 870 614 642 688;
  • 22) 0.000 000 000 025 890 784 059 461 917 870 614 642 688 × 2 = 0 + 0.000 000 000 051 781 568 118 923 835 741 229 285 376;
  • 23) 0.000 000 000 051 781 568 118 923 835 741 229 285 376 × 2 = 0 + 0.000 000 000 103 563 136 237 847 671 482 458 570 752;
  • 24) 0.000 000 000 103 563 136 237 847 671 482 458 570 752 × 2 = 0 + 0.000 000 000 207 126 272 475 695 342 964 917 141 504;
  • 25) 0.000 000 000 207 126 272 475 695 342 964 917 141 504 × 2 = 0 + 0.000 000 000 414 252 544 951 390 685 929 834 283 008;
  • 26) 0.000 000 000 414 252 544 951 390 685 929 834 283 008 × 2 = 0 + 0.000 000 000 828 505 089 902 781 371 859 668 566 016;
  • 27) 0.000 000 000 828 505 089 902 781 371 859 668 566 016 × 2 = 0 + 0.000 000 001 657 010 179 805 562 743 719 337 132 032;
  • 28) 0.000 000 001 657 010 179 805 562 743 719 337 132 032 × 2 = 0 + 0.000 000 003 314 020 359 611 125 487 438 674 264 064;
  • 29) 0.000 000 003 314 020 359 611 125 487 438 674 264 064 × 2 = 0 + 0.000 000 006 628 040 719 222 250 974 877 348 528 128;
  • 30) 0.000 000 006 628 040 719 222 250 974 877 348 528 128 × 2 = 0 + 0.000 000 013 256 081 438 444 501 949 754 697 056 256;
  • 31) 0.000 000 013 256 081 438 444 501 949 754 697 056 256 × 2 = 0 + 0.000 000 026 512 162 876 889 003 899 509 394 112 512;
  • 32) 0.000 000 026 512 162 876 889 003 899 509 394 112 512 × 2 = 0 + 0.000 000 053 024 325 753 778 007 799 018 788 225 024;
  • 33) 0.000 000 053 024 325 753 778 007 799 018 788 225 024 × 2 = 0 + 0.000 000 106 048 651 507 556 015 598 037 576 450 048;
  • 34) 0.000 000 106 048 651 507 556 015 598 037 576 450 048 × 2 = 0 + 0.000 000 212 097 303 015 112 031 196 075 152 900 096;
  • 35) 0.000 000 212 097 303 015 112 031 196 075 152 900 096 × 2 = 0 + 0.000 000 424 194 606 030 224 062 392 150 305 800 192;
  • 36) 0.000 000 424 194 606 030 224 062 392 150 305 800 192 × 2 = 0 + 0.000 000 848 389 212 060 448 124 784 300 611 600 384;
  • 37) 0.000 000 848 389 212 060 448 124 784 300 611 600 384 × 2 = 0 + 0.000 001 696 778 424 120 896 249 568 601 223 200 768;
  • 38) 0.000 001 696 778 424 120 896 249 568 601 223 200 768 × 2 = 0 + 0.000 003 393 556 848 241 792 499 137 202 446 401 536;
  • 39) 0.000 003 393 556 848 241 792 499 137 202 446 401 536 × 2 = 0 + 0.000 006 787 113 696 483 584 998 274 404 892 803 072;
  • 40) 0.000 006 787 113 696 483 584 998 274 404 892 803 072 × 2 = 0 + 0.000 013 574 227 392 967 169 996 548 809 785 606 144;
  • 41) 0.000 013 574 227 392 967 169 996 548 809 785 606 144 × 2 = 0 + 0.000 027 148 454 785 934 339 993 097 619 571 212 288;
  • 42) 0.000 027 148 454 785 934 339 993 097 619 571 212 288 × 2 = 0 + 0.000 054 296 909 571 868 679 986 195 239 142 424 576;
  • 43) 0.000 054 296 909 571 868 679 986 195 239 142 424 576 × 2 = 0 + 0.000 108 593 819 143 737 359 972 390 478 284 849 152;
  • 44) 0.000 108 593 819 143 737 359 972 390 478 284 849 152 × 2 = 0 + 0.000 217 187 638 287 474 719 944 780 956 569 698 304;
  • 45) 0.000 217 187 638 287 474 719 944 780 956 569 698 304 × 2 = 0 + 0.000 434 375 276 574 949 439 889 561 913 139 396 608;
  • 46) 0.000 434 375 276 574 949 439 889 561 913 139 396 608 × 2 = 0 + 0.000 868 750 553 149 898 879 779 123 826 278 793 216;
  • 47) 0.000 868 750 553 149 898 879 779 123 826 278 793 216 × 2 = 0 + 0.001 737 501 106 299 797 759 558 247 652 557 586 432;
  • 48) 0.001 737 501 106 299 797 759 558 247 652 557 586 432 × 2 = 0 + 0.003 475 002 212 599 595 519 116 495 305 115 172 864;
  • 49) 0.003 475 002 212 599 595 519 116 495 305 115 172 864 × 2 = 0 + 0.006 950 004 425 199 191 038 232 990 610 230 345 728;
  • 50) 0.006 950 004 425 199 191 038 232 990 610 230 345 728 × 2 = 0 + 0.013 900 008 850 398 382 076 465 981 220 460 691 456;
  • 51) 0.013 900 008 850 398 382 076 465 981 220 460 691 456 × 2 = 0 + 0.027 800 017 700 796 764 152 931 962 440 921 382 912;
  • 52) 0.027 800 017 700 796 764 152 931 962 440 921 382 912 × 2 = 0 + 0.055 600 035 401 593 528 305 863 924 881 842 765 824;
  • 53) 0.055 600 035 401 593 528 305 863 924 881 842 765 824 × 2 = 0 + 0.111 200 070 803 187 056 611 727 849 763 685 531 648;
  • 54) 0.111 200 070 803 187 056 611 727 849 763 685 531 648 × 2 = 0 + 0.222 400 141 606 374 113 223 455 699 527 371 063 296;
  • 55) 0.222 400 141 606 374 113 223 455 699 527 371 063 296 × 2 = 0 + 0.444 800 283 212 748 226 446 911 399 054 742 126 592;
  • 56) 0.444 800 283 212 748 226 446 911 399 054 742 126 592 × 2 = 0 + 0.889 600 566 425 496 452 893 822 798 109 484 253 184;
  • 57) 0.889 600 566 425 496 452 893 822 798 109 484 253 184 × 2 = 1 + 0.779 201 132 850 992 905 787 645 596 218 968 506 368;
  • 58) 0.779 201 132 850 992 905 787 645 596 218 968 506 368 × 2 = 1 + 0.558 402 265 701 985 811 575 291 192 437 937 012 736;
  • 59) 0.558 402 265 701 985 811 575 291 192 437 937 012 736 × 2 = 1 + 0.116 804 531 403 971 623 150 582 384 875 874 025 472;
  • 60) 0.116 804 531 403 971 623 150 582 384 875 874 025 472 × 2 = 0 + 0.233 609 062 807 943 246 301 164 769 751 748 050 944;
  • 61) 0.233 609 062 807 943 246 301 164 769 751 748 050 944 × 2 = 0 + 0.467 218 125 615 886 492 602 329 539 503 496 101 888;
  • 62) 0.467 218 125 615 886 492 602 329 539 503 496 101 888 × 2 = 0 + 0.934 436 251 231 772 985 204 659 079 006 992 203 776;
  • 63) 0.934 436 251 231 772 985 204 659 079 006 992 203 776 × 2 = 1 + 0.868 872 502 463 545 970 409 318 158 013 984 407 552;
  • 64) 0.868 872 502 463 545 970 409 318 158 013 984 407 552 × 2 = 1 + 0.737 745 004 927 091 940 818 636 316 027 968 815 104;
  • 65) 0.737 745 004 927 091 940 818 636 316 027 968 815 104 × 2 = 1 + 0.475 490 009 854 183 881 637 272 632 055 937 630 208;
  • 66) 0.475 490 009 854 183 881 637 272 632 055 937 630 208 × 2 = 0 + 0.950 980 019 708 367 763 274 545 264 111 875 260 416;
  • 67) 0.950 980 019 708 367 763 274 545 264 111 875 260 416 × 2 = 1 + 0.901 960 039 416 735 526 549 090 528 223 750 520 832;
  • 68) 0.901 960 039 416 735 526 549 090 528 223 750 520 832 × 2 = 1 + 0.803 920 078 833 471 053 098 181 056 447 501 041 664;
  • 69) 0.803 920 078 833 471 053 098 181 056 447 501 041 664 × 2 = 1 + 0.607 840 157 666 942 106 196 362 112 895 002 083 328;
  • 70) 0.607 840 157 666 942 106 196 362 112 895 002 083 328 × 2 = 1 + 0.215 680 315 333 884 212 392 724 225 790 004 166 656;
  • 71) 0.215 680 315 333 884 212 392 724 225 790 004 166 656 × 2 = 0 + 0.431 360 630 667 768 424 785 448 451 580 008 333 312;
  • 72) 0.431 360 630 667 768 424 785 448 451 580 008 333 312 × 2 = 0 + 0.862 721 261 335 536 849 570 896 903 160 016 666 624;
  • 73) 0.862 721 261 335 536 849 570 896 903 160 016 666 624 × 2 = 1 + 0.725 442 522 671 073 699 141 793 806 320 033 333 248;
  • 74) 0.725 442 522 671 073 699 141 793 806 320 033 333 248 × 2 = 1 + 0.450 885 045 342 147 398 283 587 612 640 066 666 496;
  • 75) 0.450 885 045 342 147 398 283 587 612 640 066 666 496 × 2 = 0 + 0.901 770 090 684 294 796 567 175 225 280 133 332 992;
  • 76) 0.901 770 090 684 294 796 567 175 225 280 133 332 992 × 2 = 1 + 0.803 540 181 368 589 593 134 350 450 560 266 665 984;
  • 77) 0.803 540 181 368 589 593 134 350 450 560 266 665 984 × 2 = 1 + 0.607 080 362 737 179 186 268 700 901 120 533 331 968;
  • 78) 0.607 080 362 737 179 186 268 700 901 120 533 331 968 × 2 = 1 + 0.214 160 725 474 358 372 537 401 802 241 066 663 936;
  • 79) 0.214 160 725 474 358 372 537 401 802 241 066 663 936 × 2 = 0 + 0.428 321 450 948 716 745 074 803 604 482 133 327 872;
  • 80) 0.428 321 450 948 716 745 074 803 604 482 133 327 872 × 2 = 0 + 0.856 642 901 897 433 490 149 607 208 964 266 655 744;
  • 81) 0.856 642 901 897 433 490 149 607 208 964 266 655 744 × 2 = 1 + 0.713 285 803 794 866 980 299 214 417 928 533 311 488;
  • 82) 0.713 285 803 794 866 980 299 214 417 928 533 311 488 × 2 = 1 + 0.426 571 607 589 733 960 598 428 835 857 066 622 976;
  • 83) 0.426 571 607 589 733 960 598 428 835 857 066 622 976 × 2 = 0 + 0.853 143 215 179 467 921 196 857 671 714 133 245 952;
  • 84) 0.853 143 215 179 467 921 196 857 671 714 133 245 952 × 2 = 1 + 0.706 286 430 358 935 842 393 715 343 428 266 491 904;
  • 85) 0.706 286 430 358 935 842 393 715 343 428 266 491 904 × 2 = 1 + 0.412 572 860 717 871 684 787 430 686 856 532 983 808;
  • 86) 0.412 572 860 717 871 684 787 430 686 856 532 983 808 × 2 = 0 + 0.825 145 721 435 743 369 574 861 373 713 065 967 616;
  • 87) 0.825 145 721 435 743 369 574 861 373 713 065 967 616 × 2 = 1 + 0.650 291 442 871 486 739 149 722 747 426 131 935 232;
  • 88) 0.650 291 442 871 486 739 149 722 747 426 131 935 232 × 2 = 1 + 0.300 582 885 742 973 478 299 445 494 852 263 870 464;
  • 89) 0.300 582 885 742 973 478 299 445 494 852 263 870 464 × 2 = 0 + 0.601 165 771 485 946 956 598 890 989 704 527 740 928;
  • 90) 0.601 165 771 485 946 956 598 890 989 704 527 740 928 × 2 = 1 + 0.202 331 542 971 893 913 197 781 979 409 055 481 856;
  • 91) 0.202 331 542 971 893 913 197 781 979 409 055 481 856 × 2 = 0 + 0.404 663 085 943 787 826 395 563 958 818 110 963 712;
  • 92) 0.404 663 085 943 787 826 395 563 958 818 110 963 712 × 2 = 0 + 0.809 326 171 887 575 652 791 127 917 636 221 927 424;
  • 93) 0.809 326 171 887 575 652 791 127 917 636 221 927 424 × 2 = 1 + 0.618 652 343 775 151 305 582 255 835 272 443 854 848;
  • 94) 0.618 652 343 775 151 305 582 255 835 272 443 854 848 × 2 = 1 + 0.237 304 687 550 302 611 164 511 670 544 887 709 696;
  • 95) 0.237 304 687 550 302 611 164 511 670 544 887 709 696 × 2 = 0 + 0.474 609 375 100 605 222 329 023 341 089 775 419 392;
  • 96) 0.474 609 375 100 605 222 329 023 341 089 775 419 392 × 2 = 0 + 0.949 218 750 201 210 444 658 046 682 179 550 838 784;
  • 97) 0.949 218 750 201 210 444 658 046 682 179 550 838 784 × 2 = 1 + 0.898 437 500 402 420 889 316 093 364 359 101 677 568;
  • 98) 0.898 437 500 402 420 889 316 093 364 359 101 677 568 × 2 = 1 + 0.796 875 000 804 841 778 632 186 728 718 203 355 136;
  • 99) 0.796 875 000 804 841 778 632 186 728 718 203 355 136 × 2 = 1 + 0.593 750 001 609 683 557 264 373 457 436 406 710 272;
  • 100) 0.593 750 001 609 683 557 264 373 457 436 406 710 272 × 2 = 1 + 0.187 500 003 219 367 114 528 746 914 872 813 420 544;
  • 101) 0.187 500 003 219 367 114 528 746 914 872 813 420 544 × 2 = 0 + 0.375 000 006 438 734 229 057 493 829 745 626 841 088;
  • 102) 0.375 000 006 438 734 229 057 493 829 745 626 841 088 × 2 = 0 + 0.750 000 012 877 468 458 114 987 659 491 253 682 176;
  • 103) 0.750 000 012 877 468 458 114 987 659 491 253 682 176 × 2 = 1 + 0.500 000 025 754 936 916 229 975 318 982 507 364 352;
  • 104) 0.500 000 025 754 936 916 229 975 318 982 507 364 352 × 2 = 1 + 0.000 000 051 509 873 832 459 950 637 965 014 728 704;
  • 105) 0.000 000 051 509 873 832 459 950 637 965 014 728 704 × 2 = 0 + 0.000 000 103 019 747 664 919 901 275 930 029 457 408;
  • 106) 0.000 000 103 019 747 664 919 901 275 930 029 457 408 × 2 = 0 + 0.000 000 206 039 495 329 839 802 551 860 058 914 816;
  • 107) 0.000 000 206 039 495 329 839 802 551 860 058 914 816 × 2 = 0 + 0.000 000 412 078 990 659 679 605 103 720 117 829 632;
  • 108) 0.000 000 412 078 990 659 679 605 103 720 117 829 632 × 2 = 0 + 0.000 000 824 157 981 319 359 210 207 440 235 659 264;
  • 109) 0.000 000 824 157 981 319 359 210 207 440 235 659 264 × 2 = 0 + 0.000 001 648 315 962 638 718 420 414 880 471 318 528;

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:


0.000 000 000 000 000 012 345 687 894 564 589 438 731 5(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 0011 0000 0(2)

5. Positive number before normalization:

0.000 000 000 000 000 012 345 687 894 564 589 438 731 5(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 0011 0000 0(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:


0.000 000 000 000 000 012 345 687 894 564 589 438 731 5(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 0011 0000 0(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 0011 0000 0(2) × 20 =

1.1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0110 0000(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 0110 0000


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 0110 0000 =

1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0110 0000


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 0110 0000


Decimal number 0.000 000 000 000 000 012 345 687 894 564 589 438 731 5 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 0110 0000

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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