0.000 000 000 000 000 000 054 210 108 624 275 61 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 054 210 108 624 275 61(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
0.000 000 000 000 000 000 054 210 108 624 275 61(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: 0.000 000 000 000 000 000 054 210 108 624 275 61.

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) 0.000 000 000 000 000 000 054 210 108 624 275 61 × 2 = 0 + 0.000 000 000 000 000 000 108 420 217 248 551 22;
  • 2) 0.000 000 000 000 000 000 108 420 217 248 551 22 × 2 = 0 + 0.000 000 000 000 000 000 216 840 434 497 102 44;
  • 3) 0.000 000 000 000 000 000 216 840 434 497 102 44 × 2 = 0 + 0.000 000 000 000 000 000 433 680 868 994 204 88;
  • 4) 0.000 000 000 000 000 000 433 680 868 994 204 88 × 2 = 0 + 0.000 000 000 000 000 000 867 361 737 988 409 76;
  • 5) 0.000 000 000 000 000 000 867 361 737 988 409 76 × 2 = 0 + 0.000 000 000 000 000 001 734 723 475 976 819 52;
  • 6) 0.000 000 000 000 000 001 734 723 475 976 819 52 × 2 = 0 + 0.000 000 000 000 000 003 469 446 951 953 639 04;
  • 7) 0.000 000 000 000 000 003 469 446 951 953 639 04 × 2 = 0 + 0.000 000 000 000 000 006 938 893 903 907 278 08;
  • 8) 0.000 000 000 000 000 006 938 893 903 907 278 08 × 2 = 0 + 0.000 000 000 000 000 013 877 787 807 814 556 16;
  • 9) 0.000 000 000 000 000 013 877 787 807 814 556 16 × 2 = 0 + 0.000 000 000 000 000 027 755 575 615 629 112 32;
  • 10) 0.000 000 000 000 000 027 755 575 615 629 112 32 × 2 = 0 + 0.000 000 000 000 000 055 511 151 231 258 224 64;
  • 11) 0.000 000 000 000 000 055 511 151 231 258 224 64 × 2 = 0 + 0.000 000 000 000 000 111 022 302 462 516 449 28;
  • 12) 0.000 000 000 000 000 111 022 302 462 516 449 28 × 2 = 0 + 0.000 000 000 000 000 222 044 604 925 032 898 56;
  • 13) 0.000 000 000 000 000 222 044 604 925 032 898 56 × 2 = 0 + 0.000 000 000 000 000 444 089 209 850 065 797 12;
  • 14) 0.000 000 000 000 000 444 089 209 850 065 797 12 × 2 = 0 + 0.000 000 000 000 000 888 178 419 700 131 594 24;
  • 15) 0.000 000 000 000 000 888 178 419 700 131 594 24 × 2 = 0 + 0.000 000 000 000 001 776 356 839 400 263 188 48;
  • 16) 0.000 000 000 000 001 776 356 839 400 263 188 48 × 2 = 0 + 0.000 000 000 000 003 552 713 678 800 526 376 96;
  • 17) 0.000 000 000 000 003 552 713 678 800 526 376 96 × 2 = 0 + 0.000 000 000 000 007 105 427 357 601 052 753 92;
  • 18) 0.000 000 000 000 007 105 427 357 601 052 753 92 × 2 = 0 + 0.000 000 000 000 014 210 854 715 202 105 507 84;
  • 19) 0.000 000 000 000 014 210 854 715 202 105 507 84 × 2 = 0 + 0.000 000 000 000 028 421 709 430 404 211 015 68;
  • 20) 0.000 000 000 000 028 421 709 430 404 211 015 68 × 2 = 0 + 0.000 000 000 000 056 843 418 860 808 422 031 36;
  • 21) 0.000 000 000 000 056 843 418 860 808 422 031 36 × 2 = 0 + 0.000 000 000 000 113 686 837 721 616 844 062 72;
  • 22) 0.000 000 000 000 113 686 837 721 616 844 062 72 × 2 = 0 + 0.000 000 000 000 227 373 675 443 233 688 125 44;
  • 23) 0.000 000 000 000 227 373 675 443 233 688 125 44 × 2 = 0 + 0.000 000 000 000 454 747 350 886 467 376 250 88;
  • 24) 0.000 000 000 000 454 747 350 886 467 376 250 88 × 2 = 0 + 0.000 000 000 000 909 494 701 772 934 752 501 76;
  • 25) 0.000 000 000 000 909 494 701 772 934 752 501 76 × 2 = 0 + 0.000 000 000 001 818 989 403 545 869 505 003 52;
  • 26) 0.000 000 000 001 818 989 403 545 869 505 003 52 × 2 = 0 + 0.000 000 000 003 637 978 807 091 739 010 007 04;
  • 27) 0.000 000 000 003 637 978 807 091 739 010 007 04 × 2 = 0 + 0.000 000 000 007 275 957 614 183 478 020 014 08;
  • 28) 0.000 000 000 007 275 957 614 183 478 020 014 08 × 2 = 0 + 0.000 000 000 014 551 915 228 366 956 040 028 16;
  • 29) 0.000 000 000 014 551 915 228 366 956 040 028 16 × 2 = 0 + 0.000 000 000 029 103 830 456 733 912 080 056 32;
  • 30) 0.000 000 000 029 103 830 456 733 912 080 056 32 × 2 = 0 + 0.000 000 000 058 207 660 913 467 824 160 112 64;
  • 31) 0.000 000 000 058 207 660 913 467 824 160 112 64 × 2 = 0 + 0.000 000 000 116 415 321 826 935 648 320 225 28;
  • 32) 0.000 000 000 116 415 321 826 935 648 320 225 28 × 2 = 0 + 0.000 000 000 232 830 643 653 871 296 640 450 56;
  • 33) 0.000 000 000 232 830 643 653 871 296 640 450 56 × 2 = 0 + 0.000 000 000 465 661 287 307 742 593 280 901 12;
  • 34) 0.000 000 000 465 661 287 307 742 593 280 901 12 × 2 = 0 + 0.000 000 000 931 322 574 615 485 186 561 802 24;
  • 35) 0.000 000 000 931 322 574 615 485 186 561 802 24 × 2 = 0 + 0.000 000 001 862 645 149 230 970 373 123 604 48;
  • 36) 0.000 000 001 862 645 149 230 970 373 123 604 48 × 2 = 0 + 0.000 000 003 725 290 298 461 940 746 247 208 96;
  • 37) 0.000 000 003 725 290 298 461 940 746 247 208 96 × 2 = 0 + 0.000 000 007 450 580 596 923 881 492 494 417 92;
  • 38) 0.000 000 007 450 580 596 923 881 492 494 417 92 × 2 = 0 + 0.000 000 014 901 161 193 847 762 984 988 835 84;
  • 39) 0.000 000 014 901 161 193 847 762 984 988 835 84 × 2 = 0 + 0.000 000 029 802 322 387 695 525 969 977 671 68;
  • 40) 0.000 000 029 802 322 387 695 525 969 977 671 68 × 2 = 0 + 0.000 000 059 604 644 775 391 051 939 955 343 36;
  • 41) 0.000 000 059 604 644 775 391 051 939 955 343 36 × 2 = 0 + 0.000 000 119 209 289 550 782 103 879 910 686 72;
  • 42) 0.000 000 119 209 289 550 782 103 879 910 686 72 × 2 = 0 + 0.000 000 238 418 579 101 564 207 759 821 373 44;
  • 43) 0.000 000 238 418 579 101 564 207 759 821 373 44 × 2 = 0 + 0.000 000 476 837 158 203 128 415 519 642 746 88;
  • 44) 0.000 000 476 837 158 203 128 415 519 642 746 88 × 2 = 0 + 0.000 000 953 674 316 406 256 831 039 285 493 76;
  • 45) 0.000 000 953 674 316 406 256 831 039 285 493 76 × 2 = 0 + 0.000 001 907 348 632 812 513 662 078 570 987 52;
  • 46) 0.000 001 907 348 632 812 513 662 078 570 987 52 × 2 = 0 + 0.000 003 814 697 265 625 027 324 157 141 975 04;
  • 47) 0.000 003 814 697 265 625 027 324 157 141 975 04 × 2 = 0 + 0.000 007 629 394 531 250 054 648 314 283 950 08;
  • 48) 0.000 007 629 394 531 250 054 648 314 283 950 08 × 2 = 0 + 0.000 015 258 789 062 500 109 296 628 567 900 16;
  • 49) 0.000 015 258 789 062 500 109 296 628 567 900 16 × 2 = 0 + 0.000 030 517 578 125 000 218 593 257 135 800 32;
  • 50) 0.000 030 517 578 125 000 218 593 257 135 800 32 × 2 = 0 + 0.000 061 035 156 250 000 437 186 514 271 600 64;
  • 51) 0.000 061 035 156 250 000 437 186 514 271 600 64 × 2 = 0 + 0.000 122 070 312 500 000 874 373 028 543 201 28;
  • 52) 0.000 122 070 312 500 000 874 373 028 543 201 28 × 2 = 0 + 0.000 244 140 625 000 001 748 746 057 086 402 56;
  • 53) 0.000 244 140 625 000 001 748 746 057 086 402 56 × 2 = 0 + 0.000 488 281 250 000 003 497 492 114 172 805 12;
  • 54) 0.000 488 281 250 000 003 497 492 114 172 805 12 × 2 = 0 + 0.000 976 562 500 000 006 994 984 228 345 610 24;
  • 55) 0.000 976 562 500 000 006 994 984 228 345 610 24 × 2 = 0 + 0.001 953 125 000 000 013 989 968 456 691 220 48;
  • 56) 0.001 953 125 000 000 013 989 968 456 691 220 48 × 2 = 0 + 0.003 906 250 000 000 027 979 936 913 382 440 96;
  • 57) 0.003 906 250 000 000 027 979 936 913 382 440 96 × 2 = 0 + 0.007 812 500 000 000 055 959 873 826 764 881 92;
  • 58) 0.007 812 500 000 000 055 959 873 826 764 881 92 × 2 = 0 + 0.015 625 000 000 000 111 919 747 653 529 763 84;
  • 59) 0.015 625 000 000 000 111 919 747 653 529 763 84 × 2 = 0 + 0.031 250 000 000 000 223 839 495 307 059 527 68;
  • 60) 0.031 250 000 000 000 223 839 495 307 059 527 68 × 2 = 0 + 0.062 500 000 000 000 447 678 990 614 119 055 36;
  • 61) 0.062 500 000 000 000 447 678 990 614 119 055 36 × 2 = 0 + 0.125 000 000 000 000 895 357 981 228 238 110 72;
  • 62) 0.125 000 000 000 000 895 357 981 228 238 110 72 × 2 = 0 + 0.250 000 000 000 001 790 715 962 456 476 221 44;
  • 63) 0.250 000 000 000 001 790 715 962 456 476 221 44 × 2 = 0 + 0.500 000 000 000 003 581 431 924 912 952 442 88;
  • 64) 0.500 000 000 000 003 581 431 924 912 952 442 88 × 2 = 1 + 0.000 000 000 000 007 162 863 849 825 904 885 76;
  • 65) 0.000 000 000 000 007 162 863 849 825 904 885 76 × 2 = 0 + 0.000 000 000 000 014 325 727 699 651 809 771 52;
  • 66) 0.000 000 000 000 014 325 727 699 651 809 771 52 × 2 = 0 + 0.000 000 000 000 028 651 455 399 303 619 543 04;
  • 67) 0.000 000 000 000 028 651 455 399 303 619 543 04 × 2 = 0 + 0.000 000 000 000 057 302 910 798 607 239 086 08;
  • 68) 0.000 000 000 000 057 302 910 798 607 239 086 08 × 2 = 0 + 0.000 000 000 000 114 605 821 597 214 478 172 16;
  • 69) 0.000 000 000 000 114 605 821 597 214 478 172 16 × 2 = 0 + 0.000 000 000 000 229 211 643 194 428 956 344 32;
  • 70) 0.000 000 000 000 229 211 643 194 428 956 344 32 × 2 = 0 + 0.000 000 000 000 458 423 286 388 857 912 688 64;
  • 71) 0.000 000 000 000 458 423 286 388 857 912 688 64 × 2 = 0 + 0.000 000 000 000 916 846 572 777 715 825 377 28;
  • 72) 0.000 000 000 000 916 846 572 777 715 825 377 28 × 2 = 0 + 0.000 000 000 001 833 693 145 555 431 650 754 56;
  • 73) 0.000 000 000 001 833 693 145 555 431 650 754 56 × 2 = 0 + 0.000 000 000 003 667 386 291 110 863 301 509 12;
  • 74) 0.000 000 000 003 667 386 291 110 863 301 509 12 × 2 = 0 + 0.000 000 000 007 334 772 582 221 726 603 018 24;
  • 75) 0.000 000 000 007 334 772 582 221 726 603 018 24 × 2 = 0 + 0.000 000 000 014 669 545 164 443 453 206 036 48;
  • 76) 0.000 000 000 014 669 545 164 443 453 206 036 48 × 2 = 0 + 0.000 000 000 029 339 090 328 886 906 412 072 96;
  • 77) 0.000 000 000 029 339 090 328 886 906 412 072 96 × 2 = 0 + 0.000 000 000 058 678 180 657 773 812 824 145 92;
  • 78) 0.000 000 000 058 678 180 657 773 812 824 145 92 × 2 = 0 + 0.000 000 000 117 356 361 315 547 625 648 291 84;
  • 79) 0.000 000 000 117 356 361 315 547 625 648 291 84 × 2 = 0 + 0.000 000 000 234 712 722 631 095 251 296 583 68;
  • 80) 0.000 000 000 234 712 722 631 095 251 296 583 68 × 2 = 0 + 0.000 000 000 469 425 445 262 190 502 593 167 36;
  • 81) 0.000 000 000 469 425 445 262 190 502 593 167 36 × 2 = 0 + 0.000 000 000 938 850 890 524 381 005 186 334 72;
  • 82) 0.000 000 000 938 850 890 524 381 005 186 334 72 × 2 = 0 + 0.000 000 001 877 701 781 048 762 010 372 669 44;
  • 83) 0.000 000 001 877 701 781 048 762 010 372 669 44 × 2 = 0 + 0.000 000 003 755 403 562 097 524 020 745 338 88;
  • 84) 0.000 000 003 755 403 562 097 524 020 745 338 88 × 2 = 0 + 0.000 000 007 510 807 124 195 048 041 490 677 76;
  • 85) 0.000 000 007 510 807 124 195 048 041 490 677 76 × 2 = 0 + 0.000 000 015 021 614 248 390 096 082 981 355 52;
  • 86) 0.000 000 015 021 614 248 390 096 082 981 355 52 × 2 = 0 + 0.000 000 030 043 228 496 780 192 165 962 711 04;
  • 87) 0.000 000 030 043 228 496 780 192 165 962 711 04 × 2 = 0 + 0.000 000 060 086 456 993 560 384 331 925 422 08;
  • 88) 0.000 000 060 086 456 993 560 384 331 925 422 08 × 2 = 0 + 0.000 000 120 172 913 987 120 768 663 850 844 16;
  • 89) 0.000 000 120 172 913 987 120 768 663 850 844 16 × 2 = 0 + 0.000 000 240 345 827 974 241 537 327 701 688 32;
  • 90) 0.000 000 240 345 827 974 241 537 327 701 688 32 × 2 = 0 + 0.000 000 480 691 655 948 483 074 655 403 376 64;
  • 91) 0.000 000 480 691 655 948 483 074 655 403 376 64 × 2 = 0 + 0.000 000 961 383 311 896 966 149 310 806 753 28;
  • 92) 0.000 000 961 383 311 896 966 149 310 806 753 28 × 2 = 0 + 0.000 001 922 766 623 793 932 298 621 613 506 56;
  • 93) 0.000 001 922 766 623 793 932 298 621 613 506 56 × 2 = 0 + 0.000 003 845 533 247 587 864 597 243 227 013 12;
  • 94) 0.000 003 845 533 247 587 864 597 243 227 013 12 × 2 = 0 + 0.000 007 691 066 495 175 729 194 486 454 026 24;
  • 95) 0.000 007 691 066 495 175 729 194 486 454 026 24 × 2 = 0 + 0.000 015 382 132 990 351 458 388 972 908 052 48;
  • 96) 0.000 015 382 132 990 351 458 388 972 908 052 48 × 2 = 0 + 0.000 030 764 265 980 702 916 777 945 816 104 96;
  • 97) 0.000 030 764 265 980 702 916 777 945 816 104 96 × 2 = 0 + 0.000 061 528 531 961 405 833 555 891 632 209 92;
  • 98) 0.000 061 528 531 961 405 833 555 891 632 209 92 × 2 = 0 + 0.000 123 057 063 922 811 667 111 783 264 419 84;
  • 99) 0.000 123 057 063 922 811 667 111 783 264 419 84 × 2 = 0 + 0.000 246 114 127 845 623 334 223 566 528 839 68;
  • 100) 0.000 246 114 127 845 623 334 223 566 528 839 68 × 2 = 0 + 0.000 492 228 255 691 246 668 447 133 057 679 36;
  • 101) 0.000 492 228 255 691 246 668 447 133 057 679 36 × 2 = 0 + 0.000 984 456 511 382 493 336 894 266 115 358 72;
  • 102) 0.000 984 456 511 382 493 336 894 266 115 358 72 × 2 = 0 + 0.001 968 913 022 764 986 673 788 532 230 717 44;
  • 103) 0.001 968 913 022 764 986 673 788 532 230 717 44 × 2 = 0 + 0.003 937 826 045 529 973 347 577 064 461 434 88;
  • 104) 0.003 937 826 045 529 973 347 577 064 461 434 88 × 2 = 0 + 0.007 875 652 091 059 946 695 154 128 922 869 76;
  • 105) 0.007 875 652 091 059 946 695 154 128 922 869 76 × 2 = 0 + 0.015 751 304 182 119 893 390 308 257 845 739 52;
  • 106) 0.015 751 304 182 119 893 390 308 257 845 739 52 × 2 = 0 + 0.031 502 608 364 239 786 780 616 515 691 479 04;
  • 107) 0.031 502 608 364 239 786 780 616 515 691 479 04 × 2 = 0 + 0.063 005 216 728 479 573 561 233 031 382 958 08;
  • 108) 0.063 005 216 728 479 573 561 233 031 382 958 08 × 2 = 0 + 0.126 010 433 456 959 147 122 466 062 765 916 16;
  • 109) 0.126 010 433 456 959 147 122 466 062 765 916 16 × 2 = 0 + 0.252 020 866 913 918 294 244 932 125 531 832 32;
  • 110) 0.252 020 866 913 918 294 244 932 125 531 832 32 × 2 = 0 + 0.504 041 733 827 836 588 489 864 251 063 664 64;
  • 111) 0.504 041 733 827 836 588 489 864 251 063 664 64 × 2 = 1 + 0.008 083 467 655 673 176 979 728 502 127 329 28;
  • 112) 0.008 083 467 655 673 176 979 728 502 127 329 28 × 2 = 0 + 0.016 166 935 311 346 353 959 457 004 254 658 56;
  • 113) 0.016 166 935 311 346 353 959 457 004 254 658 56 × 2 = 0 + 0.032 333 870 622 692 707 918 914 008 509 317 12;
  • 114) 0.032 333 870 622 692 707 918 914 008 509 317 12 × 2 = 0 + 0.064 667 741 245 385 415 837 828 017 018 634 24;
  • 115) 0.064 667 741 245 385 415 837 828 017 018 634 24 × 2 = 0 + 0.129 335 482 490 770 831 675 656 034 037 268 48;
  • 116) 0.129 335 482 490 770 831 675 656 034 037 268 48 × 2 = 0 + 0.258 670 964 981 541 663 351 312 068 074 536 96;

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:


0.000 000 000 000 000 000 054 210 108 624 275 61(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0000(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 054 210 108 624 275 61(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0000(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 64 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 054 210 108 624 275 61(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0000(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0000(2) × 20 =

1.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0000(2) × 2-64


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): -64

Mantissa (not normalized):
1.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0000


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-64 + 2(11-1) - 1 =

(-64 + 1 023)(10) =

959(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;
  • 959 ÷ 2 = 479 + 1;
  • 479 ÷ 2 = 239 + 1;
  • 239 ÷ 2 = 119 + 1;
  • 119 ÷ 2 = 59 + 1;
  • 59 ÷ 2 = 29 + 1;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 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) =

959(10) =

011 1011 1111(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. 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0000 =

0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 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 1011 1111

Mantissa (52 bits) =
0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0000


Decimal number 0.000 000 000 000 000 000 054 210 108 624 275 61 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1111 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 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