0.000 000 000 000 000 000 054 210 108 624 274 38 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 054 210 108 624 274 38(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
0.000 000 000 000 000 000 054 210 108 624 274 38(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: 0.000 000 000 000 000 000 054 210 108 624 274 38.

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) 0.000 000 000 000 000 000 054 210 108 624 274 38 × 2 = 0 + 0.000 000 000 000 000 000 108 420 217 248 548 76;
  • 2) 0.000 000 000 000 000 000 108 420 217 248 548 76 × 2 = 0 + 0.000 000 000 000 000 000 216 840 434 497 097 52;
  • 3) 0.000 000 000 000 000 000 216 840 434 497 097 52 × 2 = 0 + 0.000 000 000 000 000 000 433 680 868 994 195 04;
  • 4) 0.000 000 000 000 000 000 433 680 868 994 195 04 × 2 = 0 + 0.000 000 000 000 000 000 867 361 737 988 390 08;
  • 5) 0.000 000 000 000 000 000 867 361 737 988 390 08 × 2 = 0 + 0.000 000 000 000 000 001 734 723 475 976 780 16;
  • 6) 0.000 000 000 000 000 001 734 723 475 976 780 16 × 2 = 0 + 0.000 000 000 000 000 003 469 446 951 953 560 32;
  • 7) 0.000 000 000 000 000 003 469 446 951 953 560 32 × 2 = 0 + 0.000 000 000 000 000 006 938 893 903 907 120 64;
  • 8) 0.000 000 000 000 000 006 938 893 903 907 120 64 × 2 = 0 + 0.000 000 000 000 000 013 877 787 807 814 241 28;
  • 9) 0.000 000 000 000 000 013 877 787 807 814 241 28 × 2 = 0 + 0.000 000 000 000 000 027 755 575 615 628 482 56;
  • 10) 0.000 000 000 000 000 027 755 575 615 628 482 56 × 2 = 0 + 0.000 000 000 000 000 055 511 151 231 256 965 12;
  • 11) 0.000 000 000 000 000 055 511 151 231 256 965 12 × 2 = 0 + 0.000 000 000 000 000 111 022 302 462 513 930 24;
  • 12) 0.000 000 000 000 000 111 022 302 462 513 930 24 × 2 = 0 + 0.000 000 000 000 000 222 044 604 925 027 860 48;
  • 13) 0.000 000 000 000 000 222 044 604 925 027 860 48 × 2 = 0 + 0.000 000 000 000 000 444 089 209 850 055 720 96;
  • 14) 0.000 000 000 000 000 444 089 209 850 055 720 96 × 2 = 0 + 0.000 000 000 000 000 888 178 419 700 111 441 92;
  • 15) 0.000 000 000 000 000 888 178 419 700 111 441 92 × 2 = 0 + 0.000 000 000 000 001 776 356 839 400 222 883 84;
  • 16) 0.000 000 000 000 001 776 356 839 400 222 883 84 × 2 = 0 + 0.000 000 000 000 003 552 713 678 800 445 767 68;
  • 17) 0.000 000 000 000 003 552 713 678 800 445 767 68 × 2 = 0 + 0.000 000 000 000 007 105 427 357 600 891 535 36;
  • 18) 0.000 000 000 000 007 105 427 357 600 891 535 36 × 2 = 0 + 0.000 000 000 000 014 210 854 715 201 783 070 72;
  • 19) 0.000 000 000 000 014 210 854 715 201 783 070 72 × 2 = 0 + 0.000 000 000 000 028 421 709 430 403 566 141 44;
  • 20) 0.000 000 000 000 028 421 709 430 403 566 141 44 × 2 = 0 + 0.000 000 000 000 056 843 418 860 807 132 282 88;
  • 21) 0.000 000 000 000 056 843 418 860 807 132 282 88 × 2 = 0 + 0.000 000 000 000 113 686 837 721 614 264 565 76;
  • 22) 0.000 000 000 000 113 686 837 721 614 264 565 76 × 2 = 0 + 0.000 000 000 000 227 373 675 443 228 529 131 52;
  • 23) 0.000 000 000 000 227 373 675 443 228 529 131 52 × 2 = 0 + 0.000 000 000 000 454 747 350 886 457 058 263 04;
  • 24) 0.000 000 000 000 454 747 350 886 457 058 263 04 × 2 = 0 + 0.000 000 000 000 909 494 701 772 914 116 526 08;
  • 25) 0.000 000 000 000 909 494 701 772 914 116 526 08 × 2 = 0 + 0.000 000 000 001 818 989 403 545 828 233 052 16;
  • 26) 0.000 000 000 001 818 989 403 545 828 233 052 16 × 2 = 0 + 0.000 000 000 003 637 978 807 091 656 466 104 32;
  • 27) 0.000 000 000 003 637 978 807 091 656 466 104 32 × 2 = 0 + 0.000 000 000 007 275 957 614 183 312 932 208 64;
  • 28) 0.000 000 000 007 275 957 614 183 312 932 208 64 × 2 = 0 + 0.000 000 000 014 551 915 228 366 625 864 417 28;
  • 29) 0.000 000 000 014 551 915 228 366 625 864 417 28 × 2 = 0 + 0.000 000 000 029 103 830 456 733 251 728 834 56;
  • 30) 0.000 000 000 029 103 830 456 733 251 728 834 56 × 2 = 0 + 0.000 000 000 058 207 660 913 466 503 457 669 12;
  • 31) 0.000 000 000 058 207 660 913 466 503 457 669 12 × 2 = 0 + 0.000 000 000 116 415 321 826 933 006 915 338 24;
  • 32) 0.000 000 000 116 415 321 826 933 006 915 338 24 × 2 = 0 + 0.000 000 000 232 830 643 653 866 013 830 676 48;
  • 33) 0.000 000 000 232 830 643 653 866 013 830 676 48 × 2 = 0 + 0.000 000 000 465 661 287 307 732 027 661 352 96;
  • 34) 0.000 000 000 465 661 287 307 732 027 661 352 96 × 2 = 0 + 0.000 000 000 931 322 574 615 464 055 322 705 92;
  • 35) 0.000 000 000 931 322 574 615 464 055 322 705 92 × 2 = 0 + 0.000 000 001 862 645 149 230 928 110 645 411 84;
  • 36) 0.000 000 001 862 645 149 230 928 110 645 411 84 × 2 = 0 + 0.000 000 003 725 290 298 461 856 221 290 823 68;
  • 37) 0.000 000 003 725 290 298 461 856 221 290 823 68 × 2 = 0 + 0.000 000 007 450 580 596 923 712 442 581 647 36;
  • 38) 0.000 000 007 450 580 596 923 712 442 581 647 36 × 2 = 0 + 0.000 000 014 901 161 193 847 424 885 163 294 72;
  • 39) 0.000 000 014 901 161 193 847 424 885 163 294 72 × 2 = 0 + 0.000 000 029 802 322 387 694 849 770 326 589 44;
  • 40) 0.000 000 029 802 322 387 694 849 770 326 589 44 × 2 = 0 + 0.000 000 059 604 644 775 389 699 540 653 178 88;
  • 41) 0.000 000 059 604 644 775 389 699 540 653 178 88 × 2 = 0 + 0.000 000 119 209 289 550 779 399 081 306 357 76;
  • 42) 0.000 000 119 209 289 550 779 399 081 306 357 76 × 2 = 0 + 0.000 000 238 418 579 101 558 798 162 612 715 52;
  • 43) 0.000 000 238 418 579 101 558 798 162 612 715 52 × 2 = 0 + 0.000 000 476 837 158 203 117 596 325 225 431 04;
  • 44) 0.000 000 476 837 158 203 117 596 325 225 431 04 × 2 = 0 + 0.000 000 953 674 316 406 235 192 650 450 862 08;
  • 45) 0.000 000 953 674 316 406 235 192 650 450 862 08 × 2 = 0 + 0.000 001 907 348 632 812 470 385 300 901 724 16;
  • 46) 0.000 001 907 348 632 812 470 385 300 901 724 16 × 2 = 0 + 0.000 003 814 697 265 624 940 770 601 803 448 32;
  • 47) 0.000 003 814 697 265 624 940 770 601 803 448 32 × 2 = 0 + 0.000 007 629 394 531 249 881 541 203 606 896 64;
  • 48) 0.000 007 629 394 531 249 881 541 203 606 896 64 × 2 = 0 + 0.000 015 258 789 062 499 763 082 407 213 793 28;
  • 49) 0.000 015 258 789 062 499 763 082 407 213 793 28 × 2 = 0 + 0.000 030 517 578 124 999 526 164 814 427 586 56;
  • 50) 0.000 030 517 578 124 999 526 164 814 427 586 56 × 2 = 0 + 0.000 061 035 156 249 999 052 329 628 855 173 12;
  • 51) 0.000 061 035 156 249 999 052 329 628 855 173 12 × 2 = 0 + 0.000 122 070 312 499 998 104 659 257 710 346 24;
  • 52) 0.000 122 070 312 499 998 104 659 257 710 346 24 × 2 = 0 + 0.000 244 140 624 999 996 209 318 515 420 692 48;
  • 53) 0.000 244 140 624 999 996 209 318 515 420 692 48 × 2 = 0 + 0.000 488 281 249 999 992 418 637 030 841 384 96;
  • 54) 0.000 488 281 249 999 992 418 637 030 841 384 96 × 2 = 0 + 0.000 976 562 499 999 984 837 274 061 682 769 92;
  • 55) 0.000 976 562 499 999 984 837 274 061 682 769 92 × 2 = 0 + 0.001 953 124 999 999 969 674 548 123 365 539 84;
  • 56) 0.001 953 124 999 999 969 674 548 123 365 539 84 × 2 = 0 + 0.003 906 249 999 999 939 349 096 246 731 079 68;
  • 57) 0.003 906 249 999 999 939 349 096 246 731 079 68 × 2 = 0 + 0.007 812 499 999 999 878 698 192 493 462 159 36;
  • 58) 0.007 812 499 999 999 878 698 192 493 462 159 36 × 2 = 0 + 0.015 624 999 999 999 757 396 384 986 924 318 72;
  • 59) 0.015 624 999 999 999 757 396 384 986 924 318 72 × 2 = 0 + 0.031 249 999 999 999 514 792 769 973 848 637 44;
  • 60) 0.031 249 999 999 999 514 792 769 973 848 637 44 × 2 = 0 + 0.062 499 999 999 999 029 585 539 947 697 274 88;
  • 61) 0.062 499 999 999 999 029 585 539 947 697 274 88 × 2 = 0 + 0.124 999 999 999 998 059 171 079 895 394 549 76;
  • 62) 0.124 999 999 999 998 059 171 079 895 394 549 76 × 2 = 0 + 0.249 999 999 999 996 118 342 159 790 789 099 52;
  • 63) 0.249 999 999 999 996 118 342 159 790 789 099 52 × 2 = 0 + 0.499 999 999 999 992 236 684 319 581 578 199 04;
  • 64) 0.499 999 999 999 992 236 684 319 581 578 199 04 × 2 = 0 + 0.999 999 999 999 984 473 368 639 163 156 398 08;
  • 65) 0.999 999 999 999 984 473 368 639 163 156 398 08 × 2 = 1 + 0.999 999 999 999 968 946 737 278 326 312 796 16;
  • 66) 0.999 999 999 999 968 946 737 278 326 312 796 16 × 2 = 1 + 0.999 999 999 999 937 893 474 556 652 625 592 32;
  • 67) 0.999 999 999 999 937 893 474 556 652 625 592 32 × 2 = 1 + 0.999 999 999 999 875 786 949 113 305 251 184 64;
  • 68) 0.999 999 999 999 875 786 949 113 305 251 184 64 × 2 = 1 + 0.999 999 999 999 751 573 898 226 610 502 369 28;
  • 69) 0.999 999 999 999 751 573 898 226 610 502 369 28 × 2 = 1 + 0.999 999 999 999 503 147 796 453 221 004 738 56;
  • 70) 0.999 999 999 999 503 147 796 453 221 004 738 56 × 2 = 1 + 0.999 999 999 999 006 295 592 906 442 009 477 12;
  • 71) 0.999 999 999 999 006 295 592 906 442 009 477 12 × 2 = 1 + 0.999 999 999 998 012 591 185 812 884 018 954 24;
  • 72) 0.999 999 999 998 012 591 185 812 884 018 954 24 × 2 = 1 + 0.999 999 999 996 025 182 371 625 768 037 908 48;
  • 73) 0.999 999 999 996 025 182 371 625 768 037 908 48 × 2 = 1 + 0.999 999 999 992 050 364 743 251 536 075 816 96;
  • 74) 0.999 999 999 992 050 364 743 251 536 075 816 96 × 2 = 1 + 0.999 999 999 984 100 729 486 503 072 151 633 92;
  • 75) 0.999 999 999 984 100 729 486 503 072 151 633 92 × 2 = 1 + 0.999 999 999 968 201 458 973 006 144 303 267 84;
  • 76) 0.999 999 999 968 201 458 973 006 144 303 267 84 × 2 = 1 + 0.999 999 999 936 402 917 946 012 288 606 535 68;
  • 77) 0.999 999 999 936 402 917 946 012 288 606 535 68 × 2 = 1 + 0.999 999 999 872 805 835 892 024 577 213 071 36;
  • 78) 0.999 999 999 872 805 835 892 024 577 213 071 36 × 2 = 1 + 0.999 999 999 745 611 671 784 049 154 426 142 72;
  • 79) 0.999 999 999 745 611 671 784 049 154 426 142 72 × 2 = 1 + 0.999 999 999 491 223 343 568 098 308 852 285 44;
  • 80) 0.999 999 999 491 223 343 568 098 308 852 285 44 × 2 = 1 + 0.999 999 998 982 446 687 136 196 617 704 570 88;
  • 81) 0.999 999 998 982 446 687 136 196 617 704 570 88 × 2 = 1 + 0.999 999 997 964 893 374 272 393 235 409 141 76;
  • 82) 0.999 999 997 964 893 374 272 393 235 409 141 76 × 2 = 1 + 0.999 999 995 929 786 748 544 786 470 818 283 52;
  • 83) 0.999 999 995 929 786 748 544 786 470 818 283 52 × 2 = 1 + 0.999 999 991 859 573 497 089 572 941 636 567 04;
  • 84) 0.999 999 991 859 573 497 089 572 941 636 567 04 × 2 = 1 + 0.999 999 983 719 146 994 179 145 883 273 134 08;
  • 85) 0.999 999 983 719 146 994 179 145 883 273 134 08 × 2 = 1 + 0.999 999 967 438 293 988 358 291 766 546 268 16;
  • 86) 0.999 999 967 438 293 988 358 291 766 546 268 16 × 2 = 1 + 0.999 999 934 876 587 976 716 583 533 092 536 32;
  • 87) 0.999 999 934 876 587 976 716 583 533 092 536 32 × 2 = 1 + 0.999 999 869 753 175 953 433 167 066 185 072 64;
  • 88) 0.999 999 869 753 175 953 433 167 066 185 072 64 × 2 = 1 + 0.999 999 739 506 351 906 866 334 132 370 145 28;
  • 89) 0.999 999 739 506 351 906 866 334 132 370 145 28 × 2 = 1 + 0.999 999 479 012 703 813 732 668 264 740 290 56;
  • 90) 0.999 999 479 012 703 813 732 668 264 740 290 56 × 2 = 1 + 0.999 998 958 025 407 627 465 336 529 480 581 12;
  • 91) 0.999 998 958 025 407 627 465 336 529 480 581 12 × 2 = 1 + 0.999 997 916 050 815 254 930 673 058 961 162 24;
  • 92) 0.999 997 916 050 815 254 930 673 058 961 162 24 × 2 = 1 + 0.999 995 832 101 630 509 861 346 117 922 324 48;
  • 93) 0.999 995 832 101 630 509 861 346 117 922 324 48 × 2 = 1 + 0.999 991 664 203 261 019 722 692 235 844 648 96;
  • 94) 0.999 991 664 203 261 019 722 692 235 844 648 96 × 2 = 1 + 0.999 983 328 406 522 039 445 384 471 689 297 92;
  • 95) 0.999 983 328 406 522 039 445 384 471 689 297 92 × 2 = 1 + 0.999 966 656 813 044 078 890 768 943 378 595 84;
  • 96) 0.999 966 656 813 044 078 890 768 943 378 595 84 × 2 = 1 + 0.999 933 313 626 088 157 781 537 886 757 191 68;
  • 97) 0.999 933 313 626 088 157 781 537 886 757 191 68 × 2 = 1 + 0.999 866 627 252 176 315 563 075 773 514 383 36;
  • 98) 0.999 866 627 252 176 315 563 075 773 514 383 36 × 2 = 1 + 0.999 733 254 504 352 631 126 151 547 028 766 72;
  • 99) 0.999 733 254 504 352 631 126 151 547 028 766 72 × 2 = 1 + 0.999 466 509 008 705 262 252 303 094 057 533 44;
  • 100) 0.999 466 509 008 705 262 252 303 094 057 533 44 × 2 = 1 + 0.998 933 018 017 410 524 504 606 188 115 066 88;
  • 101) 0.998 933 018 017 410 524 504 606 188 115 066 88 × 2 = 1 + 0.997 866 036 034 821 049 009 212 376 230 133 76;
  • 102) 0.997 866 036 034 821 049 009 212 376 230 133 76 × 2 = 1 + 0.995 732 072 069 642 098 018 424 752 460 267 52;
  • 103) 0.995 732 072 069 642 098 018 424 752 460 267 52 × 2 = 1 + 0.991 464 144 139 284 196 036 849 504 920 535 04;
  • 104) 0.991 464 144 139 284 196 036 849 504 920 535 04 × 2 = 1 + 0.982 928 288 278 568 392 073 699 009 841 070 08;
  • 105) 0.982 928 288 278 568 392 073 699 009 841 070 08 × 2 = 1 + 0.965 856 576 557 136 784 147 398 019 682 140 16;
  • 106) 0.965 856 576 557 136 784 147 398 019 682 140 16 × 2 = 1 + 0.931 713 153 114 273 568 294 796 039 364 280 32;
  • 107) 0.931 713 153 114 273 568 294 796 039 364 280 32 × 2 = 1 + 0.863 426 306 228 547 136 589 592 078 728 560 64;
  • 108) 0.863 426 306 228 547 136 589 592 078 728 560 64 × 2 = 1 + 0.726 852 612 457 094 273 179 184 157 457 121 28;
  • 109) 0.726 852 612 457 094 273 179 184 157 457 121 28 × 2 = 1 + 0.453 705 224 914 188 546 358 368 314 914 242 56;
  • 110) 0.453 705 224 914 188 546 358 368 314 914 242 56 × 2 = 0 + 0.907 410 449 828 377 092 716 736 629 828 485 12;
  • 111) 0.907 410 449 828 377 092 716 736 629 828 485 12 × 2 = 1 + 0.814 820 899 656 754 185 433 473 259 656 970 24;
  • 112) 0.814 820 899 656 754 185 433 473 259 656 970 24 × 2 = 1 + 0.629 641 799 313 508 370 866 946 519 313 940 48;
  • 113) 0.629 641 799 313 508 370 866 946 519 313 940 48 × 2 = 1 + 0.259 283 598 627 016 741 733 893 038 627 880 96;
  • 114) 0.259 283 598 627 016 741 733 893 038 627 880 96 × 2 = 0 + 0.518 567 197 254 033 483 467 786 077 255 761 92;
  • 115) 0.518 567 197 254 033 483 467 786 077 255 761 92 × 2 = 1 + 0.037 134 394 508 066 966 935 572 154 511 523 84;
  • 116) 0.037 134 394 508 066 966 935 572 154 511 523 84 × 2 = 0 + 0.074 268 789 016 133 933 871 144 309 023 047 68;
  • 117) 0.074 268 789 016 133 933 871 144 309 023 047 68 × 2 = 0 + 0.148 537 578 032 267 867 742 288 618 046 095 36;

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:


0.000 000 000 000 000 000 054 210 108 624 274 38(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1011 1010 0(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 054 210 108 624 274 38(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1011 1010 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 65 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 054 210 108 624 274 38(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1011 1010 0(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1011 1010 0(2) × 20 =

1.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111 0100(2) × 2-65


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): -65

Mantissa (not normalized):
1.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111 0100


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-65 + 2(11-1) - 1 =

(-65 + 1 023)(10) =

958(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;
  • 958 ÷ 2 = 479 + 0;
  • 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) =

958(10) =

011 1011 1110(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. 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111 0100 =

1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111 0100


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 1110

Mantissa (52 bits) =
1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111 0100


Decimal number 0.000 000 000 000 000 000 054 210 108 624 274 38 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1110 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111 0100

Share

» Convert decimal 0.000 000 000 000 000 000 054 210 108 624 274 24 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 08, 2026 [August]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: August - by our visitors

» Improve your social skills: Show, Don’t Tell, the Strategic Power of Action. Gemini Notebook YouTube Video of 9.21 minutes

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