0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 356 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 356 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
0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 356 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: 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 356 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) 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 356 5 × 2 = 0 + 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 713;
  • 2) 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 713 × 2 = 0 + 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 426;
  • 3) 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 426 × 2 = 0 + 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 852;
  • 4) 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 852 × 2 = 0 + 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 704;
  • 5) 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 704 × 2 = 0 + 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 408;
  • 6) 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 408 × 2 = 0 + 0.000 000 000 000 000 790 124 025 252 133 724 078 653 462 816;
  • 7) 0.000 000 000 000 000 790 124 025 252 133 724 078 653 462 816 × 2 = 0 + 0.000 000 000 000 001 580 248 050 504 267 448 157 306 925 632;
  • 8) 0.000 000 000 000 001 580 248 050 504 267 448 157 306 925 632 × 2 = 0 + 0.000 000 000 000 003 160 496 101 008 534 896 314 613 851 264;
  • 9) 0.000 000 000 000 003 160 496 101 008 534 896 314 613 851 264 × 2 = 0 + 0.000 000 000 000 006 320 992 202 017 069 792 629 227 702 528;
  • 10) 0.000 000 000 000 006 320 992 202 017 069 792 629 227 702 528 × 2 = 0 + 0.000 000 000 000 012 641 984 404 034 139 585 258 455 405 056;
  • 11) 0.000 000 000 000 012 641 984 404 034 139 585 258 455 405 056 × 2 = 0 + 0.000 000 000 000 025 283 968 808 068 279 170 516 910 810 112;
  • 12) 0.000 000 000 000 025 283 968 808 068 279 170 516 910 810 112 × 2 = 0 + 0.000 000 000 000 050 567 937 616 136 558 341 033 821 620 224;
  • 13) 0.000 000 000 000 050 567 937 616 136 558 341 033 821 620 224 × 2 = 0 + 0.000 000 000 000 101 135 875 232 273 116 682 067 643 240 448;
  • 14) 0.000 000 000 000 101 135 875 232 273 116 682 067 643 240 448 × 2 = 0 + 0.000 000 000 000 202 271 750 464 546 233 364 135 286 480 896;
  • 15) 0.000 000 000 000 202 271 750 464 546 233 364 135 286 480 896 × 2 = 0 + 0.000 000 000 000 404 543 500 929 092 466 728 270 572 961 792;
  • 16) 0.000 000 000 000 404 543 500 929 092 466 728 270 572 961 792 × 2 = 0 + 0.000 000 000 000 809 087 001 858 184 933 456 541 145 923 584;
  • 17) 0.000 000 000 000 809 087 001 858 184 933 456 541 145 923 584 × 2 = 0 + 0.000 000 000 001 618 174 003 716 369 866 913 082 291 847 168;
  • 18) 0.000 000 000 001 618 174 003 716 369 866 913 082 291 847 168 × 2 = 0 + 0.000 000 000 003 236 348 007 432 739 733 826 164 583 694 336;
  • 19) 0.000 000 000 003 236 348 007 432 739 733 826 164 583 694 336 × 2 = 0 + 0.000 000 000 006 472 696 014 865 479 467 652 329 167 388 672;
  • 20) 0.000 000 000 006 472 696 014 865 479 467 652 329 167 388 672 × 2 = 0 + 0.000 000 000 012 945 392 029 730 958 935 304 658 334 777 344;
  • 21) 0.000 000 000 012 945 392 029 730 958 935 304 658 334 777 344 × 2 = 0 + 0.000 000 000 025 890 784 059 461 917 870 609 316 669 554 688;
  • 22) 0.000 000 000 025 890 784 059 461 917 870 609 316 669 554 688 × 2 = 0 + 0.000 000 000 051 781 568 118 923 835 741 218 633 339 109 376;
  • 23) 0.000 000 000 051 781 568 118 923 835 741 218 633 339 109 376 × 2 = 0 + 0.000 000 000 103 563 136 237 847 671 482 437 266 678 218 752;
  • 24) 0.000 000 000 103 563 136 237 847 671 482 437 266 678 218 752 × 2 = 0 + 0.000 000 000 207 126 272 475 695 342 964 874 533 356 437 504;
  • 25) 0.000 000 000 207 126 272 475 695 342 964 874 533 356 437 504 × 2 = 0 + 0.000 000 000 414 252 544 951 390 685 929 749 066 712 875 008;
  • 26) 0.000 000 000 414 252 544 951 390 685 929 749 066 712 875 008 × 2 = 0 + 0.000 000 000 828 505 089 902 781 371 859 498 133 425 750 016;
  • 27) 0.000 000 000 828 505 089 902 781 371 859 498 133 425 750 016 × 2 = 0 + 0.000 000 001 657 010 179 805 562 743 718 996 266 851 500 032;
  • 28) 0.000 000 001 657 010 179 805 562 743 718 996 266 851 500 032 × 2 = 0 + 0.000 000 003 314 020 359 611 125 487 437 992 533 703 000 064;
  • 29) 0.000 000 003 314 020 359 611 125 487 437 992 533 703 000 064 × 2 = 0 + 0.000 000 006 628 040 719 222 250 974 875 985 067 406 000 128;
  • 30) 0.000 000 006 628 040 719 222 250 974 875 985 067 406 000 128 × 2 = 0 + 0.000 000 013 256 081 438 444 501 949 751 970 134 812 000 256;
  • 31) 0.000 000 013 256 081 438 444 501 949 751 970 134 812 000 256 × 2 = 0 + 0.000 000 026 512 162 876 889 003 899 503 940 269 624 000 512;
  • 32) 0.000 000 026 512 162 876 889 003 899 503 940 269 624 000 512 × 2 = 0 + 0.000 000 053 024 325 753 778 007 799 007 880 539 248 001 024;
  • 33) 0.000 000 053 024 325 753 778 007 799 007 880 539 248 001 024 × 2 = 0 + 0.000 000 106 048 651 507 556 015 598 015 761 078 496 002 048;
  • 34) 0.000 000 106 048 651 507 556 015 598 015 761 078 496 002 048 × 2 = 0 + 0.000 000 212 097 303 015 112 031 196 031 522 156 992 004 096;
  • 35) 0.000 000 212 097 303 015 112 031 196 031 522 156 992 004 096 × 2 = 0 + 0.000 000 424 194 606 030 224 062 392 063 044 313 984 008 192;
  • 36) 0.000 000 424 194 606 030 224 062 392 063 044 313 984 008 192 × 2 = 0 + 0.000 000 848 389 212 060 448 124 784 126 088 627 968 016 384;
  • 37) 0.000 000 848 389 212 060 448 124 784 126 088 627 968 016 384 × 2 = 0 + 0.000 001 696 778 424 120 896 249 568 252 177 255 936 032 768;
  • 38) 0.000 001 696 778 424 120 896 249 568 252 177 255 936 032 768 × 2 = 0 + 0.000 003 393 556 848 241 792 499 136 504 354 511 872 065 536;
  • 39) 0.000 003 393 556 848 241 792 499 136 504 354 511 872 065 536 × 2 = 0 + 0.000 006 787 113 696 483 584 998 273 008 709 023 744 131 072;
  • 40) 0.000 006 787 113 696 483 584 998 273 008 709 023 744 131 072 × 2 = 0 + 0.000 013 574 227 392 967 169 996 546 017 418 047 488 262 144;
  • 41) 0.000 013 574 227 392 967 169 996 546 017 418 047 488 262 144 × 2 = 0 + 0.000 027 148 454 785 934 339 993 092 034 836 094 976 524 288;
  • 42) 0.000 027 148 454 785 934 339 993 092 034 836 094 976 524 288 × 2 = 0 + 0.000 054 296 909 571 868 679 986 184 069 672 189 953 048 576;
  • 43) 0.000 054 296 909 571 868 679 986 184 069 672 189 953 048 576 × 2 = 0 + 0.000 108 593 819 143 737 359 972 368 139 344 379 906 097 152;
  • 44) 0.000 108 593 819 143 737 359 972 368 139 344 379 906 097 152 × 2 = 0 + 0.000 217 187 638 287 474 719 944 736 278 688 759 812 194 304;
  • 45) 0.000 217 187 638 287 474 719 944 736 278 688 759 812 194 304 × 2 = 0 + 0.000 434 375 276 574 949 439 889 472 557 377 519 624 388 608;
  • 46) 0.000 434 375 276 574 949 439 889 472 557 377 519 624 388 608 × 2 = 0 + 0.000 868 750 553 149 898 879 778 945 114 755 039 248 777 216;
  • 47) 0.000 868 750 553 149 898 879 778 945 114 755 039 248 777 216 × 2 = 0 + 0.001 737 501 106 299 797 759 557 890 229 510 078 497 554 432;
  • 48) 0.001 737 501 106 299 797 759 557 890 229 510 078 497 554 432 × 2 = 0 + 0.003 475 002 212 599 595 519 115 780 459 020 156 995 108 864;
  • 49) 0.003 475 002 212 599 595 519 115 780 459 020 156 995 108 864 × 2 = 0 + 0.006 950 004 425 199 191 038 231 560 918 040 313 990 217 728;
  • 50) 0.006 950 004 425 199 191 038 231 560 918 040 313 990 217 728 × 2 = 0 + 0.013 900 008 850 398 382 076 463 121 836 080 627 980 435 456;
  • 51) 0.013 900 008 850 398 382 076 463 121 836 080 627 980 435 456 × 2 = 0 + 0.027 800 017 700 796 764 152 926 243 672 161 255 960 870 912;
  • 52) 0.027 800 017 700 796 764 152 926 243 672 161 255 960 870 912 × 2 = 0 + 0.055 600 035 401 593 528 305 852 487 344 322 511 921 741 824;
  • 53) 0.055 600 035 401 593 528 305 852 487 344 322 511 921 741 824 × 2 = 0 + 0.111 200 070 803 187 056 611 704 974 688 645 023 843 483 648;
  • 54) 0.111 200 070 803 187 056 611 704 974 688 645 023 843 483 648 × 2 = 0 + 0.222 400 141 606 374 113 223 409 949 377 290 047 686 967 296;
  • 55) 0.222 400 141 606 374 113 223 409 949 377 290 047 686 967 296 × 2 = 0 + 0.444 800 283 212 748 226 446 819 898 754 580 095 373 934 592;
  • 56) 0.444 800 283 212 748 226 446 819 898 754 580 095 373 934 592 × 2 = 0 + 0.889 600 566 425 496 452 893 639 797 509 160 190 747 869 184;
  • 57) 0.889 600 566 425 496 452 893 639 797 509 160 190 747 869 184 × 2 = 1 + 0.779 201 132 850 992 905 787 279 595 018 320 381 495 738 368;
  • 58) 0.779 201 132 850 992 905 787 279 595 018 320 381 495 738 368 × 2 = 1 + 0.558 402 265 701 985 811 574 559 190 036 640 762 991 476 736;
  • 59) 0.558 402 265 701 985 811 574 559 190 036 640 762 991 476 736 × 2 = 1 + 0.116 804 531 403 971 623 149 118 380 073 281 525 982 953 472;
  • 60) 0.116 804 531 403 971 623 149 118 380 073 281 525 982 953 472 × 2 = 0 + 0.233 609 062 807 943 246 298 236 760 146 563 051 965 906 944;
  • 61) 0.233 609 062 807 943 246 298 236 760 146 563 051 965 906 944 × 2 = 0 + 0.467 218 125 615 886 492 596 473 520 293 126 103 931 813 888;
  • 62) 0.467 218 125 615 886 492 596 473 520 293 126 103 931 813 888 × 2 = 0 + 0.934 436 251 231 772 985 192 947 040 586 252 207 863 627 776;
  • 63) 0.934 436 251 231 772 985 192 947 040 586 252 207 863 627 776 × 2 = 1 + 0.868 872 502 463 545 970 385 894 081 172 504 415 727 255 552;
  • 64) 0.868 872 502 463 545 970 385 894 081 172 504 415 727 255 552 × 2 = 1 + 0.737 745 004 927 091 940 771 788 162 345 008 831 454 511 104;
  • 65) 0.737 745 004 927 091 940 771 788 162 345 008 831 454 511 104 × 2 = 1 + 0.475 490 009 854 183 881 543 576 324 690 017 662 909 022 208;
  • 66) 0.475 490 009 854 183 881 543 576 324 690 017 662 909 022 208 × 2 = 0 + 0.950 980 019 708 367 763 087 152 649 380 035 325 818 044 416;
  • 67) 0.950 980 019 708 367 763 087 152 649 380 035 325 818 044 416 × 2 = 1 + 0.901 960 039 416 735 526 174 305 298 760 070 651 636 088 832;
  • 68) 0.901 960 039 416 735 526 174 305 298 760 070 651 636 088 832 × 2 = 1 + 0.803 920 078 833 471 052 348 610 597 520 141 303 272 177 664;
  • 69) 0.803 920 078 833 471 052 348 610 597 520 141 303 272 177 664 × 2 = 1 + 0.607 840 157 666 942 104 697 221 195 040 282 606 544 355 328;
  • 70) 0.607 840 157 666 942 104 697 221 195 040 282 606 544 355 328 × 2 = 1 + 0.215 680 315 333 884 209 394 442 390 080 565 213 088 710 656;
  • 71) 0.215 680 315 333 884 209 394 442 390 080 565 213 088 710 656 × 2 = 0 + 0.431 360 630 667 768 418 788 884 780 161 130 426 177 421 312;
  • 72) 0.431 360 630 667 768 418 788 884 780 161 130 426 177 421 312 × 2 = 0 + 0.862 721 261 335 536 837 577 769 560 322 260 852 354 842 624;
  • 73) 0.862 721 261 335 536 837 577 769 560 322 260 852 354 842 624 × 2 = 1 + 0.725 442 522 671 073 675 155 539 120 644 521 704 709 685 248;
  • 74) 0.725 442 522 671 073 675 155 539 120 644 521 704 709 685 248 × 2 = 1 + 0.450 885 045 342 147 350 311 078 241 289 043 409 419 370 496;
  • 75) 0.450 885 045 342 147 350 311 078 241 289 043 409 419 370 496 × 2 = 0 + 0.901 770 090 684 294 700 622 156 482 578 086 818 838 740 992;
  • 76) 0.901 770 090 684 294 700 622 156 482 578 086 818 838 740 992 × 2 = 1 + 0.803 540 181 368 589 401 244 312 965 156 173 637 677 481 984;
  • 77) 0.803 540 181 368 589 401 244 312 965 156 173 637 677 481 984 × 2 = 1 + 0.607 080 362 737 178 802 488 625 930 312 347 275 354 963 968;
  • 78) 0.607 080 362 737 178 802 488 625 930 312 347 275 354 963 968 × 2 = 1 + 0.214 160 725 474 357 604 977 251 860 624 694 550 709 927 936;
  • 79) 0.214 160 725 474 357 604 977 251 860 624 694 550 709 927 936 × 2 = 0 + 0.428 321 450 948 715 209 954 503 721 249 389 101 419 855 872;
  • 80) 0.428 321 450 948 715 209 954 503 721 249 389 101 419 855 872 × 2 = 0 + 0.856 642 901 897 430 419 909 007 442 498 778 202 839 711 744;
  • 81) 0.856 642 901 897 430 419 909 007 442 498 778 202 839 711 744 × 2 = 1 + 0.713 285 803 794 860 839 818 014 884 997 556 405 679 423 488;
  • 82) 0.713 285 803 794 860 839 818 014 884 997 556 405 679 423 488 × 2 = 1 + 0.426 571 607 589 721 679 636 029 769 995 112 811 358 846 976;
  • 83) 0.426 571 607 589 721 679 636 029 769 995 112 811 358 846 976 × 2 = 0 + 0.853 143 215 179 443 359 272 059 539 990 225 622 717 693 952;
  • 84) 0.853 143 215 179 443 359 272 059 539 990 225 622 717 693 952 × 2 = 1 + 0.706 286 430 358 886 718 544 119 079 980 451 245 435 387 904;
  • 85) 0.706 286 430 358 886 718 544 119 079 980 451 245 435 387 904 × 2 = 1 + 0.412 572 860 717 773 437 088 238 159 960 902 490 870 775 808;
  • 86) 0.412 572 860 717 773 437 088 238 159 960 902 490 870 775 808 × 2 = 0 + 0.825 145 721 435 546 874 176 476 319 921 804 981 741 551 616;
  • 87) 0.825 145 721 435 546 874 176 476 319 921 804 981 741 551 616 × 2 = 1 + 0.650 291 442 871 093 748 352 952 639 843 609 963 483 103 232;
  • 88) 0.650 291 442 871 093 748 352 952 639 843 609 963 483 103 232 × 2 = 1 + 0.300 582 885 742 187 496 705 905 279 687 219 926 966 206 464;
  • 89) 0.300 582 885 742 187 496 705 905 279 687 219 926 966 206 464 × 2 = 0 + 0.601 165 771 484 374 993 411 810 559 374 439 853 932 412 928;
  • 90) 0.601 165 771 484 374 993 411 810 559 374 439 853 932 412 928 × 2 = 1 + 0.202 331 542 968 749 986 823 621 118 748 879 707 864 825 856;
  • 91) 0.202 331 542 968 749 986 823 621 118 748 879 707 864 825 856 × 2 = 0 + 0.404 663 085 937 499 973 647 242 237 497 759 415 729 651 712;
  • 92) 0.404 663 085 937 499 973 647 242 237 497 759 415 729 651 712 × 2 = 0 + 0.809 326 171 874 999 947 294 484 474 995 518 831 459 303 424;
  • 93) 0.809 326 171 874 999 947 294 484 474 995 518 831 459 303 424 × 2 = 1 + 0.618 652 343 749 999 894 588 968 949 991 037 662 918 606 848;
  • 94) 0.618 652 343 749 999 894 588 968 949 991 037 662 918 606 848 × 2 = 1 + 0.237 304 687 499 999 789 177 937 899 982 075 325 837 213 696;
  • 95) 0.237 304 687 499 999 789 177 937 899 982 075 325 837 213 696 × 2 = 0 + 0.474 609 374 999 999 578 355 875 799 964 150 651 674 427 392;
  • 96) 0.474 609 374 999 999 578 355 875 799 964 150 651 674 427 392 × 2 = 0 + 0.949 218 749 999 999 156 711 751 599 928 301 303 348 854 784;
  • 97) 0.949 218 749 999 999 156 711 751 599 928 301 303 348 854 784 × 2 = 1 + 0.898 437 499 999 998 313 423 503 199 856 602 606 697 709 568;
  • 98) 0.898 437 499 999 998 313 423 503 199 856 602 606 697 709 568 × 2 = 1 + 0.796 874 999 999 996 626 847 006 399 713 205 213 395 419 136;
  • 99) 0.796 874 999 999 996 626 847 006 399 713 205 213 395 419 136 × 2 = 1 + 0.593 749 999 999 993 253 694 012 799 426 410 426 790 838 272;
  • 100) 0.593 749 999 999 993 253 694 012 799 426 410 426 790 838 272 × 2 = 1 + 0.187 499 999 999 986 507 388 025 598 852 820 853 581 676 544;
  • 101) 0.187 499 999 999 986 507 388 025 598 852 820 853 581 676 544 × 2 = 0 + 0.374 999 999 999 973 014 776 051 197 705 641 707 163 353 088;
  • 102) 0.374 999 999 999 973 014 776 051 197 705 641 707 163 353 088 × 2 = 0 + 0.749 999 999 999 946 029 552 102 395 411 283 414 326 706 176;
  • 103) 0.749 999 999 999 946 029 552 102 395 411 283 414 326 706 176 × 2 = 1 + 0.499 999 999 999 892 059 104 204 790 822 566 828 653 412 352;
  • 104) 0.499 999 999 999 892 059 104 204 790 822 566 828 653 412 352 × 2 = 0 + 0.999 999 999 999 784 118 208 409 581 645 133 657 306 824 704;
  • 105) 0.999 999 999 999 784 118 208 409 581 645 133 657 306 824 704 × 2 = 1 + 0.999 999 999 999 568 236 416 819 163 290 267 314 613 649 408;
  • 106) 0.999 999 999 999 568 236 416 819 163 290 267 314 613 649 408 × 2 = 1 + 0.999 999 999 999 136 472 833 638 326 580 534 629 227 298 816;
  • 107) 0.999 999 999 999 136 472 833 638 326 580 534 629 227 298 816 × 2 = 1 + 0.999 999 999 998 272 945 667 276 653 161 069 258 454 597 632;
  • 108) 0.999 999 999 998 272 945 667 276 653 161 069 258 454 597 632 × 2 = 1 + 0.999 999 999 996 545 891 334 553 306 322 138 516 909 195 264;
  • 109) 0.999 999 999 996 545 891 334 553 306 322 138 516 909 195 264 × 2 = 1 + 0.999 999 999 993 091 782 669 106 612 644 277 033 818 390 528;

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:


0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 356 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 0010 1111 1(2)

5. Positive number before normalization:

0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 356 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 0010 1111 1(2)

6. Normalize the binary representation of the number.

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


0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 356 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 0010 1111 1(2) =

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

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


7. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): -57

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


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

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

(-57 + 1 023)(10) =

966(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 966 ÷ 2 = 483 + 0;
  • 483 ÷ 2 = 241 + 1;
  • 241 ÷ 2 = 120 + 1;
  • 120 ÷ 2 = 60 + 0;
  • 60 ÷ 2 = 30 + 0;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

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

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


Exponent (adjusted) =

966(10) =

011 1100 0110(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 1111 =

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


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

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1100 0110

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


Decimal number 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 356 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 0101 1111

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

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

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100