0.000 000 000 000 755 840 011 110 101 101 010 100 109 999 999 995 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 755 840 011 110 101 101 010 100 109 999 999 995 8(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
0.000 000 000 000 755 840 011 110 101 101 010 100 109 999 999 995 8(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: 0.000 000 000 000 755 840 011 110 101 101 010 100 109 999 999 995 8.

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) 0.000 000 000 000 755 840 011 110 101 101 010 100 109 999 999 995 8 × 2 = 0 + 0.000 000 000 001 511 680 022 220 202 202 020 200 219 999 999 991 6;
  • 2) 0.000 000 000 001 511 680 022 220 202 202 020 200 219 999 999 991 6 × 2 = 0 + 0.000 000 000 003 023 360 044 440 404 404 040 400 439 999 999 983 2;
  • 3) 0.000 000 000 003 023 360 044 440 404 404 040 400 439 999 999 983 2 × 2 = 0 + 0.000 000 000 006 046 720 088 880 808 808 080 800 879 999 999 966 4;
  • 4) 0.000 000 000 006 046 720 088 880 808 808 080 800 879 999 999 966 4 × 2 = 0 + 0.000 000 000 012 093 440 177 761 617 616 161 601 759 999 999 932 8;
  • 5) 0.000 000 000 012 093 440 177 761 617 616 161 601 759 999 999 932 8 × 2 = 0 + 0.000 000 000 024 186 880 355 523 235 232 323 203 519 999 999 865 6;
  • 6) 0.000 000 000 024 186 880 355 523 235 232 323 203 519 999 999 865 6 × 2 = 0 + 0.000 000 000 048 373 760 711 046 470 464 646 407 039 999 999 731 2;
  • 7) 0.000 000 000 048 373 760 711 046 470 464 646 407 039 999 999 731 2 × 2 = 0 + 0.000 000 000 096 747 521 422 092 940 929 292 814 079 999 999 462 4;
  • 8) 0.000 000 000 096 747 521 422 092 940 929 292 814 079 999 999 462 4 × 2 = 0 + 0.000 000 000 193 495 042 844 185 881 858 585 628 159 999 998 924 8;
  • 9) 0.000 000 000 193 495 042 844 185 881 858 585 628 159 999 998 924 8 × 2 = 0 + 0.000 000 000 386 990 085 688 371 763 717 171 256 319 999 997 849 6;
  • 10) 0.000 000 000 386 990 085 688 371 763 717 171 256 319 999 997 849 6 × 2 = 0 + 0.000 000 000 773 980 171 376 743 527 434 342 512 639 999 995 699 2;
  • 11) 0.000 000 000 773 980 171 376 743 527 434 342 512 639 999 995 699 2 × 2 = 0 + 0.000 000 001 547 960 342 753 487 054 868 685 025 279 999 991 398 4;
  • 12) 0.000 000 001 547 960 342 753 487 054 868 685 025 279 999 991 398 4 × 2 = 0 + 0.000 000 003 095 920 685 506 974 109 737 370 050 559 999 982 796 8;
  • 13) 0.000 000 003 095 920 685 506 974 109 737 370 050 559 999 982 796 8 × 2 = 0 + 0.000 000 006 191 841 371 013 948 219 474 740 101 119 999 965 593 6;
  • 14) 0.000 000 006 191 841 371 013 948 219 474 740 101 119 999 965 593 6 × 2 = 0 + 0.000 000 012 383 682 742 027 896 438 949 480 202 239 999 931 187 2;
  • 15) 0.000 000 012 383 682 742 027 896 438 949 480 202 239 999 931 187 2 × 2 = 0 + 0.000 000 024 767 365 484 055 792 877 898 960 404 479 999 862 374 4;
  • 16) 0.000 000 024 767 365 484 055 792 877 898 960 404 479 999 862 374 4 × 2 = 0 + 0.000 000 049 534 730 968 111 585 755 797 920 808 959 999 724 748 8;
  • 17) 0.000 000 049 534 730 968 111 585 755 797 920 808 959 999 724 748 8 × 2 = 0 + 0.000 000 099 069 461 936 223 171 511 595 841 617 919 999 449 497 6;
  • 18) 0.000 000 099 069 461 936 223 171 511 595 841 617 919 999 449 497 6 × 2 = 0 + 0.000 000 198 138 923 872 446 343 023 191 683 235 839 998 898 995 2;
  • 19) 0.000 000 198 138 923 872 446 343 023 191 683 235 839 998 898 995 2 × 2 = 0 + 0.000 000 396 277 847 744 892 686 046 383 366 471 679 997 797 990 4;
  • 20) 0.000 000 396 277 847 744 892 686 046 383 366 471 679 997 797 990 4 × 2 = 0 + 0.000 000 792 555 695 489 785 372 092 766 732 943 359 995 595 980 8;
  • 21) 0.000 000 792 555 695 489 785 372 092 766 732 943 359 995 595 980 8 × 2 = 0 + 0.000 001 585 111 390 979 570 744 185 533 465 886 719 991 191 961 6;
  • 22) 0.000 001 585 111 390 979 570 744 185 533 465 886 719 991 191 961 6 × 2 = 0 + 0.000 003 170 222 781 959 141 488 371 066 931 773 439 982 383 923 2;
  • 23) 0.000 003 170 222 781 959 141 488 371 066 931 773 439 982 383 923 2 × 2 = 0 + 0.000 006 340 445 563 918 282 976 742 133 863 546 879 964 767 846 4;
  • 24) 0.000 006 340 445 563 918 282 976 742 133 863 546 879 964 767 846 4 × 2 = 0 + 0.000 012 680 891 127 836 565 953 484 267 727 093 759 929 535 692 8;
  • 25) 0.000 012 680 891 127 836 565 953 484 267 727 093 759 929 535 692 8 × 2 = 0 + 0.000 025 361 782 255 673 131 906 968 535 454 187 519 859 071 385 6;
  • 26) 0.000 025 361 782 255 673 131 906 968 535 454 187 519 859 071 385 6 × 2 = 0 + 0.000 050 723 564 511 346 263 813 937 070 908 375 039 718 142 771 2;
  • 27) 0.000 050 723 564 511 346 263 813 937 070 908 375 039 718 142 771 2 × 2 = 0 + 0.000 101 447 129 022 692 527 627 874 141 816 750 079 436 285 542 4;
  • 28) 0.000 101 447 129 022 692 527 627 874 141 816 750 079 436 285 542 4 × 2 = 0 + 0.000 202 894 258 045 385 055 255 748 283 633 500 158 872 571 084 8;
  • 29) 0.000 202 894 258 045 385 055 255 748 283 633 500 158 872 571 084 8 × 2 = 0 + 0.000 405 788 516 090 770 110 511 496 567 267 000 317 745 142 169 6;
  • 30) 0.000 405 788 516 090 770 110 511 496 567 267 000 317 745 142 169 6 × 2 = 0 + 0.000 811 577 032 181 540 221 022 993 134 534 000 635 490 284 339 2;
  • 31) 0.000 811 577 032 181 540 221 022 993 134 534 000 635 490 284 339 2 × 2 = 0 + 0.001 623 154 064 363 080 442 045 986 269 068 001 270 980 568 678 4;
  • 32) 0.001 623 154 064 363 080 442 045 986 269 068 001 270 980 568 678 4 × 2 = 0 + 0.003 246 308 128 726 160 884 091 972 538 136 002 541 961 137 356 8;
  • 33) 0.003 246 308 128 726 160 884 091 972 538 136 002 541 961 137 356 8 × 2 = 0 + 0.006 492 616 257 452 321 768 183 945 076 272 005 083 922 274 713 6;
  • 34) 0.006 492 616 257 452 321 768 183 945 076 272 005 083 922 274 713 6 × 2 = 0 + 0.012 985 232 514 904 643 536 367 890 152 544 010 167 844 549 427 2;
  • 35) 0.012 985 232 514 904 643 536 367 890 152 544 010 167 844 549 427 2 × 2 = 0 + 0.025 970 465 029 809 287 072 735 780 305 088 020 335 689 098 854 4;
  • 36) 0.025 970 465 029 809 287 072 735 780 305 088 020 335 689 098 854 4 × 2 = 0 + 0.051 940 930 059 618 574 145 471 560 610 176 040 671 378 197 708 8;
  • 37) 0.051 940 930 059 618 574 145 471 560 610 176 040 671 378 197 708 8 × 2 = 0 + 0.103 881 860 119 237 148 290 943 121 220 352 081 342 756 395 417 6;
  • 38) 0.103 881 860 119 237 148 290 943 121 220 352 081 342 756 395 417 6 × 2 = 0 + 0.207 763 720 238 474 296 581 886 242 440 704 162 685 512 790 835 2;
  • 39) 0.207 763 720 238 474 296 581 886 242 440 704 162 685 512 790 835 2 × 2 = 0 + 0.415 527 440 476 948 593 163 772 484 881 408 325 371 025 581 670 4;
  • 40) 0.415 527 440 476 948 593 163 772 484 881 408 325 371 025 581 670 4 × 2 = 0 + 0.831 054 880 953 897 186 327 544 969 762 816 650 742 051 163 340 8;
  • 41) 0.831 054 880 953 897 186 327 544 969 762 816 650 742 051 163 340 8 × 2 = 1 + 0.662 109 761 907 794 372 655 089 939 525 633 301 484 102 326 681 6;
  • 42) 0.662 109 761 907 794 372 655 089 939 525 633 301 484 102 326 681 6 × 2 = 1 + 0.324 219 523 815 588 745 310 179 879 051 266 602 968 204 653 363 2;
  • 43) 0.324 219 523 815 588 745 310 179 879 051 266 602 968 204 653 363 2 × 2 = 0 + 0.648 439 047 631 177 490 620 359 758 102 533 205 936 409 306 726 4;
  • 44) 0.648 439 047 631 177 490 620 359 758 102 533 205 936 409 306 726 4 × 2 = 1 + 0.296 878 095 262 354 981 240 719 516 205 066 411 872 818 613 452 8;
  • 45) 0.296 878 095 262 354 981 240 719 516 205 066 411 872 818 613 452 8 × 2 = 0 + 0.593 756 190 524 709 962 481 439 032 410 132 823 745 637 226 905 6;
  • 46) 0.593 756 190 524 709 962 481 439 032 410 132 823 745 637 226 905 6 × 2 = 1 + 0.187 512 381 049 419 924 962 878 064 820 265 647 491 274 453 811 2;
  • 47) 0.187 512 381 049 419 924 962 878 064 820 265 647 491 274 453 811 2 × 2 = 0 + 0.375 024 762 098 839 849 925 756 129 640 531 294 982 548 907 622 4;
  • 48) 0.375 024 762 098 839 849 925 756 129 640 531 294 982 548 907 622 4 × 2 = 0 + 0.750 049 524 197 679 699 851 512 259 281 062 589 965 097 815 244 8;
  • 49) 0.750 049 524 197 679 699 851 512 259 281 062 589 965 097 815 244 8 × 2 = 1 + 0.500 099 048 395 359 399 703 024 518 562 125 179 930 195 630 489 6;
  • 50) 0.500 099 048 395 359 399 703 024 518 562 125 179 930 195 630 489 6 × 2 = 1 + 0.000 198 096 790 718 799 406 049 037 124 250 359 860 391 260 979 2;
  • 51) 0.000 198 096 790 718 799 406 049 037 124 250 359 860 391 260 979 2 × 2 = 0 + 0.000 396 193 581 437 598 812 098 074 248 500 719 720 782 521 958 4;
  • 52) 0.000 396 193 581 437 598 812 098 074 248 500 719 720 782 521 958 4 × 2 = 0 + 0.000 792 387 162 875 197 624 196 148 497 001 439 441 565 043 916 8;
  • 53) 0.000 792 387 162 875 197 624 196 148 497 001 439 441 565 043 916 8 × 2 = 0 + 0.001 584 774 325 750 395 248 392 296 994 002 878 883 130 087 833 6;
  • 54) 0.001 584 774 325 750 395 248 392 296 994 002 878 883 130 087 833 6 × 2 = 0 + 0.003 169 548 651 500 790 496 784 593 988 005 757 766 260 175 667 2;
  • 55) 0.003 169 548 651 500 790 496 784 593 988 005 757 766 260 175 667 2 × 2 = 0 + 0.006 339 097 303 001 580 993 569 187 976 011 515 532 520 351 334 4;
  • 56) 0.006 339 097 303 001 580 993 569 187 976 011 515 532 520 351 334 4 × 2 = 0 + 0.012 678 194 606 003 161 987 138 375 952 023 031 065 040 702 668 8;
  • 57) 0.012 678 194 606 003 161 987 138 375 952 023 031 065 040 702 668 8 × 2 = 0 + 0.025 356 389 212 006 323 974 276 751 904 046 062 130 081 405 337 6;
  • 58) 0.025 356 389 212 006 323 974 276 751 904 046 062 130 081 405 337 6 × 2 = 0 + 0.050 712 778 424 012 647 948 553 503 808 092 124 260 162 810 675 2;
  • 59) 0.050 712 778 424 012 647 948 553 503 808 092 124 260 162 810 675 2 × 2 = 0 + 0.101 425 556 848 025 295 897 107 007 616 184 248 520 325 621 350 4;
  • 60) 0.101 425 556 848 025 295 897 107 007 616 184 248 520 325 621 350 4 × 2 = 0 + 0.202 851 113 696 050 591 794 214 015 232 368 497 040 651 242 700 8;
  • 61) 0.202 851 113 696 050 591 794 214 015 232 368 497 040 651 242 700 8 × 2 = 0 + 0.405 702 227 392 101 183 588 428 030 464 736 994 081 302 485 401 6;
  • 62) 0.405 702 227 392 101 183 588 428 030 464 736 994 081 302 485 401 6 × 2 = 0 + 0.811 404 454 784 202 367 176 856 060 929 473 988 162 604 970 803 2;
  • 63) 0.811 404 454 784 202 367 176 856 060 929 473 988 162 604 970 803 2 × 2 = 1 + 0.622 808 909 568 404 734 353 712 121 858 947 976 325 209 941 606 4;
  • 64) 0.622 808 909 568 404 734 353 712 121 858 947 976 325 209 941 606 4 × 2 = 1 + 0.245 617 819 136 809 468 707 424 243 717 895 952 650 419 883 212 8;
  • 65) 0.245 617 819 136 809 468 707 424 243 717 895 952 650 419 883 212 8 × 2 = 0 + 0.491 235 638 273 618 937 414 848 487 435 791 905 300 839 766 425 6;
  • 66) 0.491 235 638 273 618 937 414 848 487 435 791 905 300 839 766 425 6 × 2 = 0 + 0.982 471 276 547 237 874 829 696 974 871 583 810 601 679 532 851 2;
  • 67) 0.982 471 276 547 237 874 829 696 974 871 583 810 601 679 532 851 2 × 2 = 1 + 0.964 942 553 094 475 749 659 393 949 743 167 621 203 359 065 702 4;
  • 68) 0.964 942 553 094 475 749 659 393 949 743 167 621 203 359 065 702 4 × 2 = 1 + 0.929 885 106 188 951 499 318 787 899 486 335 242 406 718 131 404 8;
  • 69) 0.929 885 106 188 951 499 318 787 899 486 335 242 406 718 131 404 8 × 2 = 1 + 0.859 770 212 377 902 998 637 575 798 972 670 484 813 436 262 809 6;
  • 70) 0.859 770 212 377 902 998 637 575 798 972 670 484 813 436 262 809 6 × 2 = 1 + 0.719 540 424 755 805 997 275 151 597 945 340 969 626 872 525 619 2;
  • 71) 0.719 540 424 755 805 997 275 151 597 945 340 969 626 872 525 619 2 × 2 = 1 + 0.439 080 849 511 611 994 550 303 195 890 681 939 253 745 051 238 4;
  • 72) 0.439 080 849 511 611 994 550 303 195 890 681 939 253 745 051 238 4 × 2 = 0 + 0.878 161 699 023 223 989 100 606 391 781 363 878 507 490 102 476 8;
  • 73) 0.878 161 699 023 223 989 100 606 391 781 363 878 507 490 102 476 8 × 2 = 1 + 0.756 323 398 046 447 978 201 212 783 562 727 757 014 980 204 953 6;
  • 74) 0.756 323 398 046 447 978 201 212 783 562 727 757 014 980 204 953 6 × 2 = 1 + 0.512 646 796 092 895 956 402 425 567 125 455 514 029 960 409 907 2;
  • 75) 0.512 646 796 092 895 956 402 425 567 125 455 514 029 960 409 907 2 × 2 = 1 + 0.025 293 592 185 791 912 804 851 134 250 911 028 059 920 819 814 4;
  • 76) 0.025 293 592 185 791 912 804 851 134 250 911 028 059 920 819 814 4 × 2 = 0 + 0.050 587 184 371 583 825 609 702 268 501 822 056 119 841 639 628 8;
  • 77) 0.050 587 184 371 583 825 609 702 268 501 822 056 119 841 639 628 8 × 2 = 0 + 0.101 174 368 743 167 651 219 404 537 003 644 112 239 683 279 257 6;
  • 78) 0.101 174 368 743 167 651 219 404 537 003 644 112 239 683 279 257 6 × 2 = 0 + 0.202 348 737 486 335 302 438 809 074 007 288 224 479 366 558 515 2;
  • 79) 0.202 348 737 486 335 302 438 809 074 007 288 224 479 366 558 515 2 × 2 = 0 + 0.404 697 474 972 670 604 877 618 148 014 576 448 958 733 117 030 4;
  • 80) 0.404 697 474 972 670 604 877 618 148 014 576 448 958 733 117 030 4 × 2 = 0 + 0.809 394 949 945 341 209 755 236 296 029 152 897 917 466 234 060 8;
  • 81) 0.809 394 949 945 341 209 755 236 296 029 152 897 917 466 234 060 8 × 2 = 1 + 0.618 789 899 890 682 419 510 472 592 058 305 795 834 932 468 121 6;
  • 82) 0.618 789 899 890 682 419 510 472 592 058 305 795 834 932 468 121 6 × 2 = 1 + 0.237 579 799 781 364 839 020 945 184 116 611 591 669 864 936 243 2;
  • 83) 0.237 579 799 781 364 839 020 945 184 116 611 591 669 864 936 243 2 × 2 = 0 + 0.475 159 599 562 729 678 041 890 368 233 223 183 339 729 872 486 4;
  • 84) 0.475 159 599 562 729 678 041 890 368 233 223 183 339 729 872 486 4 × 2 = 0 + 0.950 319 199 125 459 356 083 780 736 466 446 366 679 459 744 972 8;
  • 85) 0.950 319 199 125 459 356 083 780 736 466 446 366 679 459 744 972 8 × 2 = 1 + 0.900 638 398 250 918 712 167 561 472 932 892 733 358 919 489 945 6;
  • 86) 0.900 638 398 250 918 712 167 561 472 932 892 733 358 919 489 945 6 × 2 = 1 + 0.801 276 796 501 837 424 335 122 945 865 785 466 717 838 979 891 2;
  • 87) 0.801 276 796 501 837 424 335 122 945 865 785 466 717 838 979 891 2 × 2 = 1 + 0.602 553 593 003 674 848 670 245 891 731 570 933 435 677 959 782 4;
  • 88) 0.602 553 593 003 674 848 670 245 891 731 570 933 435 677 959 782 4 × 2 = 1 + 0.205 107 186 007 349 697 340 491 783 463 141 866 871 355 919 564 8;
  • 89) 0.205 107 186 007 349 697 340 491 783 463 141 866 871 355 919 564 8 × 2 = 0 + 0.410 214 372 014 699 394 680 983 566 926 283 733 742 711 839 129 6;
  • 90) 0.410 214 372 014 699 394 680 983 566 926 283 733 742 711 839 129 6 × 2 = 0 + 0.820 428 744 029 398 789 361 967 133 852 567 467 485 423 678 259 2;
  • 91) 0.820 428 744 029 398 789 361 967 133 852 567 467 485 423 678 259 2 × 2 = 1 + 0.640 857 488 058 797 578 723 934 267 705 134 934 970 847 356 518 4;
  • 92) 0.640 857 488 058 797 578 723 934 267 705 134 934 970 847 356 518 4 × 2 = 1 + 0.281 714 976 117 595 157 447 868 535 410 269 869 941 694 713 036 8;
  • 93) 0.281 714 976 117 595 157 447 868 535 410 269 869 941 694 713 036 8 × 2 = 0 + 0.563 429 952 235 190 314 895 737 070 820 539 739 883 389 426 073 6;

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:


0.000 000 000 000 755 840 011 110 101 101 010 100 109 999 999 995 8(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1101 0100 1100 0000 0000 0011 0011 1110 1110 0000 1100 1111 0011 0(2)

5. Positive number before normalization:

0.000 000 000 000 755 840 011 110 101 101 010 100 109 999 999 995 8(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1101 0100 1100 0000 0000 0011 0011 1110 1110 0000 1100 1111 0011 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 41 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 755 840 011 110 101 101 010 100 109 999 999 995 8(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1101 0100 1100 0000 0000 0011 0011 1110 1110 0000 1100 1111 0011 0(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1101 0100 1100 0000 0000 0011 0011 1110 1110 0000 1100 1111 0011 0(2) × 20 =

1.1010 1001 1000 0000 0000 0110 0111 1101 1100 0001 1001 1110 0110(2) × 2-41


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): -41

Mantissa (not normalized):
1.1010 1001 1000 0000 0000 0110 0111 1101 1100 0001 1001 1110 0110


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-41 + 2(11-1) - 1 =

(-41 + 1 023)(10) =

982(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;
  • 982 ÷ 2 = 491 + 0;
  • 491 ÷ 2 = 245 + 1;
  • 245 ÷ 2 = 122 + 1;
  • 122 ÷ 2 = 61 + 0;
  • 61 ÷ 2 = 30 + 1;
  • 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) =

982(10) =

011 1101 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. 1010 1001 1000 0000 0000 0110 0111 1101 1100 0001 1001 1110 0110 =

1010 1001 1000 0000 0000 0110 0111 1101 1100 0001 1001 1110 0110


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 1101 0110

Mantissa (52 bits) =
1010 1001 1000 0000 0000 0110 0111 1101 1100 0001 1001 1110 0110


Decimal number 0.000 000 000 000 755 840 011 110 101 101 010 100 109 999 999 995 8 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1101 0110 - 1010 1001 1000 0000 0000 0110 0111 1101 1100 0001 1001 1110 0110

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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