0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 338 4 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 338 4(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
0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 338 4(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: 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 338 4.

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 338 4 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 234 618 868 676 8;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 234 618 868 676 8 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 469 237 737 353 6;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 469 237 737 353 6 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 284 938 475 474 707 2;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 284 938 475 474 707 2 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 569 876 950 949 414 4;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 569 876 950 949 414 4 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 139 753 901 898 828 8;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 139 753 901 898 828 8 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 279 507 803 797 657 6;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 279 507 803 797 657 6 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 559 015 607 595 315 2;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 559 015 607 595 315 2 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 118 031 215 190 630 4;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 118 031 215 190 630 4 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 236 062 430 381 260 8;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 236 062 430 381 260 8 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 472 124 860 762 521 6;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 472 124 860 762 521 6 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 944 249 721 525 043 2;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 944 249 721 525 043 2 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 888 499 443 050 086 4;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 888 499 443 050 086 4 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 776 998 886 100 172 8;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 776 998 886 100 172 8 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 553 997 772 200 345 6;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 553 997 772 200 345 6 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 559 107 995 544 400 691 2;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 559 107 995 544 400 691 2 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 118 215 991 088 801 382 4;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 118 215 991 088 801 382 4 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 236 431 982 177 602 764 8;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 236 431 982 177 602 764 8 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 472 863 964 355 205 529 6;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 472 863 964 355 205 529 6 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 945 727 928 710 411 059 2;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 945 727 928 710 411 059 2 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 891 455 857 420 822 118 4;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 891 455 857 420 822 118 4 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 782 911 714 841 644 236 8;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 782 911 714 841 644 236 8 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 565 823 429 683 288 473 6;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 565 823 429 683 288 473 6 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 131 646 859 366 576 947 2;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 131 646 859 366 576 947 2 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 263 293 718 733 153 894 4;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 263 293 718 733 153 894 4 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 900 526 587 437 466 307 788 8;
  • 26) 0.963 290 534 422 579 060 379 950 051 900 526 587 437 466 307 788 8 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 801 053 174 874 932 615 577 6;
  • 27) 0.926 581 068 845 158 120 759 900 103 801 053 174 874 932 615 577 6 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 602 106 349 749 865 231 155 2;
  • 28) 0.853 162 137 690 316 241 519 800 207 602 106 349 749 865 231 155 2 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 204 212 699 499 730 462 310 4;
  • 29) 0.706 324 275 380 632 483 039 600 415 204 212 699 499 730 462 310 4 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 408 425 398 999 460 924 620 8;
  • 30) 0.412 648 550 761 264 966 079 200 830 408 425 398 999 460 924 620 8 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 816 850 797 998 921 849 241 6;
  • 31) 0.825 297 101 522 529 932 158 401 660 816 850 797 998 921 849 241 6 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 633 701 595 997 843 698 483 2;
  • 32) 0.650 594 203 045 059 864 316 803 321 633 701 595 997 843 698 483 2 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 267 403 191 995 687 396 966 4;
  • 33) 0.301 188 406 090 119 728 633 606 643 267 403 191 995 687 396 966 4 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 534 806 383 991 374 793 932 8;
  • 34) 0.602 376 812 180 239 457 267 213 286 534 806 383 991 374 793 932 8 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 069 612 767 982 749 587 865 6;
  • 35) 0.204 753 624 360 478 914 534 426 573 069 612 767 982 749 587 865 6 × 2 = 0 + 0.409 507 248 720 957 829 068 853 146 139 225 535 965 499 175 731 2;
  • 36) 0.409 507 248 720 957 829 068 853 146 139 225 535 965 499 175 731 2 × 2 = 0 + 0.819 014 497 441 915 658 137 706 292 278 451 071 930 998 351 462 4;
  • 37) 0.819 014 497 441 915 658 137 706 292 278 451 071 930 998 351 462 4 × 2 = 1 + 0.638 028 994 883 831 316 275 412 584 556 902 143 861 996 702 924 8;
  • 38) 0.638 028 994 883 831 316 275 412 584 556 902 143 861 996 702 924 8 × 2 = 1 + 0.276 057 989 767 662 632 550 825 169 113 804 287 723 993 405 849 6;
  • 39) 0.276 057 989 767 662 632 550 825 169 113 804 287 723 993 405 849 6 × 2 = 0 + 0.552 115 979 535 325 265 101 650 338 227 608 575 447 986 811 699 2;
  • 40) 0.552 115 979 535 325 265 101 650 338 227 608 575 447 986 811 699 2 × 2 = 1 + 0.104 231 959 070 650 530 203 300 676 455 217 150 895 973 623 398 4;
  • 41) 0.104 231 959 070 650 530 203 300 676 455 217 150 895 973 623 398 4 × 2 = 0 + 0.208 463 918 141 301 060 406 601 352 910 434 301 791 947 246 796 8;
  • 42) 0.208 463 918 141 301 060 406 601 352 910 434 301 791 947 246 796 8 × 2 = 0 + 0.416 927 836 282 602 120 813 202 705 820 868 603 583 894 493 593 6;
  • 43) 0.416 927 836 282 602 120 813 202 705 820 868 603 583 894 493 593 6 × 2 = 0 + 0.833 855 672 565 204 241 626 405 411 641 737 207 167 788 987 187 2;
  • 44) 0.833 855 672 565 204 241 626 405 411 641 737 207 167 788 987 187 2 × 2 = 1 + 0.667 711 345 130 408 483 252 810 823 283 474 414 335 577 974 374 4;
  • 45) 0.667 711 345 130 408 483 252 810 823 283 474 414 335 577 974 374 4 × 2 = 1 + 0.335 422 690 260 816 966 505 621 646 566 948 828 671 155 948 748 8;
  • 46) 0.335 422 690 260 816 966 505 621 646 566 948 828 671 155 948 748 8 × 2 = 0 + 0.670 845 380 521 633 933 011 243 293 133 897 657 342 311 897 497 6;
  • 47) 0.670 845 380 521 633 933 011 243 293 133 897 657 342 311 897 497 6 × 2 = 1 + 0.341 690 761 043 267 866 022 486 586 267 795 314 684 623 794 995 2;
  • 48) 0.341 690 761 043 267 866 022 486 586 267 795 314 684 623 794 995 2 × 2 = 0 + 0.683 381 522 086 535 732 044 973 172 535 590 629 369 247 589 990 4;
  • 49) 0.683 381 522 086 535 732 044 973 172 535 590 629 369 247 589 990 4 × 2 = 1 + 0.366 763 044 173 071 464 089 946 345 071 181 258 738 495 179 980 8;
  • 50) 0.366 763 044 173 071 464 089 946 345 071 181 258 738 495 179 980 8 × 2 = 0 + 0.733 526 088 346 142 928 179 892 690 142 362 517 476 990 359 961 6;
  • 51) 0.733 526 088 346 142 928 179 892 690 142 362 517 476 990 359 961 6 × 2 = 1 + 0.467 052 176 692 285 856 359 785 380 284 725 034 953 980 719 923 2;
  • 52) 0.467 052 176 692 285 856 359 785 380 284 725 034 953 980 719 923 2 × 2 = 0 + 0.934 104 353 384 571 712 719 570 760 569 450 069 907 961 439 846 4;
  • 53) 0.934 104 353 384 571 712 719 570 760 569 450 069 907 961 439 846 4 × 2 = 1 + 0.868 208 706 769 143 425 439 141 521 138 900 139 815 922 879 692 8;
  • 54) 0.868 208 706 769 143 425 439 141 521 138 900 139 815 922 879 692 8 × 2 = 1 + 0.736 417 413 538 286 850 878 283 042 277 800 279 631 845 759 385 6;
  • 55) 0.736 417 413 538 286 850 878 283 042 277 800 279 631 845 759 385 6 × 2 = 1 + 0.472 834 827 076 573 701 756 566 084 555 600 559 263 691 518 771 2;
  • 56) 0.472 834 827 076 573 701 756 566 084 555 600 559 263 691 518 771 2 × 2 = 0 + 0.945 669 654 153 147 403 513 132 169 111 201 118 527 383 037 542 4;
  • 57) 0.945 669 654 153 147 403 513 132 169 111 201 118 527 383 037 542 4 × 2 = 1 + 0.891 339 308 306 294 807 026 264 338 222 402 237 054 766 075 084 8;
  • 58) 0.891 339 308 306 294 807 026 264 338 222 402 237 054 766 075 084 8 × 2 = 1 + 0.782 678 616 612 589 614 052 528 676 444 804 474 109 532 150 169 6;
  • 59) 0.782 678 616 612 589 614 052 528 676 444 804 474 109 532 150 169 6 × 2 = 1 + 0.565 357 233 225 179 228 105 057 352 889 608 948 219 064 300 339 2;
  • 60) 0.565 357 233 225 179 228 105 057 352 889 608 948 219 064 300 339 2 × 2 = 1 + 0.130 714 466 450 358 456 210 114 705 779 217 896 438 128 600 678 4;
  • 61) 0.130 714 466 450 358 456 210 114 705 779 217 896 438 128 600 678 4 × 2 = 0 + 0.261 428 932 900 716 912 420 229 411 558 435 792 876 257 201 356 8;
  • 62) 0.261 428 932 900 716 912 420 229 411 558 435 792 876 257 201 356 8 × 2 = 0 + 0.522 857 865 801 433 824 840 458 823 116 871 585 752 514 402 713 6;
  • 63) 0.522 857 865 801 433 824 840 458 823 116 871 585 752 514 402 713 6 × 2 = 1 + 0.045 715 731 602 867 649 680 917 646 233 743 171 505 028 805 427 2;
  • 64) 0.045 715 731 602 867 649 680 917 646 233 743 171 505 028 805 427 2 × 2 = 0 + 0.091 431 463 205 735 299 361 835 292 467 486 343 010 057 610 854 4;
  • 65) 0.091 431 463 205 735 299 361 835 292 467 486 343 010 057 610 854 4 × 2 = 0 + 0.182 862 926 411 470 598 723 670 584 934 972 686 020 115 221 708 8;
  • 66) 0.182 862 926 411 470 598 723 670 584 934 972 686 020 115 221 708 8 × 2 = 0 + 0.365 725 852 822 941 197 447 341 169 869 945 372 040 230 443 417 6;
  • 67) 0.365 725 852 822 941 197 447 341 169 869 945 372 040 230 443 417 6 × 2 = 0 + 0.731 451 705 645 882 394 894 682 339 739 890 744 080 460 886 835 2;
  • 68) 0.731 451 705 645 882 394 894 682 339 739 890 744 080 460 886 835 2 × 2 = 1 + 0.462 903 411 291 764 789 789 364 679 479 781 488 160 921 773 670 4;

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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 338 4(10) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2)

5. Positive number before normalization:

0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 338 4(10) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 16 positions to the right, so that only one non zero digit remains to the left of it:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 338 4(10) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2) × 20 =

1.0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2) × 2-16


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): -16

Mantissa (not normalized):
1.0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-16 + 2(11-1) - 1 =

(-16 + 1 023)(10) =

1 007(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;
  • 1 007 ÷ 2 = 503 + 1;
  • 503 ÷ 2 = 251 + 1;
  • 251 ÷ 2 = 125 + 1;
  • 125 ÷ 2 = 62 + 1;
  • 62 ÷ 2 = 31 + 0;
  • 31 ÷ 2 = 15 + 1;
  • 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) =

1007(10) =

011 1110 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. 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001 =

0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001


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 1110 1111

Mantissa (52 bits) =
0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001


Decimal number 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 338 4 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1110 1111 - 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001

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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