0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 854 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 854(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
0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 854(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: 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 854.

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 854 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 234 619 708;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 234 619 708 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 469 239 416;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 469 239 416 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 284 938 478 832;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 284 938 478 832 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 569 876 957 664;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 569 876 957 664 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 139 753 915 328;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 139 753 915 328 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 279 507 830 656;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 279 507 830 656 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 559 015 661 312;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 559 015 661 312 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 118 031 322 624;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 118 031 322 624 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 236 062 645 248;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 236 062 645 248 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 472 125 290 496;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 472 125 290 496 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 944 250 580 992;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 944 250 580 992 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 888 501 161 984;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 888 501 161 984 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 777 002 323 968;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 777 002 323 968 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 554 004 647 936;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 554 004 647 936 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 559 108 009 295 872;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 559 108 009 295 872 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 118 216 018 591 744;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 118 216 018 591 744 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 236 432 037 183 488;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 236 432 037 183 488 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 472 864 074 366 976;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 472 864 074 366 976 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 945 728 148 733 952;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 945 728 148 733 952 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 891 456 297 467 904;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 891 456 297 467 904 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 782 912 594 935 808;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 782 912 594 935 808 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 565 825 189 871 616;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 565 825 189 871 616 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 131 650 379 743 232;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 131 650 379 743 232 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 263 300 759 486 464;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 263 300 759 486 464 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 900 526 601 518 972 928;
  • 26) 0.963 290 534 422 579 060 379 950 051 900 526 601 518 972 928 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 801 053 203 037 945 856;
  • 27) 0.926 581 068 845 158 120 759 900 103 801 053 203 037 945 856 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 602 106 406 075 891 712;
  • 28) 0.853 162 137 690 316 241 519 800 207 602 106 406 075 891 712 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 204 212 812 151 783 424;
  • 29) 0.706 324 275 380 632 483 039 600 415 204 212 812 151 783 424 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 408 425 624 303 566 848;
  • 30) 0.412 648 550 761 264 966 079 200 830 408 425 624 303 566 848 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 816 851 248 607 133 696;
  • 31) 0.825 297 101 522 529 932 158 401 660 816 851 248 607 133 696 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 633 702 497 214 267 392;
  • 32) 0.650 594 203 045 059 864 316 803 321 633 702 497 214 267 392 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 267 404 994 428 534 784;
  • 33) 0.301 188 406 090 119 728 633 606 643 267 404 994 428 534 784 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 534 809 988 857 069 568;
  • 34) 0.602 376 812 180 239 457 267 213 286 534 809 988 857 069 568 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 069 619 977 714 139 136;
  • 35) 0.204 753 624 360 478 914 534 426 573 069 619 977 714 139 136 × 2 = 0 + 0.409 507 248 720 957 829 068 853 146 139 239 955 428 278 272;
  • 36) 0.409 507 248 720 957 829 068 853 146 139 239 955 428 278 272 × 2 = 0 + 0.819 014 497 441 915 658 137 706 292 278 479 910 856 556 544;
  • 37) 0.819 014 497 441 915 658 137 706 292 278 479 910 856 556 544 × 2 = 1 + 0.638 028 994 883 831 316 275 412 584 556 959 821 713 113 088;
  • 38) 0.638 028 994 883 831 316 275 412 584 556 959 821 713 113 088 × 2 = 1 + 0.276 057 989 767 662 632 550 825 169 113 919 643 426 226 176;
  • 39) 0.276 057 989 767 662 632 550 825 169 113 919 643 426 226 176 × 2 = 0 + 0.552 115 979 535 325 265 101 650 338 227 839 286 852 452 352;
  • 40) 0.552 115 979 535 325 265 101 650 338 227 839 286 852 452 352 × 2 = 1 + 0.104 231 959 070 650 530 203 300 676 455 678 573 704 904 704;
  • 41) 0.104 231 959 070 650 530 203 300 676 455 678 573 704 904 704 × 2 = 0 + 0.208 463 918 141 301 060 406 601 352 911 357 147 409 809 408;
  • 42) 0.208 463 918 141 301 060 406 601 352 911 357 147 409 809 408 × 2 = 0 + 0.416 927 836 282 602 120 813 202 705 822 714 294 819 618 816;
  • 43) 0.416 927 836 282 602 120 813 202 705 822 714 294 819 618 816 × 2 = 0 + 0.833 855 672 565 204 241 626 405 411 645 428 589 639 237 632;
  • 44) 0.833 855 672 565 204 241 626 405 411 645 428 589 639 237 632 × 2 = 1 + 0.667 711 345 130 408 483 252 810 823 290 857 179 278 475 264;
  • 45) 0.667 711 345 130 408 483 252 810 823 290 857 179 278 475 264 × 2 = 1 + 0.335 422 690 260 816 966 505 621 646 581 714 358 556 950 528;
  • 46) 0.335 422 690 260 816 966 505 621 646 581 714 358 556 950 528 × 2 = 0 + 0.670 845 380 521 633 933 011 243 293 163 428 717 113 901 056;
  • 47) 0.670 845 380 521 633 933 011 243 293 163 428 717 113 901 056 × 2 = 1 + 0.341 690 761 043 267 866 022 486 586 326 857 434 227 802 112;
  • 48) 0.341 690 761 043 267 866 022 486 586 326 857 434 227 802 112 × 2 = 0 + 0.683 381 522 086 535 732 044 973 172 653 714 868 455 604 224;
  • 49) 0.683 381 522 086 535 732 044 973 172 653 714 868 455 604 224 × 2 = 1 + 0.366 763 044 173 071 464 089 946 345 307 429 736 911 208 448;
  • 50) 0.366 763 044 173 071 464 089 946 345 307 429 736 911 208 448 × 2 = 0 + 0.733 526 088 346 142 928 179 892 690 614 859 473 822 416 896;
  • 51) 0.733 526 088 346 142 928 179 892 690 614 859 473 822 416 896 × 2 = 1 + 0.467 052 176 692 285 856 359 785 381 229 718 947 644 833 792;
  • 52) 0.467 052 176 692 285 856 359 785 381 229 718 947 644 833 792 × 2 = 0 + 0.934 104 353 384 571 712 719 570 762 459 437 895 289 667 584;
  • 53) 0.934 104 353 384 571 712 719 570 762 459 437 895 289 667 584 × 2 = 1 + 0.868 208 706 769 143 425 439 141 524 918 875 790 579 335 168;
  • 54) 0.868 208 706 769 143 425 439 141 524 918 875 790 579 335 168 × 2 = 1 + 0.736 417 413 538 286 850 878 283 049 837 751 581 158 670 336;
  • 55) 0.736 417 413 538 286 850 878 283 049 837 751 581 158 670 336 × 2 = 1 + 0.472 834 827 076 573 701 756 566 099 675 503 162 317 340 672;
  • 56) 0.472 834 827 076 573 701 756 566 099 675 503 162 317 340 672 × 2 = 0 + 0.945 669 654 153 147 403 513 132 199 351 006 324 634 681 344;
  • 57) 0.945 669 654 153 147 403 513 132 199 351 006 324 634 681 344 × 2 = 1 + 0.891 339 308 306 294 807 026 264 398 702 012 649 269 362 688;
  • 58) 0.891 339 308 306 294 807 026 264 398 702 012 649 269 362 688 × 2 = 1 + 0.782 678 616 612 589 614 052 528 797 404 025 298 538 725 376;
  • 59) 0.782 678 616 612 589 614 052 528 797 404 025 298 538 725 376 × 2 = 1 + 0.565 357 233 225 179 228 105 057 594 808 050 597 077 450 752;
  • 60) 0.565 357 233 225 179 228 105 057 594 808 050 597 077 450 752 × 2 = 1 + 0.130 714 466 450 358 456 210 115 189 616 101 194 154 901 504;
  • 61) 0.130 714 466 450 358 456 210 115 189 616 101 194 154 901 504 × 2 = 0 + 0.261 428 932 900 716 912 420 230 379 232 202 388 309 803 008;
  • 62) 0.261 428 932 900 716 912 420 230 379 232 202 388 309 803 008 × 2 = 0 + 0.522 857 865 801 433 824 840 460 758 464 404 776 619 606 016;
  • 63) 0.522 857 865 801 433 824 840 460 758 464 404 776 619 606 016 × 2 = 1 + 0.045 715 731 602 867 649 680 921 516 928 809 553 239 212 032;
  • 64) 0.045 715 731 602 867 649 680 921 516 928 809 553 239 212 032 × 2 = 0 + 0.091 431 463 205 735 299 361 843 033 857 619 106 478 424 064;
  • 65) 0.091 431 463 205 735 299 361 843 033 857 619 106 478 424 064 × 2 = 0 + 0.182 862 926 411 470 598 723 686 067 715 238 212 956 848 128;
  • 66) 0.182 862 926 411 470 598 723 686 067 715 238 212 956 848 128 × 2 = 0 + 0.365 725 852 822 941 197 447 372 135 430 476 425 913 696 256;
  • 67) 0.365 725 852 822 941 197 447 372 135 430 476 425 913 696 256 × 2 = 0 + 0.731 451 705 645 882 394 894 744 270 860 952 851 827 392 512;
  • 68) 0.731 451 705 645 882 394 894 744 270 860 952 851 827 392 512 × 2 = 1 + 0.462 903 411 291 764 789 789 488 541 721 905 703 654 785 024;

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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 854(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:

0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 854(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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 854(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 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 854 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