0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 778 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 778(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
0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 778(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: 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 778.

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 778 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 234 619 556;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 234 619 556 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 469 239 112;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 469 239 112 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 284 938 478 224;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 284 938 478 224 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 569 876 956 448;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 569 876 956 448 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 139 753 912 896;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 139 753 912 896 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 279 507 825 792;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 279 507 825 792 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 559 015 651 584;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 559 015 651 584 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 118 031 303 168;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 118 031 303 168 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 236 062 606 336;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 236 062 606 336 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 472 125 212 672;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 472 125 212 672 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 944 250 425 344;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 944 250 425 344 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 888 500 850 688;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 888 500 850 688 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 777 001 701 376;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 777 001 701 376 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 554 003 402 752;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 554 003 402 752 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 559 108 006 805 504;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 559 108 006 805 504 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 118 216 013 611 008;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 118 216 013 611 008 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 236 432 027 222 016;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 236 432 027 222 016 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 472 864 054 444 032;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 472 864 054 444 032 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 945 728 108 888 064;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 945 728 108 888 064 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 891 456 217 776 128;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 891 456 217 776 128 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 782 912 435 552 256;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 782 912 435 552 256 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 565 824 871 104 512;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 565 824 871 104 512 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 131 649 742 209 024;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 131 649 742 209 024 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 263 299 484 418 048;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 263 299 484 418 048 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 900 526 598 968 836 096;
  • 26) 0.963 290 534 422 579 060 379 950 051 900 526 598 968 836 096 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 801 053 197 937 672 192;
  • 27) 0.926 581 068 845 158 120 759 900 103 801 053 197 937 672 192 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 602 106 395 875 344 384;
  • 28) 0.853 162 137 690 316 241 519 800 207 602 106 395 875 344 384 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 204 212 791 750 688 768;
  • 29) 0.706 324 275 380 632 483 039 600 415 204 212 791 750 688 768 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 408 425 583 501 377 536;
  • 30) 0.412 648 550 761 264 966 079 200 830 408 425 583 501 377 536 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 816 851 167 002 755 072;
  • 31) 0.825 297 101 522 529 932 158 401 660 816 851 167 002 755 072 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 633 702 334 005 510 144;
  • 32) 0.650 594 203 045 059 864 316 803 321 633 702 334 005 510 144 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 267 404 668 011 020 288;
  • 33) 0.301 188 406 090 119 728 633 606 643 267 404 668 011 020 288 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 534 809 336 022 040 576;
  • 34) 0.602 376 812 180 239 457 267 213 286 534 809 336 022 040 576 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 069 618 672 044 081 152;
  • 35) 0.204 753 624 360 478 914 534 426 573 069 618 672 044 081 152 × 2 = 0 + 0.409 507 248 720 957 829 068 853 146 139 237 344 088 162 304;
  • 36) 0.409 507 248 720 957 829 068 853 146 139 237 344 088 162 304 × 2 = 0 + 0.819 014 497 441 915 658 137 706 292 278 474 688 176 324 608;
  • 37) 0.819 014 497 441 915 658 137 706 292 278 474 688 176 324 608 × 2 = 1 + 0.638 028 994 883 831 316 275 412 584 556 949 376 352 649 216;
  • 38) 0.638 028 994 883 831 316 275 412 584 556 949 376 352 649 216 × 2 = 1 + 0.276 057 989 767 662 632 550 825 169 113 898 752 705 298 432;
  • 39) 0.276 057 989 767 662 632 550 825 169 113 898 752 705 298 432 × 2 = 0 + 0.552 115 979 535 325 265 101 650 338 227 797 505 410 596 864;
  • 40) 0.552 115 979 535 325 265 101 650 338 227 797 505 410 596 864 × 2 = 1 + 0.104 231 959 070 650 530 203 300 676 455 595 010 821 193 728;
  • 41) 0.104 231 959 070 650 530 203 300 676 455 595 010 821 193 728 × 2 = 0 + 0.208 463 918 141 301 060 406 601 352 911 190 021 642 387 456;
  • 42) 0.208 463 918 141 301 060 406 601 352 911 190 021 642 387 456 × 2 = 0 + 0.416 927 836 282 602 120 813 202 705 822 380 043 284 774 912;
  • 43) 0.416 927 836 282 602 120 813 202 705 822 380 043 284 774 912 × 2 = 0 + 0.833 855 672 565 204 241 626 405 411 644 760 086 569 549 824;
  • 44) 0.833 855 672 565 204 241 626 405 411 644 760 086 569 549 824 × 2 = 1 + 0.667 711 345 130 408 483 252 810 823 289 520 173 139 099 648;
  • 45) 0.667 711 345 130 408 483 252 810 823 289 520 173 139 099 648 × 2 = 1 + 0.335 422 690 260 816 966 505 621 646 579 040 346 278 199 296;
  • 46) 0.335 422 690 260 816 966 505 621 646 579 040 346 278 199 296 × 2 = 0 + 0.670 845 380 521 633 933 011 243 293 158 080 692 556 398 592;
  • 47) 0.670 845 380 521 633 933 011 243 293 158 080 692 556 398 592 × 2 = 1 + 0.341 690 761 043 267 866 022 486 586 316 161 385 112 797 184;
  • 48) 0.341 690 761 043 267 866 022 486 586 316 161 385 112 797 184 × 2 = 0 + 0.683 381 522 086 535 732 044 973 172 632 322 770 225 594 368;
  • 49) 0.683 381 522 086 535 732 044 973 172 632 322 770 225 594 368 × 2 = 1 + 0.366 763 044 173 071 464 089 946 345 264 645 540 451 188 736;
  • 50) 0.366 763 044 173 071 464 089 946 345 264 645 540 451 188 736 × 2 = 0 + 0.733 526 088 346 142 928 179 892 690 529 291 080 902 377 472;
  • 51) 0.733 526 088 346 142 928 179 892 690 529 291 080 902 377 472 × 2 = 1 + 0.467 052 176 692 285 856 359 785 381 058 582 161 804 754 944;
  • 52) 0.467 052 176 692 285 856 359 785 381 058 582 161 804 754 944 × 2 = 0 + 0.934 104 353 384 571 712 719 570 762 117 164 323 609 509 888;
  • 53) 0.934 104 353 384 571 712 719 570 762 117 164 323 609 509 888 × 2 = 1 + 0.868 208 706 769 143 425 439 141 524 234 328 647 219 019 776;
  • 54) 0.868 208 706 769 143 425 439 141 524 234 328 647 219 019 776 × 2 = 1 + 0.736 417 413 538 286 850 878 283 048 468 657 294 438 039 552;
  • 55) 0.736 417 413 538 286 850 878 283 048 468 657 294 438 039 552 × 2 = 1 + 0.472 834 827 076 573 701 756 566 096 937 314 588 876 079 104;
  • 56) 0.472 834 827 076 573 701 756 566 096 937 314 588 876 079 104 × 2 = 0 + 0.945 669 654 153 147 403 513 132 193 874 629 177 752 158 208;
  • 57) 0.945 669 654 153 147 403 513 132 193 874 629 177 752 158 208 × 2 = 1 + 0.891 339 308 306 294 807 026 264 387 749 258 355 504 316 416;
  • 58) 0.891 339 308 306 294 807 026 264 387 749 258 355 504 316 416 × 2 = 1 + 0.782 678 616 612 589 614 052 528 775 498 516 711 008 632 832;
  • 59) 0.782 678 616 612 589 614 052 528 775 498 516 711 008 632 832 × 2 = 1 + 0.565 357 233 225 179 228 105 057 550 997 033 422 017 265 664;
  • 60) 0.565 357 233 225 179 228 105 057 550 997 033 422 017 265 664 × 2 = 1 + 0.130 714 466 450 358 456 210 115 101 994 066 844 034 531 328;
  • 61) 0.130 714 466 450 358 456 210 115 101 994 066 844 034 531 328 × 2 = 0 + 0.261 428 932 900 716 912 420 230 203 988 133 688 069 062 656;
  • 62) 0.261 428 932 900 716 912 420 230 203 988 133 688 069 062 656 × 2 = 0 + 0.522 857 865 801 433 824 840 460 407 976 267 376 138 125 312;
  • 63) 0.522 857 865 801 433 824 840 460 407 976 267 376 138 125 312 × 2 = 1 + 0.045 715 731 602 867 649 680 920 815 952 534 752 276 250 624;
  • 64) 0.045 715 731 602 867 649 680 920 815 952 534 752 276 250 624 × 2 = 0 + 0.091 431 463 205 735 299 361 841 631 905 069 504 552 501 248;
  • 65) 0.091 431 463 205 735 299 361 841 631 905 069 504 552 501 248 × 2 = 0 + 0.182 862 926 411 470 598 723 683 263 810 139 009 105 002 496;
  • 66) 0.182 862 926 411 470 598 723 683 263 810 139 009 105 002 496 × 2 = 0 + 0.365 725 852 822 941 197 447 366 527 620 278 018 210 004 992;
  • 67) 0.365 725 852 822 941 197 447 366 527 620 278 018 210 004 992 × 2 = 0 + 0.731 451 705 645 882 394 894 733 055 240 556 036 420 009 984;
  • 68) 0.731 451 705 645 882 394 894 733 055 240 556 036 420 009 984 × 2 = 1 + 0.462 903 411 291 764 789 789 466 110 481 112 072 840 019 968;

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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 778(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:

0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 778(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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 778(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 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 778 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