0.000 020 830 729 321 671 205 134 999 154 509 660 726 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 726(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
0.000 020 830 729 321 671 205 134 999 154 509 660 726(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: 0.000 020 830 729 321 671 205 134 999 154 509 660 726.

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 726 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 452;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 452 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 904;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 904 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 285 808;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 285 808 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 571 616;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 571 616 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 143 232;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 143 232 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 286 464;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 286 464 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 572 928;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 572 928 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 145 856;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 145 856 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 291 712;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 291 712 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 583 424;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 583 424 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 785 166 848;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 785 166 848 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 570 333 696;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 570 333 696 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 140 667 392;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 140 667 392 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 281 334 784;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 281 334 784 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 562 669 568;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 562 669 568 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 125 339 136;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 125 339 136 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 250 678 272;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 250 678 272 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 501 356 544;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 501 356 544 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 561 002 713 088;
  • 20) 0.921 301 414 600 352 797 818 436 719 561 002 713 088 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 122 005 426 176;
  • 21) 0.842 602 829 200 705 595 636 873 439 122 005 426 176 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 244 010 852 352;
  • 22) 0.685 205 658 401 411 191 273 746 878 244 010 852 352 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 488 021 704 704;
  • 23) 0.370 411 316 802 822 382 547 493 756 488 021 704 704 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 976 043 409 408;
  • 24) 0.740 822 633 605 644 765 094 987 512 976 043 409 408 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 952 086 818 816;
  • 25) 0.481 645 267 211 289 530 189 975 025 952 086 818 816 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 904 173 637 632;
  • 26) 0.963 290 534 422 579 060 379 950 051 904 173 637 632 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 808 347 275 264;
  • 27) 0.926 581 068 845 158 120 759 900 103 808 347 275 264 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 616 694 550 528;
  • 28) 0.853 162 137 690 316 241 519 800 207 616 694 550 528 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 233 389 101 056;
  • 29) 0.706 324 275 380 632 483 039 600 415 233 389 101 056 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 466 778 202 112;
  • 30) 0.412 648 550 761 264 966 079 200 830 466 778 202 112 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 933 556 404 224;
  • 31) 0.825 297 101 522 529 932 158 401 660 933 556 404 224 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 867 112 808 448;
  • 32) 0.650 594 203 045 059 864 316 803 321 867 112 808 448 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 734 225 616 896;
  • 33) 0.301 188 406 090 119 728 633 606 643 734 225 616 896 × 2 = 0 + 0.602 376 812 180 239 457 267 213 287 468 451 233 792;
  • 34) 0.602 376 812 180 239 457 267 213 287 468 451 233 792 × 2 = 1 + 0.204 753 624 360 478 914 534 426 574 936 902 467 584;
  • 35) 0.204 753 624 360 478 914 534 426 574 936 902 467 584 × 2 = 0 + 0.409 507 248 720 957 829 068 853 149 873 804 935 168;
  • 36) 0.409 507 248 720 957 829 068 853 149 873 804 935 168 × 2 = 0 + 0.819 014 497 441 915 658 137 706 299 747 609 870 336;
  • 37) 0.819 014 497 441 915 658 137 706 299 747 609 870 336 × 2 = 1 + 0.638 028 994 883 831 316 275 412 599 495 219 740 672;
  • 38) 0.638 028 994 883 831 316 275 412 599 495 219 740 672 × 2 = 1 + 0.276 057 989 767 662 632 550 825 198 990 439 481 344;
  • 39) 0.276 057 989 767 662 632 550 825 198 990 439 481 344 × 2 = 0 + 0.552 115 979 535 325 265 101 650 397 980 878 962 688;
  • 40) 0.552 115 979 535 325 265 101 650 397 980 878 962 688 × 2 = 1 + 0.104 231 959 070 650 530 203 300 795 961 757 925 376;
  • 41) 0.104 231 959 070 650 530 203 300 795 961 757 925 376 × 2 = 0 + 0.208 463 918 141 301 060 406 601 591 923 515 850 752;
  • 42) 0.208 463 918 141 301 060 406 601 591 923 515 850 752 × 2 = 0 + 0.416 927 836 282 602 120 813 203 183 847 031 701 504;
  • 43) 0.416 927 836 282 602 120 813 203 183 847 031 701 504 × 2 = 0 + 0.833 855 672 565 204 241 626 406 367 694 063 403 008;
  • 44) 0.833 855 672 565 204 241 626 406 367 694 063 403 008 × 2 = 1 + 0.667 711 345 130 408 483 252 812 735 388 126 806 016;
  • 45) 0.667 711 345 130 408 483 252 812 735 388 126 806 016 × 2 = 1 + 0.335 422 690 260 816 966 505 625 470 776 253 612 032;
  • 46) 0.335 422 690 260 816 966 505 625 470 776 253 612 032 × 2 = 0 + 0.670 845 380 521 633 933 011 250 941 552 507 224 064;
  • 47) 0.670 845 380 521 633 933 011 250 941 552 507 224 064 × 2 = 1 + 0.341 690 761 043 267 866 022 501 883 105 014 448 128;
  • 48) 0.341 690 761 043 267 866 022 501 883 105 014 448 128 × 2 = 0 + 0.683 381 522 086 535 732 045 003 766 210 028 896 256;
  • 49) 0.683 381 522 086 535 732 045 003 766 210 028 896 256 × 2 = 1 + 0.366 763 044 173 071 464 090 007 532 420 057 792 512;
  • 50) 0.366 763 044 173 071 464 090 007 532 420 057 792 512 × 2 = 0 + 0.733 526 088 346 142 928 180 015 064 840 115 585 024;
  • 51) 0.733 526 088 346 142 928 180 015 064 840 115 585 024 × 2 = 1 + 0.467 052 176 692 285 856 360 030 129 680 231 170 048;
  • 52) 0.467 052 176 692 285 856 360 030 129 680 231 170 048 × 2 = 0 + 0.934 104 353 384 571 712 720 060 259 360 462 340 096;
  • 53) 0.934 104 353 384 571 712 720 060 259 360 462 340 096 × 2 = 1 + 0.868 208 706 769 143 425 440 120 518 720 924 680 192;
  • 54) 0.868 208 706 769 143 425 440 120 518 720 924 680 192 × 2 = 1 + 0.736 417 413 538 286 850 880 241 037 441 849 360 384;
  • 55) 0.736 417 413 538 286 850 880 241 037 441 849 360 384 × 2 = 1 + 0.472 834 827 076 573 701 760 482 074 883 698 720 768;
  • 56) 0.472 834 827 076 573 701 760 482 074 883 698 720 768 × 2 = 0 + 0.945 669 654 153 147 403 520 964 149 767 397 441 536;
  • 57) 0.945 669 654 153 147 403 520 964 149 767 397 441 536 × 2 = 1 + 0.891 339 308 306 294 807 041 928 299 534 794 883 072;
  • 58) 0.891 339 308 306 294 807 041 928 299 534 794 883 072 × 2 = 1 + 0.782 678 616 612 589 614 083 856 599 069 589 766 144;
  • 59) 0.782 678 616 612 589 614 083 856 599 069 589 766 144 × 2 = 1 + 0.565 357 233 225 179 228 167 713 198 139 179 532 288;
  • 60) 0.565 357 233 225 179 228 167 713 198 139 179 532 288 × 2 = 1 + 0.130 714 466 450 358 456 335 426 396 278 359 064 576;
  • 61) 0.130 714 466 450 358 456 335 426 396 278 359 064 576 × 2 = 0 + 0.261 428 932 900 716 912 670 852 792 556 718 129 152;
  • 62) 0.261 428 932 900 716 912 670 852 792 556 718 129 152 × 2 = 0 + 0.522 857 865 801 433 825 341 705 585 113 436 258 304;
  • 63) 0.522 857 865 801 433 825 341 705 585 113 436 258 304 × 2 = 1 + 0.045 715 731 602 867 650 683 411 170 226 872 516 608;
  • 64) 0.045 715 731 602 867 650 683 411 170 226 872 516 608 × 2 = 0 + 0.091 431 463 205 735 301 366 822 340 453 745 033 216;
  • 65) 0.091 431 463 205 735 301 366 822 340 453 745 033 216 × 2 = 0 + 0.182 862 926 411 470 602 733 644 680 907 490 066 432;
  • 66) 0.182 862 926 411 470 602 733 644 680 907 490 066 432 × 2 = 0 + 0.365 725 852 822 941 205 467 289 361 814 980 132 864;
  • 67) 0.365 725 852 822 941 205 467 289 361 814 980 132 864 × 2 = 0 + 0.731 451 705 645 882 410 934 578 723 629 960 265 728;
  • 68) 0.731 451 705 645 882 410 934 578 723 629 960 265 728 × 2 = 1 + 0.462 903 411 291 764 821 869 157 447 259 920 531 456;

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:


0.000 020 830 729 321 671 205 134 999 154 509 660 726(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:

0.000 020 830 729 321 671 205 134 999 154 509 660 726(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:


0.000 020 830 729 321 671 205 134 999 154 509 660 726(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 0.000 020 830 729 321 671 205 134 999 154 509 660 726 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