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

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

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 778 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 556;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 556 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 643 112;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 643 112 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 286 224;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 286 224 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 572 448;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 572 448 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 144 896;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 144 896 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 289 792;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 289 792 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 579 584;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 579 584 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 159 168;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 159 168 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 318 336;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 318 336 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 636 672;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 636 672 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 785 273 344;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 785 273 344 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 570 546 688;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 570 546 688 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 141 093 376;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 141 093 376 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 282 186 752;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 282 186 752 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 564 373 504;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 564 373 504 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 128 747 008;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 128 747 008 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 257 494 016;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 257 494 016 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 514 988 032;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 514 988 032 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 561 029 976 064;
  • 20) 0.921 301 414 600 352 797 818 436 719 561 029 976 064 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 122 059 952 128;
  • 21) 0.842 602 829 200 705 595 636 873 439 122 059 952 128 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 244 119 904 256;
  • 22) 0.685 205 658 401 411 191 273 746 878 244 119 904 256 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 488 239 808 512;
  • 23) 0.370 411 316 802 822 382 547 493 756 488 239 808 512 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 976 479 617 024;
  • 24) 0.740 822 633 605 644 765 094 987 512 976 479 617 024 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 952 959 234 048;
  • 25) 0.481 645 267 211 289 530 189 975 025 952 959 234 048 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 905 918 468 096;
  • 26) 0.963 290 534 422 579 060 379 950 051 905 918 468 096 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 811 836 936 192;
  • 27) 0.926 581 068 845 158 120 759 900 103 811 836 936 192 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 623 673 872 384;
  • 28) 0.853 162 137 690 316 241 519 800 207 623 673 872 384 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 247 347 744 768;
  • 29) 0.706 324 275 380 632 483 039 600 415 247 347 744 768 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 494 695 489 536;
  • 30) 0.412 648 550 761 264 966 079 200 830 494 695 489 536 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 989 390 979 072;
  • 31) 0.825 297 101 522 529 932 158 401 660 989 390 979 072 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 978 781 958 144;
  • 32) 0.650 594 203 045 059 864 316 803 321 978 781 958 144 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 957 563 916 288;
  • 33) 0.301 188 406 090 119 728 633 606 643 957 563 916 288 × 2 = 0 + 0.602 376 812 180 239 457 267 213 287 915 127 832 576;
  • 34) 0.602 376 812 180 239 457 267 213 287 915 127 832 576 × 2 = 1 + 0.204 753 624 360 478 914 534 426 575 830 255 665 152;
  • 35) 0.204 753 624 360 478 914 534 426 575 830 255 665 152 × 2 = 0 + 0.409 507 248 720 957 829 068 853 151 660 511 330 304;
  • 36) 0.409 507 248 720 957 829 068 853 151 660 511 330 304 × 2 = 0 + 0.819 014 497 441 915 658 137 706 303 321 022 660 608;
  • 37) 0.819 014 497 441 915 658 137 706 303 321 022 660 608 × 2 = 1 + 0.638 028 994 883 831 316 275 412 606 642 045 321 216;
  • 38) 0.638 028 994 883 831 316 275 412 606 642 045 321 216 × 2 = 1 + 0.276 057 989 767 662 632 550 825 213 284 090 642 432;
  • 39) 0.276 057 989 767 662 632 550 825 213 284 090 642 432 × 2 = 0 + 0.552 115 979 535 325 265 101 650 426 568 181 284 864;
  • 40) 0.552 115 979 535 325 265 101 650 426 568 181 284 864 × 2 = 1 + 0.104 231 959 070 650 530 203 300 853 136 362 569 728;
  • 41) 0.104 231 959 070 650 530 203 300 853 136 362 569 728 × 2 = 0 + 0.208 463 918 141 301 060 406 601 706 272 725 139 456;
  • 42) 0.208 463 918 141 301 060 406 601 706 272 725 139 456 × 2 = 0 + 0.416 927 836 282 602 120 813 203 412 545 450 278 912;
  • 43) 0.416 927 836 282 602 120 813 203 412 545 450 278 912 × 2 = 0 + 0.833 855 672 565 204 241 626 406 825 090 900 557 824;
  • 44) 0.833 855 672 565 204 241 626 406 825 090 900 557 824 × 2 = 1 + 0.667 711 345 130 408 483 252 813 650 181 801 115 648;
  • 45) 0.667 711 345 130 408 483 252 813 650 181 801 115 648 × 2 = 1 + 0.335 422 690 260 816 966 505 627 300 363 602 231 296;
  • 46) 0.335 422 690 260 816 966 505 627 300 363 602 231 296 × 2 = 0 + 0.670 845 380 521 633 933 011 254 600 727 204 462 592;
  • 47) 0.670 845 380 521 633 933 011 254 600 727 204 462 592 × 2 = 1 + 0.341 690 761 043 267 866 022 509 201 454 408 925 184;
  • 48) 0.341 690 761 043 267 866 022 509 201 454 408 925 184 × 2 = 0 + 0.683 381 522 086 535 732 045 018 402 908 817 850 368;
  • 49) 0.683 381 522 086 535 732 045 018 402 908 817 850 368 × 2 = 1 + 0.366 763 044 173 071 464 090 036 805 817 635 700 736;
  • 50) 0.366 763 044 173 071 464 090 036 805 817 635 700 736 × 2 = 0 + 0.733 526 088 346 142 928 180 073 611 635 271 401 472;
  • 51) 0.733 526 088 346 142 928 180 073 611 635 271 401 472 × 2 = 1 + 0.467 052 176 692 285 856 360 147 223 270 542 802 944;
  • 52) 0.467 052 176 692 285 856 360 147 223 270 542 802 944 × 2 = 0 + 0.934 104 353 384 571 712 720 294 446 541 085 605 888;
  • 53) 0.934 104 353 384 571 712 720 294 446 541 085 605 888 × 2 = 1 + 0.868 208 706 769 143 425 440 588 893 082 171 211 776;
  • 54) 0.868 208 706 769 143 425 440 588 893 082 171 211 776 × 2 = 1 + 0.736 417 413 538 286 850 881 177 786 164 342 423 552;
  • 55) 0.736 417 413 538 286 850 881 177 786 164 342 423 552 × 2 = 1 + 0.472 834 827 076 573 701 762 355 572 328 684 847 104;
  • 56) 0.472 834 827 076 573 701 762 355 572 328 684 847 104 × 2 = 0 + 0.945 669 654 153 147 403 524 711 144 657 369 694 208;
  • 57) 0.945 669 654 153 147 403 524 711 144 657 369 694 208 × 2 = 1 + 0.891 339 308 306 294 807 049 422 289 314 739 388 416;
  • 58) 0.891 339 308 306 294 807 049 422 289 314 739 388 416 × 2 = 1 + 0.782 678 616 612 589 614 098 844 578 629 478 776 832;
  • 59) 0.782 678 616 612 589 614 098 844 578 629 478 776 832 × 2 = 1 + 0.565 357 233 225 179 228 197 689 157 258 957 553 664;
  • 60) 0.565 357 233 225 179 228 197 689 157 258 957 553 664 × 2 = 1 + 0.130 714 466 450 358 456 395 378 314 517 915 107 328;
  • 61) 0.130 714 466 450 358 456 395 378 314 517 915 107 328 × 2 = 0 + 0.261 428 932 900 716 912 790 756 629 035 830 214 656;
  • 62) 0.261 428 932 900 716 912 790 756 629 035 830 214 656 × 2 = 0 + 0.522 857 865 801 433 825 581 513 258 071 660 429 312;
  • 63) 0.522 857 865 801 433 825 581 513 258 071 660 429 312 × 2 = 1 + 0.045 715 731 602 867 651 163 026 516 143 320 858 624;
  • 64) 0.045 715 731 602 867 651 163 026 516 143 320 858 624 × 2 = 0 + 0.091 431 463 205 735 302 326 053 032 286 641 717 248;
  • 65) 0.091 431 463 205 735 302 326 053 032 286 641 717 248 × 2 = 0 + 0.182 862 926 411 470 604 652 106 064 573 283 434 496;
  • 66) 0.182 862 926 411 470 604 652 106 064 573 283 434 496 × 2 = 0 + 0.365 725 852 822 941 209 304 212 129 146 566 868 992;
  • 67) 0.365 725 852 822 941 209 304 212 129 146 566 868 992 × 2 = 0 + 0.731 451 705 645 882 418 608 424 258 293 133 737 984;
  • 68) 0.731 451 705 645 882 418 608 424 258 293 133 737 984 × 2 = 1 + 0.462 903 411 291 764 837 216 848 516 586 267 475 968;

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:


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

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


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