0.000 020 830 729 321 671 205 134 999 154 509 616 1 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 616 1(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
0.000 020 830 729 321 671 205 134 999 154 509 616 1(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: 0.000 020 830 729 321 671 205 134 999 154 509 616 1.

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) 0.000 020 830 729 321 671 205 134 999 154 509 616 1 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 232 2;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 232 2 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 464 4;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 464 4 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 076 928 8;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 076 928 8 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 153 857 6;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 153 857 6 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 307 715 2;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 307 715 2 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 615 430 4;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 615 430 4 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 230 860 8;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 230 860 8 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 461 721 6;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 461 721 6 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 923 443 2;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 923 443 2 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 846 886 4;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 846 886 4 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 693 772 8;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 693 772 8 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 387 545 6;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 387 545 6 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 742 775 091 2;
  • 14) 0.170 645 334 603 130 512 465 913 073 742 775 091 2 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 485 550 182 4;
  • 15) 0.341 290 669 206 261 024 931 826 147 485 550 182 4 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 971 100 364 8;
  • 16) 0.682 581 338 412 522 049 863 652 294 971 100 364 8 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 942 200 729 6;
  • 17) 0.365 162 676 825 044 099 727 304 589 942 200 729 6 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 884 401 459 2;
  • 18) 0.730 325 353 650 088 199 454 609 179 884 401 459 2 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 768 802 918 4;
  • 19) 0.460 650 707 300 176 398 909 218 359 768 802 918 4 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 537 605 836 8;
  • 20) 0.921 301 414 600 352 797 818 436 719 537 605 836 8 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 075 211 673 6;
  • 21) 0.842 602 829 200 705 595 636 873 439 075 211 673 6 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 150 423 347 2;
  • 22) 0.685 205 658 401 411 191 273 746 878 150 423 347 2 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 300 846 694 4;
  • 23) 0.370 411 316 802 822 382 547 493 756 300 846 694 4 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 601 693 388 8;
  • 24) 0.740 822 633 605 644 765 094 987 512 601 693 388 8 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 203 386 777 6;
  • 25) 0.481 645 267 211 289 530 189 975 025 203 386 777 6 × 2 = 0 + 0.963 290 534 422 579 060 379 950 050 406 773 555 2;
  • 26) 0.963 290 534 422 579 060 379 950 050 406 773 555 2 × 2 = 1 + 0.926 581 068 845 158 120 759 900 100 813 547 110 4;
  • 27) 0.926 581 068 845 158 120 759 900 100 813 547 110 4 × 2 = 1 + 0.853 162 137 690 316 241 519 800 201 627 094 220 8;
  • 28) 0.853 162 137 690 316 241 519 800 201 627 094 220 8 × 2 = 1 + 0.706 324 275 380 632 483 039 600 403 254 188 441 6;
  • 29) 0.706 324 275 380 632 483 039 600 403 254 188 441 6 × 2 = 1 + 0.412 648 550 761 264 966 079 200 806 508 376 883 2;
  • 30) 0.412 648 550 761 264 966 079 200 806 508 376 883 2 × 2 = 0 + 0.825 297 101 522 529 932 158 401 613 016 753 766 4;
  • 31) 0.825 297 101 522 529 932 158 401 613 016 753 766 4 × 2 = 1 + 0.650 594 203 045 059 864 316 803 226 033 507 532 8;
  • 32) 0.650 594 203 045 059 864 316 803 226 033 507 532 8 × 2 = 1 + 0.301 188 406 090 119 728 633 606 452 067 015 065 6;
  • 33) 0.301 188 406 090 119 728 633 606 452 067 015 065 6 × 2 = 0 + 0.602 376 812 180 239 457 267 212 904 134 030 131 2;
  • 34) 0.602 376 812 180 239 457 267 212 904 134 030 131 2 × 2 = 1 + 0.204 753 624 360 478 914 534 425 808 268 060 262 4;
  • 35) 0.204 753 624 360 478 914 534 425 808 268 060 262 4 × 2 = 0 + 0.409 507 248 720 957 829 068 851 616 536 120 524 8;
  • 36) 0.409 507 248 720 957 829 068 851 616 536 120 524 8 × 2 = 0 + 0.819 014 497 441 915 658 137 703 233 072 241 049 6;
  • 37) 0.819 014 497 441 915 658 137 703 233 072 241 049 6 × 2 = 1 + 0.638 028 994 883 831 316 275 406 466 144 482 099 2;
  • 38) 0.638 028 994 883 831 316 275 406 466 144 482 099 2 × 2 = 1 + 0.276 057 989 767 662 632 550 812 932 288 964 198 4;
  • 39) 0.276 057 989 767 662 632 550 812 932 288 964 198 4 × 2 = 0 + 0.552 115 979 535 325 265 101 625 864 577 928 396 8;
  • 40) 0.552 115 979 535 325 265 101 625 864 577 928 396 8 × 2 = 1 + 0.104 231 959 070 650 530 203 251 729 155 856 793 6;
  • 41) 0.104 231 959 070 650 530 203 251 729 155 856 793 6 × 2 = 0 + 0.208 463 918 141 301 060 406 503 458 311 713 587 2;
  • 42) 0.208 463 918 141 301 060 406 503 458 311 713 587 2 × 2 = 0 + 0.416 927 836 282 602 120 813 006 916 623 427 174 4;
  • 43) 0.416 927 836 282 602 120 813 006 916 623 427 174 4 × 2 = 0 + 0.833 855 672 565 204 241 626 013 833 246 854 348 8;
  • 44) 0.833 855 672 565 204 241 626 013 833 246 854 348 8 × 2 = 1 + 0.667 711 345 130 408 483 252 027 666 493 708 697 6;
  • 45) 0.667 711 345 130 408 483 252 027 666 493 708 697 6 × 2 = 1 + 0.335 422 690 260 816 966 504 055 332 987 417 395 2;
  • 46) 0.335 422 690 260 816 966 504 055 332 987 417 395 2 × 2 = 0 + 0.670 845 380 521 633 933 008 110 665 974 834 790 4;
  • 47) 0.670 845 380 521 633 933 008 110 665 974 834 790 4 × 2 = 1 + 0.341 690 761 043 267 866 016 221 331 949 669 580 8;
  • 48) 0.341 690 761 043 267 866 016 221 331 949 669 580 8 × 2 = 0 + 0.683 381 522 086 535 732 032 442 663 899 339 161 6;
  • 49) 0.683 381 522 086 535 732 032 442 663 899 339 161 6 × 2 = 1 + 0.366 763 044 173 071 464 064 885 327 798 678 323 2;
  • 50) 0.366 763 044 173 071 464 064 885 327 798 678 323 2 × 2 = 0 + 0.733 526 088 346 142 928 129 770 655 597 356 646 4;
  • 51) 0.733 526 088 346 142 928 129 770 655 597 356 646 4 × 2 = 1 + 0.467 052 176 692 285 856 259 541 311 194 713 292 8;
  • 52) 0.467 052 176 692 285 856 259 541 311 194 713 292 8 × 2 = 0 + 0.934 104 353 384 571 712 519 082 622 389 426 585 6;
  • 53) 0.934 104 353 384 571 712 519 082 622 389 426 585 6 × 2 = 1 + 0.868 208 706 769 143 425 038 165 244 778 853 171 2;
  • 54) 0.868 208 706 769 143 425 038 165 244 778 853 171 2 × 2 = 1 + 0.736 417 413 538 286 850 076 330 489 557 706 342 4;
  • 55) 0.736 417 413 538 286 850 076 330 489 557 706 342 4 × 2 = 1 + 0.472 834 827 076 573 700 152 660 979 115 412 684 8;
  • 56) 0.472 834 827 076 573 700 152 660 979 115 412 684 8 × 2 = 0 + 0.945 669 654 153 147 400 305 321 958 230 825 369 6;
  • 57) 0.945 669 654 153 147 400 305 321 958 230 825 369 6 × 2 = 1 + 0.891 339 308 306 294 800 610 643 916 461 650 739 2;
  • 58) 0.891 339 308 306 294 800 610 643 916 461 650 739 2 × 2 = 1 + 0.782 678 616 612 589 601 221 287 832 923 301 478 4;
  • 59) 0.782 678 616 612 589 601 221 287 832 923 301 478 4 × 2 = 1 + 0.565 357 233 225 179 202 442 575 665 846 602 956 8;
  • 60) 0.565 357 233 225 179 202 442 575 665 846 602 956 8 × 2 = 1 + 0.130 714 466 450 358 404 885 151 331 693 205 913 6;
  • 61) 0.130 714 466 450 358 404 885 151 331 693 205 913 6 × 2 = 0 + 0.261 428 932 900 716 809 770 302 663 386 411 827 2;
  • 62) 0.261 428 932 900 716 809 770 302 663 386 411 827 2 × 2 = 0 + 0.522 857 865 801 433 619 540 605 326 772 823 654 4;
  • 63) 0.522 857 865 801 433 619 540 605 326 772 823 654 4 × 2 = 1 + 0.045 715 731 602 867 239 081 210 653 545 647 308 8;
  • 64) 0.045 715 731 602 867 239 081 210 653 545 647 308 8 × 2 = 0 + 0.091 431 463 205 734 478 162 421 307 091 294 617 6;
  • 65) 0.091 431 463 205 734 478 162 421 307 091 294 617 6 × 2 = 0 + 0.182 862 926 411 468 956 324 842 614 182 589 235 2;
  • 66) 0.182 862 926 411 468 956 324 842 614 182 589 235 2 × 2 = 0 + 0.365 725 852 822 937 912 649 685 228 365 178 470 4;
  • 67) 0.365 725 852 822 937 912 649 685 228 365 178 470 4 × 2 = 0 + 0.731 451 705 645 875 825 299 370 456 730 356 940 8;
  • 68) 0.731 451 705 645 875 825 299 370 456 730 356 940 8 × 2 = 1 + 0.462 903 411 291 751 650 598 740 913 460 713 881 6;

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:


0.000 020 830 729 321 671 205 134 999 154 509 616 1(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:

0.000 020 830 729 321 671 205 134 999 154 509 616 1(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:


0.000 020 830 729 321 671 205 134 999 154 509 616 1(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 0.000 020 830 729 321 671 205 134 999 154 509 616 1 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