0.000 020 830 729 321 671 205 134 999 154 509 647 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 647 2(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
0.000 020 830 729 321 671 205 134 999 154 509 647 2(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: 0.000 020 830 729 321 671 205 134 999 154 509 647 2.

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) 0.000 020 830 729 321 671 205 134 999 154 509 647 2 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 294 4;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 294 4 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 588 8;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 588 8 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 177 6;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 177 6 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 355 2;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 355 2 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 308 710 4;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 308 710 4 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 617 420 8;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 617 420 8 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 234 841 6;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 234 841 6 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 469 683 2;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 469 683 2 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 939 366 4;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 939 366 4 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 878 732 8;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 878 732 8 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 757 465 6;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 757 465 6 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 514 931 2;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 514 931 2 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 029 862 4;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 029 862 4 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 059 724 8;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 059 724 8 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 119 449 6;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 119 449 6 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 944 238 899 2;
  • 17) 0.365 162 676 825 044 099 727 304 589 944 238 899 2 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 888 477 798 4;
  • 18) 0.730 325 353 650 088 199 454 609 179 888 477 798 4 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 776 955 596 8;
  • 19) 0.460 650 707 300 176 398 909 218 359 776 955 596 8 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 553 911 193 6;
  • 20) 0.921 301 414 600 352 797 818 436 719 553 911 193 6 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 107 822 387 2;
  • 21) 0.842 602 829 200 705 595 636 873 439 107 822 387 2 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 215 644 774 4;
  • 22) 0.685 205 658 401 411 191 273 746 878 215 644 774 4 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 431 289 548 8;
  • 23) 0.370 411 316 802 822 382 547 493 756 431 289 548 8 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 862 579 097 6;
  • 24) 0.740 822 633 605 644 765 094 987 512 862 579 097 6 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 725 158 195 2;
  • 25) 0.481 645 267 211 289 530 189 975 025 725 158 195 2 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 450 316 390 4;
  • 26) 0.963 290 534 422 579 060 379 950 051 450 316 390 4 × 2 = 1 + 0.926 581 068 845 158 120 759 900 102 900 632 780 8;
  • 27) 0.926 581 068 845 158 120 759 900 102 900 632 780 8 × 2 = 1 + 0.853 162 137 690 316 241 519 800 205 801 265 561 6;
  • 28) 0.853 162 137 690 316 241 519 800 205 801 265 561 6 × 2 = 1 + 0.706 324 275 380 632 483 039 600 411 602 531 123 2;
  • 29) 0.706 324 275 380 632 483 039 600 411 602 531 123 2 × 2 = 1 + 0.412 648 550 761 264 966 079 200 823 205 062 246 4;
  • 30) 0.412 648 550 761 264 966 079 200 823 205 062 246 4 × 2 = 0 + 0.825 297 101 522 529 932 158 401 646 410 124 492 8;
  • 31) 0.825 297 101 522 529 932 158 401 646 410 124 492 8 × 2 = 1 + 0.650 594 203 045 059 864 316 803 292 820 248 985 6;
  • 32) 0.650 594 203 045 059 864 316 803 292 820 248 985 6 × 2 = 1 + 0.301 188 406 090 119 728 633 606 585 640 497 971 2;
  • 33) 0.301 188 406 090 119 728 633 606 585 640 497 971 2 × 2 = 0 + 0.602 376 812 180 239 457 267 213 171 280 995 942 4;
  • 34) 0.602 376 812 180 239 457 267 213 171 280 995 942 4 × 2 = 1 + 0.204 753 624 360 478 914 534 426 342 561 991 884 8;
  • 35) 0.204 753 624 360 478 914 534 426 342 561 991 884 8 × 2 = 0 + 0.409 507 248 720 957 829 068 852 685 123 983 769 6;
  • 36) 0.409 507 248 720 957 829 068 852 685 123 983 769 6 × 2 = 0 + 0.819 014 497 441 915 658 137 705 370 247 967 539 2;
  • 37) 0.819 014 497 441 915 658 137 705 370 247 967 539 2 × 2 = 1 + 0.638 028 994 883 831 316 275 410 740 495 935 078 4;
  • 38) 0.638 028 994 883 831 316 275 410 740 495 935 078 4 × 2 = 1 + 0.276 057 989 767 662 632 550 821 480 991 870 156 8;
  • 39) 0.276 057 989 767 662 632 550 821 480 991 870 156 8 × 2 = 0 + 0.552 115 979 535 325 265 101 642 961 983 740 313 6;
  • 40) 0.552 115 979 535 325 265 101 642 961 983 740 313 6 × 2 = 1 + 0.104 231 959 070 650 530 203 285 923 967 480 627 2;
  • 41) 0.104 231 959 070 650 530 203 285 923 967 480 627 2 × 2 = 0 + 0.208 463 918 141 301 060 406 571 847 934 961 254 4;
  • 42) 0.208 463 918 141 301 060 406 571 847 934 961 254 4 × 2 = 0 + 0.416 927 836 282 602 120 813 143 695 869 922 508 8;
  • 43) 0.416 927 836 282 602 120 813 143 695 869 922 508 8 × 2 = 0 + 0.833 855 672 565 204 241 626 287 391 739 845 017 6;
  • 44) 0.833 855 672 565 204 241 626 287 391 739 845 017 6 × 2 = 1 + 0.667 711 345 130 408 483 252 574 783 479 690 035 2;
  • 45) 0.667 711 345 130 408 483 252 574 783 479 690 035 2 × 2 = 1 + 0.335 422 690 260 816 966 505 149 566 959 380 070 4;
  • 46) 0.335 422 690 260 816 966 505 149 566 959 380 070 4 × 2 = 0 + 0.670 845 380 521 633 933 010 299 133 918 760 140 8;
  • 47) 0.670 845 380 521 633 933 010 299 133 918 760 140 8 × 2 = 1 + 0.341 690 761 043 267 866 020 598 267 837 520 281 6;
  • 48) 0.341 690 761 043 267 866 020 598 267 837 520 281 6 × 2 = 0 + 0.683 381 522 086 535 732 041 196 535 675 040 563 2;
  • 49) 0.683 381 522 086 535 732 041 196 535 675 040 563 2 × 2 = 1 + 0.366 763 044 173 071 464 082 393 071 350 081 126 4;
  • 50) 0.366 763 044 173 071 464 082 393 071 350 081 126 4 × 2 = 0 + 0.733 526 088 346 142 928 164 786 142 700 162 252 8;
  • 51) 0.733 526 088 346 142 928 164 786 142 700 162 252 8 × 2 = 1 + 0.467 052 176 692 285 856 329 572 285 400 324 505 6;
  • 52) 0.467 052 176 692 285 856 329 572 285 400 324 505 6 × 2 = 0 + 0.934 104 353 384 571 712 659 144 570 800 649 011 2;
  • 53) 0.934 104 353 384 571 712 659 144 570 800 649 011 2 × 2 = 1 + 0.868 208 706 769 143 425 318 289 141 601 298 022 4;
  • 54) 0.868 208 706 769 143 425 318 289 141 601 298 022 4 × 2 = 1 + 0.736 417 413 538 286 850 636 578 283 202 596 044 8;
  • 55) 0.736 417 413 538 286 850 636 578 283 202 596 044 8 × 2 = 1 + 0.472 834 827 076 573 701 273 156 566 405 192 089 6;
  • 56) 0.472 834 827 076 573 701 273 156 566 405 192 089 6 × 2 = 0 + 0.945 669 654 153 147 402 546 313 132 810 384 179 2;
  • 57) 0.945 669 654 153 147 402 546 313 132 810 384 179 2 × 2 = 1 + 0.891 339 308 306 294 805 092 626 265 620 768 358 4;
  • 58) 0.891 339 308 306 294 805 092 626 265 620 768 358 4 × 2 = 1 + 0.782 678 616 612 589 610 185 252 531 241 536 716 8;
  • 59) 0.782 678 616 612 589 610 185 252 531 241 536 716 8 × 2 = 1 + 0.565 357 233 225 179 220 370 505 062 483 073 433 6;
  • 60) 0.565 357 233 225 179 220 370 505 062 483 073 433 6 × 2 = 1 + 0.130 714 466 450 358 440 741 010 124 966 146 867 2;
  • 61) 0.130 714 466 450 358 440 741 010 124 966 146 867 2 × 2 = 0 + 0.261 428 932 900 716 881 482 020 249 932 293 734 4;
  • 62) 0.261 428 932 900 716 881 482 020 249 932 293 734 4 × 2 = 0 + 0.522 857 865 801 433 762 964 040 499 864 587 468 8;
  • 63) 0.522 857 865 801 433 762 964 040 499 864 587 468 8 × 2 = 1 + 0.045 715 731 602 867 525 928 080 999 729 174 937 6;
  • 64) 0.045 715 731 602 867 525 928 080 999 729 174 937 6 × 2 = 0 + 0.091 431 463 205 735 051 856 161 999 458 349 875 2;
  • 65) 0.091 431 463 205 735 051 856 161 999 458 349 875 2 × 2 = 0 + 0.182 862 926 411 470 103 712 323 998 916 699 750 4;
  • 66) 0.182 862 926 411 470 103 712 323 998 916 699 750 4 × 2 = 0 + 0.365 725 852 822 940 207 424 647 997 833 399 500 8;
  • 67) 0.365 725 852 822 940 207 424 647 997 833 399 500 8 × 2 = 0 + 0.731 451 705 645 880 414 849 295 995 666 799 001 6;
  • 68) 0.731 451 705 645 880 414 849 295 995 666 799 001 6 × 2 = 1 + 0.462 903 411 291 760 829 698 591 991 333 598 003 2;

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:


0.000 020 830 729 321 671 205 134 999 154 509 647 2(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:

0.000 020 830 729 321 671 205 134 999 154 509 647 2(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:


0.000 020 830 729 321 671 205 134 999 154 509 647 2(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 0.000 020 830 729 321 671 205 134 999 154 509 647 2 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