0.000 020 830 729 321 671 205 134 999 154 507 12 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 507 12(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
0.000 020 830 729 321 671 205 134 999 154 507 12(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: 0.000 020 830 729 321 671 205 134 999 154 507 12.

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) 0.000 020 830 729 321 671 205 134 999 154 507 12 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 014 24;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 014 24 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 028 48;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 028 48 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 056 96;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 056 96 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 113 92;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 113 92 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 227 84;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 227 84 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 455 68;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 455 68 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 776 911 36;
  • 8) 0.002 666 333 353 173 914 257 279 891 776 911 36 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 553 822 72;
  • 9) 0.005 332 666 706 347 828 514 559 783 553 822 72 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 107 645 44;
  • 10) 0.010 665 333 412 695 657 029 119 567 107 645 44 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 215 290 88;
  • 11) 0.021 330 666 825 391 314 058 239 134 215 290 88 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 430 581 76;
  • 12) 0.042 661 333 650 782 628 116 478 268 430 581 76 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 861 163 52;
  • 13) 0.085 322 667 301 565 256 232 956 536 861 163 52 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 722 327 04;
  • 14) 0.170 645 334 603 130 512 465 913 073 722 327 04 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 444 654 08;
  • 15) 0.341 290 669 206 261 024 931 826 147 444 654 08 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 889 308 16;
  • 16) 0.682 581 338 412 522 049 863 652 294 889 308 16 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 778 616 32;
  • 17) 0.365 162 676 825 044 099 727 304 589 778 616 32 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 557 232 64;
  • 18) 0.730 325 353 650 088 199 454 609 179 557 232 64 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 114 465 28;
  • 19) 0.460 650 707 300 176 398 909 218 359 114 465 28 × 2 = 0 + 0.921 301 414 600 352 797 818 436 718 228 930 56;
  • 20) 0.921 301 414 600 352 797 818 436 718 228 930 56 × 2 = 1 + 0.842 602 829 200 705 595 636 873 436 457 861 12;
  • 21) 0.842 602 829 200 705 595 636 873 436 457 861 12 × 2 = 1 + 0.685 205 658 401 411 191 273 746 872 915 722 24;
  • 22) 0.685 205 658 401 411 191 273 746 872 915 722 24 × 2 = 1 + 0.370 411 316 802 822 382 547 493 745 831 444 48;
  • 23) 0.370 411 316 802 822 382 547 493 745 831 444 48 × 2 = 0 + 0.740 822 633 605 644 765 094 987 491 662 888 96;
  • 24) 0.740 822 633 605 644 765 094 987 491 662 888 96 × 2 = 1 + 0.481 645 267 211 289 530 189 974 983 325 777 92;
  • 25) 0.481 645 267 211 289 530 189 974 983 325 777 92 × 2 = 0 + 0.963 290 534 422 579 060 379 949 966 651 555 84;
  • 26) 0.963 290 534 422 579 060 379 949 966 651 555 84 × 2 = 1 + 0.926 581 068 845 158 120 759 899 933 303 111 68;
  • 27) 0.926 581 068 845 158 120 759 899 933 303 111 68 × 2 = 1 + 0.853 162 137 690 316 241 519 799 866 606 223 36;
  • 28) 0.853 162 137 690 316 241 519 799 866 606 223 36 × 2 = 1 + 0.706 324 275 380 632 483 039 599 733 212 446 72;
  • 29) 0.706 324 275 380 632 483 039 599 733 212 446 72 × 2 = 1 + 0.412 648 550 761 264 966 079 199 466 424 893 44;
  • 30) 0.412 648 550 761 264 966 079 199 466 424 893 44 × 2 = 0 + 0.825 297 101 522 529 932 158 398 932 849 786 88;
  • 31) 0.825 297 101 522 529 932 158 398 932 849 786 88 × 2 = 1 + 0.650 594 203 045 059 864 316 797 865 699 573 76;
  • 32) 0.650 594 203 045 059 864 316 797 865 699 573 76 × 2 = 1 + 0.301 188 406 090 119 728 633 595 731 399 147 52;
  • 33) 0.301 188 406 090 119 728 633 595 731 399 147 52 × 2 = 0 + 0.602 376 812 180 239 457 267 191 462 798 295 04;
  • 34) 0.602 376 812 180 239 457 267 191 462 798 295 04 × 2 = 1 + 0.204 753 624 360 478 914 534 382 925 596 590 08;
  • 35) 0.204 753 624 360 478 914 534 382 925 596 590 08 × 2 = 0 + 0.409 507 248 720 957 829 068 765 851 193 180 16;
  • 36) 0.409 507 248 720 957 829 068 765 851 193 180 16 × 2 = 0 + 0.819 014 497 441 915 658 137 531 702 386 360 32;
  • 37) 0.819 014 497 441 915 658 137 531 702 386 360 32 × 2 = 1 + 0.638 028 994 883 831 316 275 063 404 772 720 64;
  • 38) 0.638 028 994 883 831 316 275 063 404 772 720 64 × 2 = 1 + 0.276 057 989 767 662 632 550 126 809 545 441 28;
  • 39) 0.276 057 989 767 662 632 550 126 809 545 441 28 × 2 = 0 + 0.552 115 979 535 325 265 100 253 619 090 882 56;
  • 40) 0.552 115 979 535 325 265 100 253 619 090 882 56 × 2 = 1 + 0.104 231 959 070 650 530 200 507 238 181 765 12;
  • 41) 0.104 231 959 070 650 530 200 507 238 181 765 12 × 2 = 0 + 0.208 463 918 141 301 060 401 014 476 363 530 24;
  • 42) 0.208 463 918 141 301 060 401 014 476 363 530 24 × 2 = 0 + 0.416 927 836 282 602 120 802 028 952 727 060 48;
  • 43) 0.416 927 836 282 602 120 802 028 952 727 060 48 × 2 = 0 + 0.833 855 672 565 204 241 604 057 905 454 120 96;
  • 44) 0.833 855 672 565 204 241 604 057 905 454 120 96 × 2 = 1 + 0.667 711 345 130 408 483 208 115 810 908 241 92;
  • 45) 0.667 711 345 130 408 483 208 115 810 908 241 92 × 2 = 1 + 0.335 422 690 260 816 966 416 231 621 816 483 84;
  • 46) 0.335 422 690 260 816 966 416 231 621 816 483 84 × 2 = 0 + 0.670 845 380 521 633 932 832 463 243 632 967 68;
  • 47) 0.670 845 380 521 633 932 832 463 243 632 967 68 × 2 = 1 + 0.341 690 761 043 267 865 664 926 487 265 935 36;
  • 48) 0.341 690 761 043 267 865 664 926 487 265 935 36 × 2 = 0 + 0.683 381 522 086 535 731 329 852 974 531 870 72;
  • 49) 0.683 381 522 086 535 731 329 852 974 531 870 72 × 2 = 1 + 0.366 763 044 173 071 462 659 705 949 063 741 44;
  • 50) 0.366 763 044 173 071 462 659 705 949 063 741 44 × 2 = 0 + 0.733 526 088 346 142 925 319 411 898 127 482 88;
  • 51) 0.733 526 088 346 142 925 319 411 898 127 482 88 × 2 = 1 + 0.467 052 176 692 285 850 638 823 796 254 965 76;
  • 52) 0.467 052 176 692 285 850 638 823 796 254 965 76 × 2 = 0 + 0.934 104 353 384 571 701 277 647 592 509 931 52;
  • 53) 0.934 104 353 384 571 701 277 647 592 509 931 52 × 2 = 1 + 0.868 208 706 769 143 402 555 295 185 019 863 04;
  • 54) 0.868 208 706 769 143 402 555 295 185 019 863 04 × 2 = 1 + 0.736 417 413 538 286 805 110 590 370 039 726 08;
  • 55) 0.736 417 413 538 286 805 110 590 370 039 726 08 × 2 = 1 + 0.472 834 827 076 573 610 221 180 740 079 452 16;
  • 56) 0.472 834 827 076 573 610 221 180 740 079 452 16 × 2 = 0 + 0.945 669 654 153 147 220 442 361 480 158 904 32;
  • 57) 0.945 669 654 153 147 220 442 361 480 158 904 32 × 2 = 1 + 0.891 339 308 306 294 440 884 722 960 317 808 64;
  • 58) 0.891 339 308 306 294 440 884 722 960 317 808 64 × 2 = 1 + 0.782 678 616 612 588 881 769 445 920 635 617 28;
  • 59) 0.782 678 616 612 588 881 769 445 920 635 617 28 × 2 = 1 + 0.565 357 233 225 177 763 538 891 841 271 234 56;
  • 60) 0.565 357 233 225 177 763 538 891 841 271 234 56 × 2 = 1 + 0.130 714 466 450 355 527 077 783 682 542 469 12;
  • 61) 0.130 714 466 450 355 527 077 783 682 542 469 12 × 2 = 0 + 0.261 428 932 900 711 054 155 567 365 084 938 24;
  • 62) 0.261 428 932 900 711 054 155 567 365 084 938 24 × 2 = 0 + 0.522 857 865 801 422 108 311 134 730 169 876 48;
  • 63) 0.522 857 865 801 422 108 311 134 730 169 876 48 × 2 = 1 + 0.045 715 731 602 844 216 622 269 460 339 752 96;
  • 64) 0.045 715 731 602 844 216 622 269 460 339 752 96 × 2 = 0 + 0.091 431 463 205 688 433 244 538 920 679 505 92;
  • 65) 0.091 431 463 205 688 433 244 538 920 679 505 92 × 2 = 0 + 0.182 862 926 411 376 866 489 077 841 359 011 84;
  • 66) 0.182 862 926 411 376 866 489 077 841 359 011 84 × 2 = 0 + 0.365 725 852 822 753 732 978 155 682 718 023 68;
  • 67) 0.365 725 852 822 753 732 978 155 682 718 023 68 × 2 = 0 + 0.731 451 705 645 507 465 956 311 365 436 047 36;
  • 68) 0.731 451 705 645 507 465 956 311 365 436 047 36 × 2 = 1 + 0.462 903 411 291 014 931 912 622 730 872 094 72;

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:


0.000 020 830 729 321 671 205 134 999 154 507 12(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:

0.000 020 830 729 321 671 205 134 999 154 507 12(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:


0.000 020 830 729 321 671 205 134 999 154 507 12(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 0.000 020 830 729 321 671 205 134 999 154 507 12 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