0.000 020 830 729 321 671 205 134 999 154 511 52 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 511 52(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
0.000 020 830 729 321 671 205 134 999 154 511 52(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: 0.000 020 830 729 321 671 205 134 999 154 511 52.

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) 0.000 020 830 729 321 671 205 134 999 154 511 52 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 023 04;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 023 04 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 046 08;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 046 08 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 092 16;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 092 16 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 184 32;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 184 32 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 368 64;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 368 64 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 737 28;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 737 28 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 474 56;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 474 56 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 949 12;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 949 12 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 109 898 24;
  • 10) 0.010 665 333 412 695 657 029 119 567 109 898 24 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 219 796 48;
  • 11) 0.021 330 666 825 391 314 058 239 134 219 796 48 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 439 592 96;
  • 12) 0.042 661 333 650 782 628 116 478 268 439 592 96 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 879 185 92;
  • 13) 0.085 322 667 301 565 256 232 956 536 879 185 92 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 758 371 84;
  • 14) 0.170 645 334 603 130 512 465 913 073 758 371 84 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 516 743 68;
  • 15) 0.341 290 669 206 261 024 931 826 147 516 743 68 × 2 = 0 + 0.682 581 338 412 522 049 863 652 295 033 487 36;
  • 16) 0.682 581 338 412 522 049 863 652 295 033 487 36 × 2 = 1 + 0.365 162 676 825 044 099 727 304 590 066 974 72;
  • 17) 0.365 162 676 825 044 099 727 304 590 066 974 72 × 2 = 0 + 0.730 325 353 650 088 199 454 609 180 133 949 44;
  • 18) 0.730 325 353 650 088 199 454 609 180 133 949 44 × 2 = 1 + 0.460 650 707 300 176 398 909 218 360 267 898 88;
  • 19) 0.460 650 707 300 176 398 909 218 360 267 898 88 × 2 = 0 + 0.921 301 414 600 352 797 818 436 720 535 797 76;
  • 20) 0.921 301 414 600 352 797 818 436 720 535 797 76 × 2 = 1 + 0.842 602 829 200 705 595 636 873 441 071 595 52;
  • 21) 0.842 602 829 200 705 595 636 873 441 071 595 52 × 2 = 1 + 0.685 205 658 401 411 191 273 746 882 143 191 04;
  • 22) 0.685 205 658 401 411 191 273 746 882 143 191 04 × 2 = 1 + 0.370 411 316 802 822 382 547 493 764 286 382 08;
  • 23) 0.370 411 316 802 822 382 547 493 764 286 382 08 × 2 = 0 + 0.740 822 633 605 644 765 094 987 528 572 764 16;
  • 24) 0.740 822 633 605 644 765 094 987 528 572 764 16 × 2 = 1 + 0.481 645 267 211 289 530 189 975 057 145 528 32;
  • 25) 0.481 645 267 211 289 530 189 975 057 145 528 32 × 2 = 0 + 0.963 290 534 422 579 060 379 950 114 291 056 64;
  • 26) 0.963 290 534 422 579 060 379 950 114 291 056 64 × 2 = 1 + 0.926 581 068 845 158 120 759 900 228 582 113 28;
  • 27) 0.926 581 068 845 158 120 759 900 228 582 113 28 × 2 = 1 + 0.853 162 137 690 316 241 519 800 457 164 226 56;
  • 28) 0.853 162 137 690 316 241 519 800 457 164 226 56 × 2 = 1 + 0.706 324 275 380 632 483 039 600 914 328 453 12;
  • 29) 0.706 324 275 380 632 483 039 600 914 328 453 12 × 2 = 1 + 0.412 648 550 761 264 966 079 201 828 656 906 24;
  • 30) 0.412 648 550 761 264 966 079 201 828 656 906 24 × 2 = 0 + 0.825 297 101 522 529 932 158 403 657 313 812 48;
  • 31) 0.825 297 101 522 529 932 158 403 657 313 812 48 × 2 = 1 + 0.650 594 203 045 059 864 316 807 314 627 624 96;
  • 32) 0.650 594 203 045 059 864 316 807 314 627 624 96 × 2 = 1 + 0.301 188 406 090 119 728 633 614 629 255 249 92;
  • 33) 0.301 188 406 090 119 728 633 614 629 255 249 92 × 2 = 0 + 0.602 376 812 180 239 457 267 229 258 510 499 84;
  • 34) 0.602 376 812 180 239 457 267 229 258 510 499 84 × 2 = 1 + 0.204 753 624 360 478 914 534 458 517 020 999 68;
  • 35) 0.204 753 624 360 478 914 534 458 517 020 999 68 × 2 = 0 + 0.409 507 248 720 957 829 068 917 034 041 999 36;
  • 36) 0.409 507 248 720 957 829 068 917 034 041 999 36 × 2 = 0 + 0.819 014 497 441 915 658 137 834 068 083 998 72;
  • 37) 0.819 014 497 441 915 658 137 834 068 083 998 72 × 2 = 1 + 0.638 028 994 883 831 316 275 668 136 167 997 44;
  • 38) 0.638 028 994 883 831 316 275 668 136 167 997 44 × 2 = 1 + 0.276 057 989 767 662 632 551 336 272 335 994 88;
  • 39) 0.276 057 989 767 662 632 551 336 272 335 994 88 × 2 = 0 + 0.552 115 979 535 325 265 102 672 544 671 989 76;
  • 40) 0.552 115 979 535 325 265 102 672 544 671 989 76 × 2 = 1 + 0.104 231 959 070 650 530 205 345 089 343 979 52;
  • 41) 0.104 231 959 070 650 530 205 345 089 343 979 52 × 2 = 0 + 0.208 463 918 141 301 060 410 690 178 687 959 04;
  • 42) 0.208 463 918 141 301 060 410 690 178 687 959 04 × 2 = 0 + 0.416 927 836 282 602 120 821 380 357 375 918 08;
  • 43) 0.416 927 836 282 602 120 821 380 357 375 918 08 × 2 = 0 + 0.833 855 672 565 204 241 642 760 714 751 836 16;
  • 44) 0.833 855 672 565 204 241 642 760 714 751 836 16 × 2 = 1 + 0.667 711 345 130 408 483 285 521 429 503 672 32;
  • 45) 0.667 711 345 130 408 483 285 521 429 503 672 32 × 2 = 1 + 0.335 422 690 260 816 966 571 042 859 007 344 64;
  • 46) 0.335 422 690 260 816 966 571 042 859 007 344 64 × 2 = 0 + 0.670 845 380 521 633 933 142 085 718 014 689 28;
  • 47) 0.670 845 380 521 633 933 142 085 718 014 689 28 × 2 = 1 + 0.341 690 761 043 267 866 284 171 436 029 378 56;
  • 48) 0.341 690 761 043 267 866 284 171 436 029 378 56 × 2 = 0 + 0.683 381 522 086 535 732 568 342 872 058 757 12;
  • 49) 0.683 381 522 086 535 732 568 342 872 058 757 12 × 2 = 1 + 0.366 763 044 173 071 465 136 685 744 117 514 24;
  • 50) 0.366 763 044 173 071 465 136 685 744 117 514 24 × 2 = 0 + 0.733 526 088 346 142 930 273 371 488 235 028 48;
  • 51) 0.733 526 088 346 142 930 273 371 488 235 028 48 × 2 = 1 + 0.467 052 176 692 285 860 546 742 976 470 056 96;
  • 52) 0.467 052 176 692 285 860 546 742 976 470 056 96 × 2 = 0 + 0.934 104 353 384 571 721 093 485 952 940 113 92;
  • 53) 0.934 104 353 384 571 721 093 485 952 940 113 92 × 2 = 1 + 0.868 208 706 769 143 442 186 971 905 880 227 84;
  • 54) 0.868 208 706 769 143 442 186 971 905 880 227 84 × 2 = 1 + 0.736 417 413 538 286 884 373 943 811 760 455 68;
  • 55) 0.736 417 413 538 286 884 373 943 811 760 455 68 × 2 = 1 + 0.472 834 827 076 573 768 747 887 623 520 911 36;
  • 56) 0.472 834 827 076 573 768 747 887 623 520 911 36 × 2 = 0 + 0.945 669 654 153 147 537 495 775 247 041 822 72;
  • 57) 0.945 669 654 153 147 537 495 775 247 041 822 72 × 2 = 1 + 0.891 339 308 306 295 074 991 550 494 083 645 44;
  • 58) 0.891 339 308 306 295 074 991 550 494 083 645 44 × 2 = 1 + 0.782 678 616 612 590 149 983 100 988 167 290 88;
  • 59) 0.782 678 616 612 590 149 983 100 988 167 290 88 × 2 = 1 + 0.565 357 233 225 180 299 966 201 976 334 581 76;
  • 60) 0.565 357 233 225 180 299 966 201 976 334 581 76 × 2 = 1 + 0.130 714 466 450 360 599 932 403 952 669 163 52;
  • 61) 0.130 714 466 450 360 599 932 403 952 669 163 52 × 2 = 0 + 0.261 428 932 900 721 199 864 807 905 338 327 04;
  • 62) 0.261 428 932 900 721 199 864 807 905 338 327 04 × 2 = 0 + 0.522 857 865 801 442 399 729 615 810 676 654 08;
  • 63) 0.522 857 865 801 442 399 729 615 810 676 654 08 × 2 = 1 + 0.045 715 731 602 884 799 459 231 621 353 308 16;
  • 64) 0.045 715 731 602 884 799 459 231 621 353 308 16 × 2 = 0 + 0.091 431 463 205 769 598 918 463 242 706 616 32;
  • 65) 0.091 431 463 205 769 598 918 463 242 706 616 32 × 2 = 0 + 0.182 862 926 411 539 197 836 926 485 413 232 64;
  • 66) 0.182 862 926 411 539 197 836 926 485 413 232 64 × 2 = 0 + 0.365 725 852 823 078 395 673 852 970 826 465 28;
  • 67) 0.365 725 852 823 078 395 673 852 970 826 465 28 × 2 = 0 + 0.731 451 705 646 156 791 347 705 941 652 930 56;
  • 68) 0.731 451 705 646 156 791 347 705 941 652 930 56 × 2 = 1 + 0.462 903 411 292 313 582 695 411 883 305 861 12;

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:


0.000 020 830 729 321 671 205 134 999 154 511 52(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:

0.000 020 830 729 321 671 205 134 999 154 511 52(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:


0.000 020 830 729 321 671 205 134 999 154 511 52(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 0.000 020 830 729 321 671 205 134 999 154 511 52 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