0.000 020 830 729 321 671 205 134 999 154 508 22 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 508 22(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
0.000 020 830 729 321 671 205 134 999 154 508 22(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: 0.000 020 830 729 321 671 205 134 999 154 508 22.

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) 0.000 020 830 729 321 671 205 134 999 154 508 22 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 016 44;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 016 44 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 032 88;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 032 88 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 065 76;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 065 76 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 131 52;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 131 52 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 263 04;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 263 04 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 526 08;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 526 08 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 052 16;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 052 16 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 104 32;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 104 32 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 208 64;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 208 64 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 216 417 28;
  • 11) 0.021 330 666 825 391 314 058 239 134 216 417 28 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 432 834 56;
  • 12) 0.042 661 333 650 782 628 116 478 268 432 834 56 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 865 669 12;
  • 13) 0.085 322 667 301 565 256 232 956 536 865 669 12 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 731 338 24;
  • 14) 0.170 645 334 603 130 512 465 913 073 731 338 24 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 462 676 48;
  • 15) 0.341 290 669 206 261 024 931 826 147 462 676 48 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 925 352 96;
  • 16) 0.682 581 338 412 522 049 863 652 294 925 352 96 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 850 705 92;
  • 17) 0.365 162 676 825 044 099 727 304 589 850 705 92 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 701 411 84;
  • 18) 0.730 325 353 650 088 199 454 609 179 701 411 84 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 402 823 68;
  • 19) 0.460 650 707 300 176 398 909 218 359 402 823 68 × 2 = 0 + 0.921 301 414 600 352 797 818 436 718 805 647 36;
  • 20) 0.921 301 414 600 352 797 818 436 718 805 647 36 × 2 = 1 + 0.842 602 829 200 705 595 636 873 437 611 294 72;
  • 21) 0.842 602 829 200 705 595 636 873 437 611 294 72 × 2 = 1 + 0.685 205 658 401 411 191 273 746 875 222 589 44;
  • 22) 0.685 205 658 401 411 191 273 746 875 222 589 44 × 2 = 1 + 0.370 411 316 802 822 382 547 493 750 445 178 88;
  • 23) 0.370 411 316 802 822 382 547 493 750 445 178 88 × 2 = 0 + 0.740 822 633 605 644 765 094 987 500 890 357 76;
  • 24) 0.740 822 633 605 644 765 094 987 500 890 357 76 × 2 = 1 + 0.481 645 267 211 289 530 189 975 001 780 715 52;
  • 25) 0.481 645 267 211 289 530 189 975 001 780 715 52 × 2 = 0 + 0.963 290 534 422 579 060 379 950 003 561 431 04;
  • 26) 0.963 290 534 422 579 060 379 950 003 561 431 04 × 2 = 1 + 0.926 581 068 845 158 120 759 900 007 122 862 08;
  • 27) 0.926 581 068 845 158 120 759 900 007 122 862 08 × 2 = 1 + 0.853 162 137 690 316 241 519 800 014 245 724 16;
  • 28) 0.853 162 137 690 316 241 519 800 014 245 724 16 × 2 = 1 + 0.706 324 275 380 632 483 039 600 028 491 448 32;
  • 29) 0.706 324 275 380 632 483 039 600 028 491 448 32 × 2 = 1 + 0.412 648 550 761 264 966 079 200 056 982 896 64;
  • 30) 0.412 648 550 761 264 966 079 200 056 982 896 64 × 2 = 0 + 0.825 297 101 522 529 932 158 400 113 965 793 28;
  • 31) 0.825 297 101 522 529 932 158 400 113 965 793 28 × 2 = 1 + 0.650 594 203 045 059 864 316 800 227 931 586 56;
  • 32) 0.650 594 203 045 059 864 316 800 227 931 586 56 × 2 = 1 + 0.301 188 406 090 119 728 633 600 455 863 173 12;
  • 33) 0.301 188 406 090 119 728 633 600 455 863 173 12 × 2 = 0 + 0.602 376 812 180 239 457 267 200 911 726 346 24;
  • 34) 0.602 376 812 180 239 457 267 200 911 726 346 24 × 2 = 1 + 0.204 753 624 360 478 914 534 401 823 452 692 48;
  • 35) 0.204 753 624 360 478 914 534 401 823 452 692 48 × 2 = 0 + 0.409 507 248 720 957 829 068 803 646 905 384 96;
  • 36) 0.409 507 248 720 957 829 068 803 646 905 384 96 × 2 = 0 + 0.819 014 497 441 915 658 137 607 293 810 769 92;
  • 37) 0.819 014 497 441 915 658 137 607 293 810 769 92 × 2 = 1 + 0.638 028 994 883 831 316 275 214 587 621 539 84;
  • 38) 0.638 028 994 883 831 316 275 214 587 621 539 84 × 2 = 1 + 0.276 057 989 767 662 632 550 429 175 243 079 68;
  • 39) 0.276 057 989 767 662 632 550 429 175 243 079 68 × 2 = 0 + 0.552 115 979 535 325 265 100 858 350 486 159 36;
  • 40) 0.552 115 979 535 325 265 100 858 350 486 159 36 × 2 = 1 + 0.104 231 959 070 650 530 201 716 700 972 318 72;
  • 41) 0.104 231 959 070 650 530 201 716 700 972 318 72 × 2 = 0 + 0.208 463 918 141 301 060 403 433 401 944 637 44;
  • 42) 0.208 463 918 141 301 060 403 433 401 944 637 44 × 2 = 0 + 0.416 927 836 282 602 120 806 866 803 889 274 88;
  • 43) 0.416 927 836 282 602 120 806 866 803 889 274 88 × 2 = 0 + 0.833 855 672 565 204 241 613 733 607 778 549 76;
  • 44) 0.833 855 672 565 204 241 613 733 607 778 549 76 × 2 = 1 + 0.667 711 345 130 408 483 227 467 215 557 099 52;
  • 45) 0.667 711 345 130 408 483 227 467 215 557 099 52 × 2 = 1 + 0.335 422 690 260 816 966 454 934 431 114 199 04;
  • 46) 0.335 422 690 260 816 966 454 934 431 114 199 04 × 2 = 0 + 0.670 845 380 521 633 932 909 868 862 228 398 08;
  • 47) 0.670 845 380 521 633 932 909 868 862 228 398 08 × 2 = 1 + 0.341 690 761 043 267 865 819 737 724 456 796 16;
  • 48) 0.341 690 761 043 267 865 819 737 724 456 796 16 × 2 = 0 + 0.683 381 522 086 535 731 639 475 448 913 592 32;
  • 49) 0.683 381 522 086 535 731 639 475 448 913 592 32 × 2 = 1 + 0.366 763 044 173 071 463 278 950 897 827 184 64;
  • 50) 0.366 763 044 173 071 463 278 950 897 827 184 64 × 2 = 0 + 0.733 526 088 346 142 926 557 901 795 654 369 28;
  • 51) 0.733 526 088 346 142 926 557 901 795 654 369 28 × 2 = 1 + 0.467 052 176 692 285 853 115 803 591 308 738 56;
  • 52) 0.467 052 176 692 285 853 115 803 591 308 738 56 × 2 = 0 + 0.934 104 353 384 571 706 231 607 182 617 477 12;
  • 53) 0.934 104 353 384 571 706 231 607 182 617 477 12 × 2 = 1 + 0.868 208 706 769 143 412 463 214 365 234 954 24;
  • 54) 0.868 208 706 769 143 412 463 214 365 234 954 24 × 2 = 1 + 0.736 417 413 538 286 824 926 428 730 469 908 48;
  • 55) 0.736 417 413 538 286 824 926 428 730 469 908 48 × 2 = 1 + 0.472 834 827 076 573 649 852 857 460 939 816 96;
  • 56) 0.472 834 827 076 573 649 852 857 460 939 816 96 × 2 = 0 + 0.945 669 654 153 147 299 705 714 921 879 633 92;
  • 57) 0.945 669 654 153 147 299 705 714 921 879 633 92 × 2 = 1 + 0.891 339 308 306 294 599 411 429 843 759 267 84;
  • 58) 0.891 339 308 306 294 599 411 429 843 759 267 84 × 2 = 1 + 0.782 678 616 612 589 198 822 859 687 518 535 68;
  • 59) 0.782 678 616 612 589 198 822 859 687 518 535 68 × 2 = 1 + 0.565 357 233 225 178 397 645 719 375 037 071 36;
  • 60) 0.565 357 233 225 178 397 645 719 375 037 071 36 × 2 = 1 + 0.130 714 466 450 356 795 291 438 750 074 142 72;
  • 61) 0.130 714 466 450 356 795 291 438 750 074 142 72 × 2 = 0 + 0.261 428 932 900 713 590 582 877 500 148 285 44;
  • 62) 0.261 428 932 900 713 590 582 877 500 148 285 44 × 2 = 0 + 0.522 857 865 801 427 181 165 755 000 296 570 88;
  • 63) 0.522 857 865 801 427 181 165 755 000 296 570 88 × 2 = 1 + 0.045 715 731 602 854 362 331 510 000 593 141 76;
  • 64) 0.045 715 731 602 854 362 331 510 000 593 141 76 × 2 = 0 + 0.091 431 463 205 708 724 663 020 001 186 283 52;
  • 65) 0.091 431 463 205 708 724 663 020 001 186 283 52 × 2 = 0 + 0.182 862 926 411 417 449 326 040 002 372 567 04;
  • 66) 0.182 862 926 411 417 449 326 040 002 372 567 04 × 2 = 0 + 0.365 725 852 822 834 898 652 080 004 745 134 08;
  • 67) 0.365 725 852 822 834 898 652 080 004 745 134 08 × 2 = 0 + 0.731 451 705 645 669 797 304 160 009 490 268 16;
  • 68) 0.731 451 705 645 669 797 304 160 009 490 268 16 × 2 = 1 + 0.462 903 411 291 339 594 608 320 018 980 536 32;

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:


0.000 020 830 729 321 671 205 134 999 154 508 22(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:

0.000 020 830 729 321 671 205 134 999 154 508 22(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:


0.000 020 830 729 321 671 205 134 999 154 508 22(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 0.000 020 830 729 321 671 205 134 999 154 508 22 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