0.000 020 830 729 321 671 205 134 999 154 508 43 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 43(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 43(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 43.

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 43 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 016 86;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 016 86 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 033 72;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 033 72 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 067 44;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 067 44 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 134 88;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 134 88 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 269 76;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 269 76 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 539 52;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 539 52 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 079 04;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 079 04 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 158 08;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 158 08 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 316 16;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 316 16 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 216 632 32;
  • 11) 0.021 330 666 825 391 314 058 239 134 216 632 32 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 433 264 64;
  • 12) 0.042 661 333 650 782 628 116 478 268 433 264 64 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 866 529 28;
  • 13) 0.085 322 667 301 565 256 232 956 536 866 529 28 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 733 058 56;
  • 14) 0.170 645 334 603 130 512 465 913 073 733 058 56 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 466 117 12;
  • 15) 0.341 290 669 206 261 024 931 826 147 466 117 12 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 932 234 24;
  • 16) 0.682 581 338 412 522 049 863 652 294 932 234 24 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 864 468 48;
  • 17) 0.365 162 676 825 044 099 727 304 589 864 468 48 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 728 936 96;
  • 18) 0.730 325 353 650 088 199 454 609 179 728 936 96 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 457 873 92;
  • 19) 0.460 650 707 300 176 398 909 218 359 457 873 92 × 2 = 0 + 0.921 301 414 600 352 797 818 436 718 915 747 84;
  • 20) 0.921 301 414 600 352 797 818 436 718 915 747 84 × 2 = 1 + 0.842 602 829 200 705 595 636 873 437 831 495 68;
  • 21) 0.842 602 829 200 705 595 636 873 437 831 495 68 × 2 = 1 + 0.685 205 658 401 411 191 273 746 875 662 991 36;
  • 22) 0.685 205 658 401 411 191 273 746 875 662 991 36 × 2 = 1 + 0.370 411 316 802 822 382 547 493 751 325 982 72;
  • 23) 0.370 411 316 802 822 382 547 493 751 325 982 72 × 2 = 0 + 0.740 822 633 605 644 765 094 987 502 651 965 44;
  • 24) 0.740 822 633 605 644 765 094 987 502 651 965 44 × 2 = 1 + 0.481 645 267 211 289 530 189 975 005 303 930 88;
  • 25) 0.481 645 267 211 289 530 189 975 005 303 930 88 × 2 = 0 + 0.963 290 534 422 579 060 379 950 010 607 861 76;
  • 26) 0.963 290 534 422 579 060 379 950 010 607 861 76 × 2 = 1 + 0.926 581 068 845 158 120 759 900 021 215 723 52;
  • 27) 0.926 581 068 845 158 120 759 900 021 215 723 52 × 2 = 1 + 0.853 162 137 690 316 241 519 800 042 431 447 04;
  • 28) 0.853 162 137 690 316 241 519 800 042 431 447 04 × 2 = 1 + 0.706 324 275 380 632 483 039 600 084 862 894 08;
  • 29) 0.706 324 275 380 632 483 039 600 084 862 894 08 × 2 = 1 + 0.412 648 550 761 264 966 079 200 169 725 788 16;
  • 30) 0.412 648 550 761 264 966 079 200 169 725 788 16 × 2 = 0 + 0.825 297 101 522 529 932 158 400 339 451 576 32;
  • 31) 0.825 297 101 522 529 932 158 400 339 451 576 32 × 2 = 1 + 0.650 594 203 045 059 864 316 800 678 903 152 64;
  • 32) 0.650 594 203 045 059 864 316 800 678 903 152 64 × 2 = 1 + 0.301 188 406 090 119 728 633 601 357 806 305 28;
  • 33) 0.301 188 406 090 119 728 633 601 357 806 305 28 × 2 = 0 + 0.602 376 812 180 239 457 267 202 715 612 610 56;
  • 34) 0.602 376 812 180 239 457 267 202 715 612 610 56 × 2 = 1 + 0.204 753 624 360 478 914 534 405 431 225 221 12;
  • 35) 0.204 753 624 360 478 914 534 405 431 225 221 12 × 2 = 0 + 0.409 507 248 720 957 829 068 810 862 450 442 24;
  • 36) 0.409 507 248 720 957 829 068 810 862 450 442 24 × 2 = 0 + 0.819 014 497 441 915 658 137 621 724 900 884 48;
  • 37) 0.819 014 497 441 915 658 137 621 724 900 884 48 × 2 = 1 + 0.638 028 994 883 831 316 275 243 449 801 768 96;
  • 38) 0.638 028 994 883 831 316 275 243 449 801 768 96 × 2 = 1 + 0.276 057 989 767 662 632 550 486 899 603 537 92;
  • 39) 0.276 057 989 767 662 632 550 486 899 603 537 92 × 2 = 0 + 0.552 115 979 535 325 265 100 973 799 207 075 84;
  • 40) 0.552 115 979 535 325 265 100 973 799 207 075 84 × 2 = 1 + 0.104 231 959 070 650 530 201 947 598 414 151 68;
  • 41) 0.104 231 959 070 650 530 201 947 598 414 151 68 × 2 = 0 + 0.208 463 918 141 301 060 403 895 196 828 303 36;
  • 42) 0.208 463 918 141 301 060 403 895 196 828 303 36 × 2 = 0 + 0.416 927 836 282 602 120 807 790 393 656 606 72;
  • 43) 0.416 927 836 282 602 120 807 790 393 656 606 72 × 2 = 0 + 0.833 855 672 565 204 241 615 580 787 313 213 44;
  • 44) 0.833 855 672 565 204 241 615 580 787 313 213 44 × 2 = 1 + 0.667 711 345 130 408 483 231 161 574 626 426 88;
  • 45) 0.667 711 345 130 408 483 231 161 574 626 426 88 × 2 = 1 + 0.335 422 690 260 816 966 462 323 149 252 853 76;
  • 46) 0.335 422 690 260 816 966 462 323 149 252 853 76 × 2 = 0 + 0.670 845 380 521 633 932 924 646 298 505 707 52;
  • 47) 0.670 845 380 521 633 932 924 646 298 505 707 52 × 2 = 1 + 0.341 690 761 043 267 865 849 292 597 011 415 04;
  • 48) 0.341 690 761 043 267 865 849 292 597 011 415 04 × 2 = 0 + 0.683 381 522 086 535 731 698 585 194 022 830 08;
  • 49) 0.683 381 522 086 535 731 698 585 194 022 830 08 × 2 = 1 + 0.366 763 044 173 071 463 397 170 388 045 660 16;
  • 50) 0.366 763 044 173 071 463 397 170 388 045 660 16 × 2 = 0 + 0.733 526 088 346 142 926 794 340 776 091 320 32;
  • 51) 0.733 526 088 346 142 926 794 340 776 091 320 32 × 2 = 1 + 0.467 052 176 692 285 853 588 681 552 182 640 64;
  • 52) 0.467 052 176 692 285 853 588 681 552 182 640 64 × 2 = 0 + 0.934 104 353 384 571 707 177 363 104 365 281 28;
  • 53) 0.934 104 353 384 571 707 177 363 104 365 281 28 × 2 = 1 + 0.868 208 706 769 143 414 354 726 208 730 562 56;
  • 54) 0.868 208 706 769 143 414 354 726 208 730 562 56 × 2 = 1 + 0.736 417 413 538 286 828 709 452 417 461 125 12;
  • 55) 0.736 417 413 538 286 828 709 452 417 461 125 12 × 2 = 1 + 0.472 834 827 076 573 657 418 904 834 922 250 24;
  • 56) 0.472 834 827 076 573 657 418 904 834 922 250 24 × 2 = 0 + 0.945 669 654 153 147 314 837 809 669 844 500 48;
  • 57) 0.945 669 654 153 147 314 837 809 669 844 500 48 × 2 = 1 + 0.891 339 308 306 294 629 675 619 339 689 000 96;
  • 58) 0.891 339 308 306 294 629 675 619 339 689 000 96 × 2 = 1 + 0.782 678 616 612 589 259 351 238 679 378 001 92;
  • 59) 0.782 678 616 612 589 259 351 238 679 378 001 92 × 2 = 1 + 0.565 357 233 225 178 518 702 477 358 756 003 84;
  • 60) 0.565 357 233 225 178 518 702 477 358 756 003 84 × 2 = 1 + 0.130 714 466 450 357 037 404 954 717 512 007 68;
  • 61) 0.130 714 466 450 357 037 404 954 717 512 007 68 × 2 = 0 + 0.261 428 932 900 714 074 809 909 435 024 015 36;
  • 62) 0.261 428 932 900 714 074 809 909 435 024 015 36 × 2 = 0 + 0.522 857 865 801 428 149 619 818 870 048 030 72;
  • 63) 0.522 857 865 801 428 149 619 818 870 048 030 72 × 2 = 1 + 0.045 715 731 602 856 299 239 637 740 096 061 44;
  • 64) 0.045 715 731 602 856 299 239 637 740 096 061 44 × 2 = 0 + 0.091 431 463 205 712 598 479 275 480 192 122 88;
  • 65) 0.091 431 463 205 712 598 479 275 480 192 122 88 × 2 = 0 + 0.182 862 926 411 425 196 958 550 960 384 245 76;
  • 66) 0.182 862 926 411 425 196 958 550 960 384 245 76 × 2 = 0 + 0.365 725 852 822 850 393 917 101 920 768 491 52;
  • 67) 0.365 725 852 822 850 393 917 101 920 768 491 52 × 2 = 0 + 0.731 451 705 645 700 787 834 203 841 536 983 04;
  • 68) 0.731 451 705 645 700 787 834 203 841 536 983 04 × 2 = 1 + 0.462 903 411 291 401 575 668 407 683 073 966 08;

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 43(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 43(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 43(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 43 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