0.000 020 830 729 321 671 205 134 999 154 509 14 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 14(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 14(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 14.

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 14 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 018 28;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 018 28 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 036 56;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 036 56 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 073 12;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 073 12 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 146 24;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 146 24 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 292 48;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 292 48 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 584 96;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 584 96 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 169 92;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 169 92 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 339 84;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 339 84 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 679 68;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 679 68 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 359 36;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 359 36 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 434 718 72;
  • 12) 0.042 661 333 650 782 628 116 478 268 434 718 72 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 869 437 44;
  • 13) 0.085 322 667 301 565 256 232 956 536 869 437 44 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 738 874 88;
  • 14) 0.170 645 334 603 130 512 465 913 073 738 874 88 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 477 749 76;
  • 15) 0.341 290 669 206 261 024 931 826 147 477 749 76 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 955 499 52;
  • 16) 0.682 581 338 412 522 049 863 652 294 955 499 52 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 910 999 04;
  • 17) 0.365 162 676 825 044 099 727 304 589 910 999 04 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 821 998 08;
  • 18) 0.730 325 353 650 088 199 454 609 179 821 998 08 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 643 996 16;
  • 19) 0.460 650 707 300 176 398 909 218 359 643 996 16 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 287 992 32;
  • 20) 0.921 301 414 600 352 797 818 436 719 287 992 32 × 2 = 1 + 0.842 602 829 200 705 595 636 873 438 575 984 64;
  • 21) 0.842 602 829 200 705 595 636 873 438 575 984 64 × 2 = 1 + 0.685 205 658 401 411 191 273 746 877 151 969 28;
  • 22) 0.685 205 658 401 411 191 273 746 877 151 969 28 × 2 = 1 + 0.370 411 316 802 822 382 547 493 754 303 938 56;
  • 23) 0.370 411 316 802 822 382 547 493 754 303 938 56 × 2 = 0 + 0.740 822 633 605 644 765 094 987 508 607 877 12;
  • 24) 0.740 822 633 605 644 765 094 987 508 607 877 12 × 2 = 1 + 0.481 645 267 211 289 530 189 975 017 215 754 24;
  • 25) 0.481 645 267 211 289 530 189 975 017 215 754 24 × 2 = 0 + 0.963 290 534 422 579 060 379 950 034 431 508 48;
  • 26) 0.963 290 534 422 579 060 379 950 034 431 508 48 × 2 = 1 + 0.926 581 068 845 158 120 759 900 068 863 016 96;
  • 27) 0.926 581 068 845 158 120 759 900 068 863 016 96 × 2 = 1 + 0.853 162 137 690 316 241 519 800 137 726 033 92;
  • 28) 0.853 162 137 690 316 241 519 800 137 726 033 92 × 2 = 1 + 0.706 324 275 380 632 483 039 600 275 452 067 84;
  • 29) 0.706 324 275 380 632 483 039 600 275 452 067 84 × 2 = 1 + 0.412 648 550 761 264 966 079 200 550 904 135 68;
  • 30) 0.412 648 550 761 264 966 079 200 550 904 135 68 × 2 = 0 + 0.825 297 101 522 529 932 158 401 101 808 271 36;
  • 31) 0.825 297 101 522 529 932 158 401 101 808 271 36 × 2 = 1 + 0.650 594 203 045 059 864 316 802 203 616 542 72;
  • 32) 0.650 594 203 045 059 864 316 802 203 616 542 72 × 2 = 1 + 0.301 188 406 090 119 728 633 604 407 233 085 44;
  • 33) 0.301 188 406 090 119 728 633 604 407 233 085 44 × 2 = 0 + 0.602 376 812 180 239 457 267 208 814 466 170 88;
  • 34) 0.602 376 812 180 239 457 267 208 814 466 170 88 × 2 = 1 + 0.204 753 624 360 478 914 534 417 628 932 341 76;
  • 35) 0.204 753 624 360 478 914 534 417 628 932 341 76 × 2 = 0 + 0.409 507 248 720 957 829 068 835 257 864 683 52;
  • 36) 0.409 507 248 720 957 829 068 835 257 864 683 52 × 2 = 0 + 0.819 014 497 441 915 658 137 670 515 729 367 04;
  • 37) 0.819 014 497 441 915 658 137 670 515 729 367 04 × 2 = 1 + 0.638 028 994 883 831 316 275 341 031 458 734 08;
  • 38) 0.638 028 994 883 831 316 275 341 031 458 734 08 × 2 = 1 + 0.276 057 989 767 662 632 550 682 062 917 468 16;
  • 39) 0.276 057 989 767 662 632 550 682 062 917 468 16 × 2 = 0 + 0.552 115 979 535 325 265 101 364 125 834 936 32;
  • 40) 0.552 115 979 535 325 265 101 364 125 834 936 32 × 2 = 1 + 0.104 231 959 070 650 530 202 728 251 669 872 64;
  • 41) 0.104 231 959 070 650 530 202 728 251 669 872 64 × 2 = 0 + 0.208 463 918 141 301 060 405 456 503 339 745 28;
  • 42) 0.208 463 918 141 301 060 405 456 503 339 745 28 × 2 = 0 + 0.416 927 836 282 602 120 810 913 006 679 490 56;
  • 43) 0.416 927 836 282 602 120 810 913 006 679 490 56 × 2 = 0 + 0.833 855 672 565 204 241 621 826 013 358 981 12;
  • 44) 0.833 855 672 565 204 241 621 826 013 358 981 12 × 2 = 1 + 0.667 711 345 130 408 483 243 652 026 717 962 24;
  • 45) 0.667 711 345 130 408 483 243 652 026 717 962 24 × 2 = 1 + 0.335 422 690 260 816 966 487 304 053 435 924 48;
  • 46) 0.335 422 690 260 816 966 487 304 053 435 924 48 × 2 = 0 + 0.670 845 380 521 633 932 974 608 106 871 848 96;
  • 47) 0.670 845 380 521 633 932 974 608 106 871 848 96 × 2 = 1 + 0.341 690 761 043 267 865 949 216 213 743 697 92;
  • 48) 0.341 690 761 043 267 865 949 216 213 743 697 92 × 2 = 0 + 0.683 381 522 086 535 731 898 432 427 487 395 84;
  • 49) 0.683 381 522 086 535 731 898 432 427 487 395 84 × 2 = 1 + 0.366 763 044 173 071 463 796 864 854 974 791 68;
  • 50) 0.366 763 044 173 071 463 796 864 854 974 791 68 × 2 = 0 + 0.733 526 088 346 142 927 593 729 709 949 583 36;
  • 51) 0.733 526 088 346 142 927 593 729 709 949 583 36 × 2 = 1 + 0.467 052 176 692 285 855 187 459 419 899 166 72;
  • 52) 0.467 052 176 692 285 855 187 459 419 899 166 72 × 2 = 0 + 0.934 104 353 384 571 710 374 918 839 798 333 44;
  • 53) 0.934 104 353 384 571 710 374 918 839 798 333 44 × 2 = 1 + 0.868 208 706 769 143 420 749 837 679 596 666 88;
  • 54) 0.868 208 706 769 143 420 749 837 679 596 666 88 × 2 = 1 + 0.736 417 413 538 286 841 499 675 359 193 333 76;
  • 55) 0.736 417 413 538 286 841 499 675 359 193 333 76 × 2 = 1 + 0.472 834 827 076 573 682 999 350 718 386 667 52;
  • 56) 0.472 834 827 076 573 682 999 350 718 386 667 52 × 2 = 0 + 0.945 669 654 153 147 365 998 701 436 773 335 04;
  • 57) 0.945 669 654 153 147 365 998 701 436 773 335 04 × 2 = 1 + 0.891 339 308 306 294 731 997 402 873 546 670 08;
  • 58) 0.891 339 308 306 294 731 997 402 873 546 670 08 × 2 = 1 + 0.782 678 616 612 589 463 994 805 747 093 340 16;
  • 59) 0.782 678 616 612 589 463 994 805 747 093 340 16 × 2 = 1 + 0.565 357 233 225 178 927 989 611 494 186 680 32;
  • 60) 0.565 357 233 225 178 927 989 611 494 186 680 32 × 2 = 1 + 0.130 714 466 450 357 855 979 222 988 373 360 64;
  • 61) 0.130 714 466 450 357 855 979 222 988 373 360 64 × 2 = 0 + 0.261 428 932 900 715 711 958 445 976 746 721 28;
  • 62) 0.261 428 932 900 715 711 958 445 976 746 721 28 × 2 = 0 + 0.522 857 865 801 431 423 916 891 953 493 442 56;
  • 63) 0.522 857 865 801 431 423 916 891 953 493 442 56 × 2 = 1 + 0.045 715 731 602 862 847 833 783 906 986 885 12;
  • 64) 0.045 715 731 602 862 847 833 783 906 986 885 12 × 2 = 0 + 0.091 431 463 205 725 695 667 567 813 973 770 24;
  • 65) 0.091 431 463 205 725 695 667 567 813 973 770 24 × 2 = 0 + 0.182 862 926 411 451 391 335 135 627 947 540 48;
  • 66) 0.182 862 926 411 451 391 335 135 627 947 540 48 × 2 = 0 + 0.365 725 852 822 902 782 670 271 255 895 080 96;
  • 67) 0.365 725 852 822 902 782 670 271 255 895 080 96 × 2 = 0 + 0.731 451 705 645 805 565 340 542 511 790 161 92;
  • 68) 0.731 451 705 645 805 565 340 542 511 790 161 92 × 2 = 1 + 0.462 903 411 291 611 130 681 085 023 580 323 84;

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 14(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 14(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 14(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 14 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