0.000 020 830 729 321 671 205 134 999 154 510 55 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 510 55(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
0.000 020 830 729 321 671 205 134 999 154 510 55(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: 0.000 020 830 729 321 671 205 134 999 154 510 55.

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) 0.000 020 830 729 321 671 205 134 999 154 510 55 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 021 1;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 021 1 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 042 2;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 042 2 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 084 4;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 084 4 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 168 8;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 168 8 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 337 6;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 337 6 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 675 2;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 675 2 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 350 4;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 350 4 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 700 8;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 700 8 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 109 401 6;
  • 10) 0.010 665 333 412 695 657 029 119 567 109 401 6 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 218 803 2;
  • 11) 0.021 330 666 825 391 314 058 239 134 218 803 2 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 437 606 4;
  • 12) 0.042 661 333 650 782 628 116 478 268 437 606 4 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 875 212 8;
  • 13) 0.085 322 667 301 565 256 232 956 536 875 212 8 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 750 425 6;
  • 14) 0.170 645 334 603 130 512 465 913 073 750 425 6 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 500 851 2;
  • 15) 0.341 290 669 206 261 024 931 826 147 500 851 2 × 2 = 0 + 0.682 581 338 412 522 049 863 652 295 001 702 4;
  • 16) 0.682 581 338 412 522 049 863 652 295 001 702 4 × 2 = 1 + 0.365 162 676 825 044 099 727 304 590 003 404 8;
  • 17) 0.365 162 676 825 044 099 727 304 590 003 404 8 × 2 = 0 + 0.730 325 353 650 088 199 454 609 180 006 809 6;
  • 18) 0.730 325 353 650 088 199 454 609 180 006 809 6 × 2 = 1 + 0.460 650 707 300 176 398 909 218 360 013 619 2;
  • 19) 0.460 650 707 300 176 398 909 218 360 013 619 2 × 2 = 0 + 0.921 301 414 600 352 797 818 436 720 027 238 4;
  • 20) 0.921 301 414 600 352 797 818 436 720 027 238 4 × 2 = 1 + 0.842 602 829 200 705 595 636 873 440 054 476 8;
  • 21) 0.842 602 829 200 705 595 636 873 440 054 476 8 × 2 = 1 + 0.685 205 658 401 411 191 273 746 880 108 953 6;
  • 22) 0.685 205 658 401 411 191 273 746 880 108 953 6 × 2 = 1 + 0.370 411 316 802 822 382 547 493 760 217 907 2;
  • 23) 0.370 411 316 802 822 382 547 493 760 217 907 2 × 2 = 0 + 0.740 822 633 605 644 765 094 987 520 435 814 4;
  • 24) 0.740 822 633 605 644 765 094 987 520 435 814 4 × 2 = 1 + 0.481 645 267 211 289 530 189 975 040 871 628 8;
  • 25) 0.481 645 267 211 289 530 189 975 040 871 628 8 × 2 = 0 + 0.963 290 534 422 579 060 379 950 081 743 257 6;
  • 26) 0.963 290 534 422 579 060 379 950 081 743 257 6 × 2 = 1 + 0.926 581 068 845 158 120 759 900 163 486 515 2;
  • 27) 0.926 581 068 845 158 120 759 900 163 486 515 2 × 2 = 1 + 0.853 162 137 690 316 241 519 800 326 973 030 4;
  • 28) 0.853 162 137 690 316 241 519 800 326 973 030 4 × 2 = 1 + 0.706 324 275 380 632 483 039 600 653 946 060 8;
  • 29) 0.706 324 275 380 632 483 039 600 653 946 060 8 × 2 = 1 + 0.412 648 550 761 264 966 079 201 307 892 121 6;
  • 30) 0.412 648 550 761 264 966 079 201 307 892 121 6 × 2 = 0 + 0.825 297 101 522 529 932 158 402 615 784 243 2;
  • 31) 0.825 297 101 522 529 932 158 402 615 784 243 2 × 2 = 1 + 0.650 594 203 045 059 864 316 805 231 568 486 4;
  • 32) 0.650 594 203 045 059 864 316 805 231 568 486 4 × 2 = 1 + 0.301 188 406 090 119 728 633 610 463 136 972 8;
  • 33) 0.301 188 406 090 119 728 633 610 463 136 972 8 × 2 = 0 + 0.602 376 812 180 239 457 267 220 926 273 945 6;
  • 34) 0.602 376 812 180 239 457 267 220 926 273 945 6 × 2 = 1 + 0.204 753 624 360 478 914 534 441 852 547 891 2;
  • 35) 0.204 753 624 360 478 914 534 441 852 547 891 2 × 2 = 0 + 0.409 507 248 720 957 829 068 883 705 095 782 4;
  • 36) 0.409 507 248 720 957 829 068 883 705 095 782 4 × 2 = 0 + 0.819 014 497 441 915 658 137 767 410 191 564 8;
  • 37) 0.819 014 497 441 915 658 137 767 410 191 564 8 × 2 = 1 + 0.638 028 994 883 831 316 275 534 820 383 129 6;
  • 38) 0.638 028 994 883 831 316 275 534 820 383 129 6 × 2 = 1 + 0.276 057 989 767 662 632 551 069 640 766 259 2;
  • 39) 0.276 057 989 767 662 632 551 069 640 766 259 2 × 2 = 0 + 0.552 115 979 535 325 265 102 139 281 532 518 4;
  • 40) 0.552 115 979 535 325 265 102 139 281 532 518 4 × 2 = 1 + 0.104 231 959 070 650 530 204 278 563 065 036 8;
  • 41) 0.104 231 959 070 650 530 204 278 563 065 036 8 × 2 = 0 + 0.208 463 918 141 301 060 408 557 126 130 073 6;
  • 42) 0.208 463 918 141 301 060 408 557 126 130 073 6 × 2 = 0 + 0.416 927 836 282 602 120 817 114 252 260 147 2;
  • 43) 0.416 927 836 282 602 120 817 114 252 260 147 2 × 2 = 0 + 0.833 855 672 565 204 241 634 228 504 520 294 4;
  • 44) 0.833 855 672 565 204 241 634 228 504 520 294 4 × 2 = 1 + 0.667 711 345 130 408 483 268 457 009 040 588 8;
  • 45) 0.667 711 345 130 408 483 268 457 009 040 588 8 × 2 = 1 + 0.335 422 690 260 816 966 536 914 018 081 177 6;
  • 46) 0.335 422 690 260 816 966 536 914 018 081 177 6 × 2 = 0 + 0.670 845 380 521 633 933 073 828 036 162 355 2;
  • 47) 0.670 845 380 521 633 933 073 828 036 162 355 2 × 2 = 1 + 0.341 690 761 043 267 866 147 656 072 324 710 4;
  • 48) 0.341 690 761 043 267 866 147 656 072 324 710 4 × 2 = 0 + 0.683 381 522 086 535 732 295 312 144 649 420 8;
  • 49) 0.683 381 522 086 535 732 295 312 144 649 420 8 × 2 = 1 + 0.366 763 044 173 071 464 590 624 289 298 841 6;
  • 50) 0.366 763 044 173 071 464 590 624 289 298 841 6 × 2 = 0 + 0.733 526 088 346 142 929 181 248 578 597 683 2;
  • 51) 0.733 526 088 346 142 929 181 248 578 597 683 2 × 2 = 1 + 0.467 052 176 692 285 858 362 497 157 195 366 4;
  • 52) 0.467 052 176 692 285 858 362 497 157 195 366 4 × 2 = 0 + 0.934 104 353 384 571 716 724 994 314 390 732 8;
  • 53) 0.934 104 353 384 571 716 724 994 314 390 732 8 × 2 = 1 + 0.868 208 706 769 143 433 449 988 628 781 465 6;
  • 54) 0.868 208 706 769 143 433 449 988 628 781 465 6 × 2 = 1 + 0.736 417 413 538 286 866 899 977 257 562 931 2;
  • 55) 0.736 417 413 538 286 866 899 977 257 562 931 2 × 2 = 1 + 0.472 834 827 076 573 733 799 954 515 125 862 4;
  • 56) 0.472 834 827 076 573 733 799 954 515 125 862 4 × 2 = 0 + 0.945 669 654 153 147 467 599 909 030 251 724 8;
  • 57) 0.945 669 654 153 147 467 599 909 030 251 724 8 × 2 = 1 + 0.891 339 308 306 294 935 199 818 060 503 449 6;
  • 58) 0.891 339 308 306 294 935 199 818 060 503 449 6 × 2 = 1 + 0.782 678 616 612 589 870 399 636 121 006 899 2;
  • 59) 0.782 678 616 612 589 870 399 636 121 006 899 2 × 2 = 1 + 0.565 357 233 225 179 740 799 272 242 013 798 4;
  • 60) 0.565 357 233 225 179 740 799 272 242 013 798 4 × 2 = 1 + 0.130 714 466 450 359 481 598 544 484 027 596 8;
  • 61) 0.130 714 466 450 359 481 598 544 484 027 596 8 × 2 = 0 + 0.261 428 932 900 718 963 197 088 968 055 193 6;
  • 62) 0.261 428 932 900 718 963 197 088 968 055 193 6 × 2 = 0 + 0.522 857 865 801 437 926 394 177 936 110 387 2;
  • 63) 0.522 857 865 801 437 926 394 177 936 110 387 2 × 2 = 1 + 0.045 715 731 602 875 852 788 355 872 220 774 4;
  • 64) 0.045 715 731 602 875 852 788 355 872 220 774 4 × 2 = 0 + 0.091 431 463 205 751 705 576 711 744 441 548 8;
  • 65) 0.091 431 463 205 751 705 576 711 744 441 548 8 × 2 = 0 + 0.182 862 926 411 503 411 153 423 488 883 097 6;
  • 66) 0.182 862 926 411 503 411 153 423 488 883 097 6 × 2 = 0 + 0.365 725 852 823 006 822 306 846 977 766 195 2;
  • 67) 0.365 725 852 823 006 822 306 846 977 766 195 2 × 2 = 0 + 0.731 451 705 646 013 644 613 693 955 532 390 4;
  • 68) 0.731 451 705 646 013 644 613 693 955 532 390 4 × 2 = 1 + 0.462 903 411 292 027 289 227 387 911 064 780 8;

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:


0.000 020 830 729 321 671 205 134 999 154 510 55(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:

0.000 020 830 729 321 671 205 134 999 154 510 55(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:


0.000 020 830 729 321 671 205 134 999 154 510 55(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 0.000 020 830 729 321 671 205 134 999 154 510 55 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