0.000 020 830 729 321 671 205 134 999 154 510 002 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 510 002(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
0.000 020 830 729 321 671 205 134 999 154 510 002(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: 0.000 020 830 729 321 671 205 134 999 154 510 002.

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) 0.000 020 830 729 321 671 205 134 999 154 510 002 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 020 004;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 020 004 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 040 008;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 040 008 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 080 016;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 080 016 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 160 032;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 160 032 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 320 064;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 320 064 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 640 128;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 640 128 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 280 256;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 280 256 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 560 512;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 560 512 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 109 121 024;
  • 10) 0.010 665 333 412 695 657 029 119 567 109 121 024 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 218 242 048;
  • 11) 0.021 330 666 825 391 314 058 239 134 218 242 048 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 436 484 096;
  • 12) 0.042 661 333 650 782 628 116 478 268 436 484 096 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 872 968 192;
  • 13) 0.085 322 667 301 565 256 232 956 536 872 968 192 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 745 936 384;
  • 14) 0.170 645 334 603 130 512 465 913 073 745 936 384 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 491 872 768;
  • 15) 0.341 290 669 206 261 024 931 826 147 491 872 768 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 983 745 536;
  • 16) 0.682 581 338 412 522 049 863 652 294 983 745 536 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 967 491 072;
  • 17) 0.365 162 676 825 044 099 727 304 589 967 491 072 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 934 982 144;
  • 18) 0.730 325 353 650 088 199 454 609 179 934 982 144 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 869 964 288;
  • 19) 0.460 650 707 300 176 398 909 218 359 869 964 288 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 739 928 576;
  • 20) 0.921 301 414 600 352 797 818 436 719 739 928 576 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 479 857 152;
  • 21) 0.842 602 829 200 705 595 636 873 439 479 857 152 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 959 714 304;
  • 22) 0.685 205 658 401 411 191 273 746 878 959 714 304 × 2 = 1 + 0.370 411 316 802 822 382 547 493 757 919 428 608;
  • 23) 0.370 411 316 802 822 382 547 493 757 919 428 608 × 2 = 0 + 0.740 822 633 605 644 765 094 987 515 838 857 216;
  • 24) 0.740 822 633 605 644 765 094 987 515 838 857 216 × 2 = 1 + 0.481 645 267 211 289 530 189 975 031 677 714 432;
  • 25) 0.481 645 267 211 289 530 189 975 031 677 714 432 × 2 = 0 + 0.963 290 534 422 579 060 379 950 063 355 428 864;
  • 26) 0.963 290 534 422 579 060 379 950 063 355 428 864 × 2 = 1 + 0.926 581 068 845 158 120 759 900 126 710 857 728;
  • 27) 0.926 581 068 845 158 120 759 900 126 710 857 728 × 2 = 1 + 0.853 162 137 690 316 241 519 800 253 421 715 456;
  • 28) 0.853 162 137 690 316 241 519 800 253 421 715 456 × 2 = 1 + 0.706 324 275 380 632 483 039 600 506 843 430 912;
  • 29) 0.706 324 275 380 632 483 039 600 506 843 430 912 × 2 = 1 + 0.412 648 550 761 264 966 079 201 013 686 861 824;
  • 30) 0.412 648 550 761 264 966 079 201 013 686 861 824 × 2 = 0 + 0.825 297 101 522 529 932 158 402 027 373 723 648;
  • 31) 0.825 297 101 522 529 932 158 402 027 373 723 648 × 2 = 1 + 0.650 594 203 045 059 864 316 804 054 747 447 296;
  • 32) 0.650 594 203 045 059 864 316 804 054 747 447 296 × 2 = 1 + 0.301 188 406 090 119 728 633 608 109 494 894 592;
  • 33) 0.301 188 406 090 119 728 633 608 109 494 894 592 × 2 = 0 + 0.602 376 812 180 239 457 267 216 218 989 789 184;
  • 34) 0.602 376 812 180 239 457 267 216 218 989 789 184 × 2 = 1 + 0.204 753 624 360 478 914 534 432 437 979 578 368;
  • 35) 0.204 753 624 360 478 914 534 432 437 979 578 368 × 2 = 0 + 0.409 507 248 720 957 829 068 864 875 959 156 736;
  • 36) 0.409 507 248 720 957 829 068 864 875 959 156 736 × 2 = 0 + 0.819 014 497 441 915 658 137 729 751 918 313 472;
  • 37) 0.819 014 497 441 915 658 137 729 751 918 313 472 × 2 = 1 + 0.638 028 994 883 831 316 275 459 503 836 626 944;
  • 38) 0.638 028 994 883 831 316 275 459 503 836 626 944 × 2 = 1 + 0.276 057 989 767 662 632 550 919 007 673 253 888;
  • 39) 0.276 057 989 767 662 632 550 919 007 673 253 888 × 2 = 0 + 0.552 115 979 535 325 265 101 838 015 346 507 776;
  • 40) 0.552 115 979 535 325 265 101 838 015 346 507 776 × 2 = 1 + 0.104 231 959 070 650 530 203 676 030 693 015 552;
  • 41) 0.104 231 959 070 650 530 203 676 030 693 015 552 × 2 = 0 + 0.208 463 918 141 301 060 407 352 061 386 031 104;
  • 42) 0.208 463 918 141 301 060 407 352 061 386 031 104 × 2 = 0 + 0.416 927 836 282 602 120 814 704 122 772 062 208;
  • 43) 0.416 927 836 282 602 120 814 704 122 772 062 208 × 2 = 0 + 0.833 855 672 565 204 241 629 408 245 544 124 416;
  • 44) 0.833 855 672 565 204 241 629 408 245 544 124 416 × 2 = 1 + 0.667 711 345 130 408 483 258 816 491 088 248 832;
  • 45) 0.667 711 345 130 408 483 258 816 491 088 248 832 × 2 = 1 + 0.335 422 690 260 816 966 517 632 982 176 497 664;
  • 46) 0.335 422 690 260 816 966 517 632 982 176 497 664 × 2 = 0 + 0.670 845 380 521 633 933 035 265 964 352 995 328;
  • 47) 0.670 845 380 521 633 933 035 265 964 352 995 328 × 2 = 1 + 0.341 690 761 043 267 866 070 531 928 705 990 656;
  • 48) 0.341 690 761 043 267 866 070 531 928 705 990 656 × 2 = 0 + 0.683 381 522 086 535 732 141 063 857 411 981 312;
  • 49) 0.683 381 522 086 535 732 141 063 857 411 981 312 × 2 = 1 + 0.366 763 044 173 071 464 282 127 714 823 962 624;
  • 50) 0.366 763 044 173 071 464 282 127 714 823 962 624 × 2 = 0 + 0.733 526 088 346 142 928 564 255 429 647 925 248;
  • 51) 0.733 526 088 346 142 928 564 255 429 647 925 248 × 2 = 1 + 0.467 052 176 692 285 857 128 510 859 295 850 496;
  • 52) 0.467 052 176 692 285 857 128 510 859 295 850 496 × 2 = 0 + 0.934 104 353 384 571 714 257 021 718 591 700 992;
  • 53) 0.934 104 353 384 571 714 257 021 718 591 700 992 × 2 = 1 + 0.868 208 706 769 143 428 514 043 437 183 401 984;
  • 54) 0.868 208 706 769 143 428 514 043 437 183 401 984 × 2 = 1 + 0.736 417 413 538 286 857 028 086 874 366 803 968;
  • 55) 0.736 417 413 538 286 857 028 086 874 366 803 968 × 2 = 1 + 0.472 834 827 076 573 714 056 173 748 733 607 936;
  • 56) 0.472 834 827 076 573 714 056 173 748 733 607 936 × 2 = 0 + 0.945 669 654 153 147 428 112 347 497 467 215 872;
  • 57) 0.945 669 654 153 147 428 112 347 497 467 215 872 × 2 = 1 + 0.891 339 308 306 294 856 224 694 994 934 431 744;
  • 58) 0.891 339 308 306 294 856 224 694 994 934 431 744 × 2 = 1 + 0.782 678 616 612 589 712 449 389 989 868 863 488;
  • 59) 0.782 678 616 612 589 712 449 389 989 868 863 488 × 2 = 1 + 0.565 357 233 225 179 424 898 779 979 737 726 976;
  • 60) 0.565 357 233 225 179 424 898 779 979 737 726 976 × 2 = 1 + 0.130 714 466 450 358 849 797 559 959 475 453 952;
  • 61) 0.130 714 466 450 358 849 797 559 959 475 453 952 × 2 = 0 + 0.261 428 932 900 717 699 595 119 918 950 907 904;
  • 62) 0.261 428 932 900 717 699 595 119 918 950 907 904 × 2 = 0 + 0.522 857 865 801 435 399 190 239 837 901 815 808;
  • 63) 0.522 857 865 801 435 399 190 239 837 901 815 808 × 2 = 1 + 0.045 715 731 602 870 798 380 479 675 803 631 616;
  • 64) 0.045 715 731 602 870 798 380 479 675 803 631 616 × 2 = 0 + 0.091 431 463 205 741 596 760 959 351 607 263 232;
  • 65) 0.091 431 463 205 741 596 760 959 351 607 263 232 × 2 = 0 + 0.182 862 926 411 483 193 521 918 703 214 526 464;
  • 66) 0.182 862 926 411 483 193 521 918 703 214 526 464 × 2 = 0 + 0.365 725 852 822 966 387 043 837 406 429 052 928;
  • 67) 0.365 725 852 822 966 387 043 837 406 429 052 928 × 2 = 0 + 0.731 451 705 645 932 774 087 674 812 858 105 856;
  • 68) 0.731 451 705 645 932 774 087 674 812 858 105 856 × 2 = 1 + 0.462 903 411 291 865 548 175 349 625 716 211 712;

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:


0.000 020 830 729 321 671 205 134 999 154 510 002(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:

0.000 020 830 729 321 671 205 134 999 154 510 002(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:


0.000 020 830 729 321 671 205 134 999 154 510 002(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 0.000 020 830 729 321 671 205 134 999 154 510 002 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