0.000 020 830 729 321 671 205 134 999 154 509 762 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 762(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
0.000 020 830 729 321 671 205 134 999 154 509 762(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: 0.000 020 830 729 321 671 205 134 999 154 509 762.

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) 0.000 020 830 729 321 671 205 134 999 154 509 762 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 524;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 524 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 039 048;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 039 048 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 078 096;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 078 096 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 156 192;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 156 192 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 312 384;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 312 384 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 624 768;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 624 768 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 249 536;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 249 536 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 499 072;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 499 072 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 998 144;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 998 144 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 996 288;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 996 288 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 992 576;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 992 576 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 985 152;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 985 152 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 970 304;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 970 304 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 487 940 608;
  • 15) 0.341 290 669 206 261 024 931 826 147 487 940 608 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 975 881 216;
  • 16) 0.682 581 338 412 522 049 863 652 294 975 881 216 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 951 762 432;
  • 17) 0.365 162 676 825 044 099 727 304 589 951 762 432 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 903 524 864;
  • 18) 0.730 325 353 650 088 199 454 609 179 903 524 864 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 807 049 728;
  • 19) 0.460 650 707 300 176 398 909 218 359 807 049 728 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 614 099 456;
  • 20) 0.921 301 414 600 352 797 818 436 719 614 099 456 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 228 198 912;
  • 21) 0.842 602 829 200 705 595 636 873 439 228 198 912 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 456 397 824;
  • 22) 0.685 205 658 401 411 191 273 746 878 456 397 824 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 912 795 648;
  • 23) 0.370 411 316 802 822 382 547 493 756 912 795 648 × 2 = 0 + 0.740 822 633 605 644 765 094 987 513 825 591 296;
  • 24) 0.740 822 633 605 644 765 094 987 513 825 591 296 × 2 = 1 + 0.481 645 267 211 289 530 189 975 027 651 182 592;
  • 25) 0.481 645 267 211 289 530 189 975 027 651 182 592 × 2 = 0 + 0.963 290 534 422 579 060 379 950 055 302 365 184;
  • 26) 0.963 290 534 422 579 060 379 950 055 302 365 184 × 2 = 1 + 0.926 581 068 845 158 120 759 900 110 604 730 368;
  • 27) 0.926 581 068 845 158 120 759 900 110 604 730 368 × 2 = 1 + 0.853 162 137 690 316 241 519 800 221 209 460 736;
  • 28) 0.853 162 137 690 316 241 519 800 221 209 460 736 × 2 = 1 + 0.706 324 275 380 632 483 039 600 442 418 921 472;
  • 29) 0.706 324 275 380 632 483 039 600 442 418 921 472 × 2 = 1 + 0.412 648 550 761 264 966 079 200 884 837 842 944;
  • 30) 0.412 648 550 761 264 966 079 200 884 837 842 944 × 2 = 0 + 0.825 297 101 522 529 932 158 401 769 675 685 888;
  • 31) 0.825 297 101 522 529 932 158 401 769 675 685 888 × 2 = 1 + 0.650 594 203 045 059 864 316 803 539 351 371 776;
  • 32) 0.650 594 203 045 059 864 316 803 539 351 371 776 × 2 = 1 + 0.301 188 406 090 119 728 633 607 078 702 743 552;
  • 33) 0.301 188 406 090 119 728 633 607 078 702 743 552 × 2 = 0 + 0.602 376 812 180 239 457 267 214 157 405 487 104;
  • 34) 0.602 376 812 180 239 457 267 214 157 405 487 104 × 2 = 1 + 0.204 753 624 360 478 914 534 428 314 810 974 208;
  • 35) 0.204 753 624 360 478 914 534 428 314 810 974 208 × 2 = 0 + 0.409 507 248 720 957 829 068 856 629 621 948 416;
  • 36) 0.409 507 248 720 957 829 068 856 629 621 948 416 × 2 = 0 + 0.819 014 497 441 915 658 137 713 259 243 896 832;
  • 37) 0.819 014 497 441 915 658 137 713 259 243 896 832 × 2 = 1 + 0.638 028 994 883 831 316 275 426 518 487 793 664;
  • 38) 0.638 028 994 883 831 316 275 426 518 487 793 664 × 2 = 1 + 0.276 057 989 767 662 632 550 853 036 975 587 328;
  • 39) 0.276 057 989 767 662 632 550 853 036 975 587 328 × 2 = 0 + 0.552 115 979 535 325 265 101 706 073 951 174 656;
  • 40) 0.552 115 979 535 325 265 101 706 073 951 174 656 × 2 = 1 + 0.104 231 959 070 650 530 203 412 147 902 349 312;
  • 41) 0.104 231 959 070 650 530 203 412 147 902 349 312 × 2 = 0 + 0.208 463 918 141 301 060 406 824 295 804 698 624;
  • 42) 0.208 463 918 141 301 060 406 824 295 804 698 624 × 2 = 0 + 0.416 927 836 282 602 120 813 648 591 609 397 248;
  • 43) 0.416 927 836 282 602 120 813 648 591 609 397 248 × 2 = 0 + 0.833 855 672 565 204 241 627 297 183 218 794 496;
  • 44) 0.833 855 672 565 204 241 627 297 183 218 794 496 × 2 = 1 + 0.667 711 345 130 408 483 254 594 366 437 588 992;
  • 45) 0.667 711 345 130 408 483 254 594 366 437 588 992 × 2 = 1 + 0.335 422 690 260 816 966 509 188 732 875 177 984;
  • 46) 0.335 422 690 260 816 966 509 188 732 875 177 984 × 2 = 0 + 0.670 845 380 521 633 933 018 377 465 750 355 968;
  • 47) 0.670 845 380 521 633 933 018 377 465 750 355 968 × 2 = 1 + 0.341 690 761 043 267 866 036 754 931 500 711 936;
  • 48) 0.341 690 761 043 267 866 036 754 931 500 711 936 × 2 = 0 + 0.683 381 522 086 535 732 073 509 863 001 423 872;
  • 49) 0.683 381 522 086 535 732 073 509 863 001 423 872 × 2 = 1 + 0.366 763 044 173 071 464 147 019 726 002 847 744;
  • 50) 0.366 763 044 173 071 464 147 019 726 002 847 744 × 2 = 0 + 0.733 526 088 346 142 928 294 039 452 005 695 488;
  • 51) 0.733 526 088 346 142 928 294 039 452 005 695 488 × 2 = 1 + 0.467 052 176 692 285 856 588 078 904 011 390 976;
  • 52) 0.467 052 176 692 285 856 588 078 904 011 390 976 × 2 = 0 + 0.934 104 353 384 571 713 176 157 808 022 781 952;
  • 53) 0.934 104 353 384 571 713 176 157 808 022 781 952 × 2 = 1 + 0.868 208 706 769 143 426 352 315 616 045 563 904;
  • 54) 0.868 208 706 769 143 426 352 315 616 045 563 904 × 2 = 1 + 0.736 417 413 538 286 852 704 631 232 091 127 808;
  • 55) 0.736 417 413 538 286 852 704 631 232 091 127 808 × 2 = 1 + 0.472 834 827 076 573 705 409 262 464 182 255 616;
  • 56) 0.472 834 827 076 573 705 409 262 464 182 255 616 × 2 = 0 + 0.945 669 654 153 147 410 818 524 928 364 511 232;
  • 57) 0.945 669 654 153 147 410 818 524 928 364 511 232 × 2 = 1 + 0.891 339 308 306 294 821 637 049 856 729 022 464;
  • 58) 0.891 339 308 306 294 821 637 049 856 729 022 464 × 2 = 1 + 0.782 678 616 612 589 643 274 099 713 458 044 928;
  • 59) 0.782 678 616 612 589 643 274 099 713 458 044 928 × 2 = 1 + 0.565 357 233 225 179 286 548 199 426 916 089 856;
  • 60) 0.565 357 233 225 179 286 548 199 426 916 089 856 × 2 = 1 + 0.130 714 466 450 358 573 096 398 853 832 179 712;
  • 61) 0.130 714 466 450 358 573 096 398 853 832 179 712 × 2 = 0 + 0.261 428 932 900 717 146 192 797 707 664 359 424;
  • 62) 0.261 428 932 900 717 146 192 797 707 664 359 424 × 2 = 0 + 0.522 857 865 801 434 292 385 595 415 328 718 848;
  • 63) 0.522 857 865 801 434 292 385 595 415 328 718 848 × 2 = 1 + 0.045 715 731 602 868 584 771 190 830 657 437 696;
  • 64) 0.045 715 731 602 868 584 771 190 830 657 437 696 × 2 = 0 + 0.091 431 463 205 737 169 542 381 661 314 875 392;
  • 65) 0.091 431 463 205 737 169 542 381 661 314 875 392 × 2 = 0 + 0.182 862 926 411 474 339 084 763 322 629 750 784;
  • 66) 0.182 862 926 411 474 339 084 763 322 629 750 784 × 2 = 0 + 0.365 725 852 822 948 678 169 526 645 259 501 568;
  • 67) 0.365 725 852 822 948 678 169 526 645 259 501 568 × 2 = 0 + 0.731 451 705 645 897 356 339 053 290 519 003 136;
  • 68) 0.731 451 705 645 897 356 339 053 290 519 003 136 × 2 = 1 + 0.462 903 411 291 794 712 678 106 581 038 006 272;

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:


0.000 020 830 729 321 671 205 134 999 154 509 762(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:

0.000 020 830 729 321 671 205 134 999 154 509 762(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:


0.000 020 830 729 321 671 205 134 999 154 509 762(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 0.000 020 830 729 321 671 205 134 999 154 509 762 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