0.000 020 830 729 321 671 205 134 999 154 509 938 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 938(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
0.000 020 830 729 321 671 205 134 999 154 509 938(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: 0.000 020 830 729 321 671 205 134 999 154 509 938.

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) 0.000 020 830 729 321 671 205 134 999 154 509 938 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 876;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 876 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 039 752;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 039 752 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 079 504;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 079 504 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 159 008;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 159 008 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 318 016;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 318 016 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 636 032;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 636 032 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 272 064;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 272 064 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 544 128;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 544 128 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 109 088 256;
  • 10) 0.010 665 333 412 695 657 029 119 567 109 088 256 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 218 176 512;
  • 11) 0.021 330 666 825 391 314 058 239 134 218 176 512 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 436 353 024;
  • 12) 0.042 661 333 650 782 628 116 478 268 436 353 024 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 872 706 048;
  • 13) 0.085 322 667 301 565 256 232 956 536 872 706 048 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 745 412 096;
  • 14) 0.170 645 334 603 130 512 465 913 073 745 412 096 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 490 824 192;
  • 15) 0.341 290 669 206 261 024 931 826 147 490 824 192 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 981 648 384;
  • 16) 0.682 581 338 412 522 049 863 652 294 981 648 384 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 963 296 768;
  • 17) 0.365 162 676 825 044 099 727 304 589 963 296 768 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 926 593 536;
  • 18) 0.730 325 353 650 088 199 454 609 179 926 593 536 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 853 187 072;
  • 19) 0.460 650 707 300 176 398 909 218 359 853 187 072 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 706 374 144;
  • 20) 0.921 301 414 600 352 797 818 436 719 706 374 144 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 412 748 288;
  • 21) 0.842 602 829 200 705 595 636 873 439 412 748 288 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 825 496 576;
  • 22) 0.685 205 658 401 411 191 273 746 878 825 496 576 × 2 = 1 + 0.370 411 316 802 822 382 547 493 757 650 993 152;
  • 23) 0.370 411 316 802 822 382 547 493 757 650 993 152 × 2 = 0 + 0.740 822 633 605 644 765 094 987 515 301 986 304;
  • 24) 0.740 822 633 605 644 765 094 987 515 301 986 304 × 2 = 1 + 0.481 645 267 211 289 530 189 975 030 603 972 608;
  • 25) 0.481 645 267 211 289 530 189 975 030 603 972 608 × 2 = 0 + 0.963 290 534 422 579 060 379 950 061 207 945 216;
  • 26) 0.963 290 534 422 579 060 379 950 061 207 945 216 × 2 = 1 + 0.926 581 068 845 158 120 759 900 122 415 890 432;
  • 27) 0.926 581 068 845 158 120 759 900 122 415 890 432 × 2 = 1 + 0.853 162 137 690 316 241 519 800 244 831 780 864;
  • 28) 0.853 162 137 690 316 241 519 800 244 831 780 864 × 2 = 1 + 0.706 324 275 380 632 483 039 600 489 663 561 728;
  • 29) 0.706 324 275 380 632 483 039 600 489 663 561 728 × 2 = 1 + 0.412 648 550 761 264 966 079 200 979 327 123 456;
  • 30) 0.412 648 550 761 264 966 079 200 979 327 123 456 × 2 = 0 + 0.825 297 101 522 529 932 158 401 958 654 246 912;
  • 31) 0.825 297 101 522 529 932 158 401 958 654 246 912 × 2 = 1 + 0.650 594 203 045 059 864 316 803 917 308 493 824;
  • 32) 0.650 594 203 045 059 864 316 803 917 308 493 824 × 2 = 1 + 0.301 188 406 090 119 728 633 607 834 616 987 648;
  • 33) 0.301 188 406 090 119 728 633 607 834 616 987 648 × 2 = 0 + 0.602 376 812 180 239 457 267 215 669 233 975 296;
  • 34) 0.602 376 812 180 239 457 267 215 669 233 975 296 × 2 = 1 + 0.204 753 624 360 478 914 534 431 338 467 950 592;
  • 35) 0.204 753 624 360 478 914 534 431 338 467 950 592 × 2 = 0 + 0.409 507 248 720 957 829 068 862 676 935 901 184;
  • 36) 0.409 507 248 720 957 829 068 862 676 935 901 184 × 2 = 0 + 0.819 014 497 441 915 658 137 725 353 871 802 368;
  • 37) 0.819 014 497 441 915 658 137 725 353 871 802 368 × 2 = 1 + 0.638 028 994 883 831 316 275 450 707 743 604 736;
  • 38) 0.638 028 994 883 831 316 275 450 707 743 604 736 × 2 = 1 + 0.276 057 989 767 662 632 550 901 415 487 209 472;
  • 39) 0.276 057 989 767 662 632 550 901 415 487 209 472 × 2 = 0 + 0.552 115 979 535 325 265 101 802 830 974 418 944;
  • 40) 0.552 115 979 535 325 265 101 802 830 974 418 944 × 2 = 1 + 0.104 231 959 070 650 530 203 605 661 948 837 888;
  • 41) 0.104 231 959 070 650 530 203 605 661 948 837 888 × 2 = 0 + 0.208 463 918 141 301 060 407 211 323 897 675 776;
  • 42) 0.208 463 918 141 301 060 407 211 323 897 675 776 × 2 = 0 + 0.416 927 836 282 602 120 814 422 647 795 351 552;
  • 43) 0.416 927 836 282 602 120 814 422 647 795 351 552 × 2 = 0 + 0.833 855 672 565 204 241 628 845 295 590 703 104;
  • 44) 0.833 855 672 565 204 241 628 845 295 590 703 104 × 2 = 1 + 0.667 711 345 130 408 483 257 690 591 181 406 208;
  • 45) 0.667 711 345 130 408 483 257 690 591 181 406 208 × 2 = 1 + 0.335 422 690 260 816 966 515 381 182 362 812 416;
  • 46) 0.335 422 690 260 816 966 515 381 182 362 812 416 × 2 = 0 + 0.670 845 380 521 633 933 030 762 364 725 624 832;
  • 47) 0.670 845 380 521 633 933 030 762 364 725 624 832 × 2 = 1 + 0.341 690 761 043 267 866 061 524 729 451 249 664;
  • 48) 0.341 690 761 043 267 866 061 524 729 451 249 664 × 2 = 0 + 0.683 381 522 086 535 732 123 049 458 902 499 328;
  • 49) 0.683 381 522 086 535 732 123 049 458 902 499 328 × 2 = 1 + 0.366 763 044 173 071 464 246 098 917 804 998 656;
  • 50) 0.366 763 044 173 071 464 246 098 917 804 998 656 × 2 = 0 + 0.733 526 088 346 142 928 492 197 835 609 997 312;
  • 51) 0.733 526 088 346 142 928 492 197 835 609 997 312 × 2 = 1 + 0.467 052 176 692 285 856 984 395 671 219 994 624;
  • 52) 0.467 052 176 692 285 856 984 395 671 219 994 624 × 2 = 0 + 0.934 104 353 384 571 713 968 791 342 439 989 248;
  • 53) 0.934 104 353 384 571 713 968 791 342 439 989 248 × 2 = 1 + 0.868 208 706 769 143 427 937 582 684 879 978 496;
  • 54) 0.868 208 706 769 143 427 937 582 684 879 978 496 × 2 = 1 + 0.736 417 413 538 286 855 875 165 369 759 956 992;
  • 55) 0.736 417 413 538 286 855 875 165 369 759 956 992 × 2 = 1 + 0.472 834 827 076 573 711 750 330 739 519 913 984;
  • 56) 0.472 834 827 076 573 711 750 330 739 519 913 984 × 2 = 0 + 0.945 669 654 153 147 423 500 661 479 039 827 968;
  • 57) 0.945 669 654 153 147 423 500 661 479 039 827 968 × 2 = 1 + 0.891 339 308 306 294 847 001 322 958 079 655 936;
  • 58) 0.891 339 308 306 294 847 001 322 958 079 655 936 × 2 = 1 + 0.782 678 616 612 589 694 002 645 916 159 311 872;
  • 59) 0.782 678 616 612 589 694 002 645 916 159 311 872 × 2 = 1 + 0.565 357 233 225 179 388 005 291 832 318 623 744;
  • 60) 0.565 357 233 225 179 388 005 291 832 318 623 744 × 2 = 1 + 0.130 714 466 450 358 776 010 583 664 637 247 488;
  • 61) 0.130 714 466 450 358 776 010 583 664 637 247 488 × 2 = 0 + 0.261 428 932 900 717 552 021 167 329 274 494 976;
  • 62) 0.261 428 932 900 717 552 021 167 329 274 494 976 × 2 = 0 + 0.522 857 865 801 435 104 042 334 658 548 989 952;
  • 63) 0.522 857 865 801 435 104 042 334 658 548 989 952 × 2 = 1 + 0.045 715 731 602 870 208 084 669 317 097 979 904;
  • 64) 0.045 715 731 602 870 208 084 669 317 097 979 904 × 2 = 0 + 0.091 431 463 205 740 416 169 338 634 195 959 808;
  • 65) 0.091 431 463 205 740 416 169 338 634 195 959 808 × 2 = 0 + 0.182 862 926 411 480 832 338 677 268 391 919 616;
  • 66) 0.182 862 926 411 480 832 338 677 268 391 919 616 × 2 = 0 + 0.365 725 852 822 961 664 677 354 536 783 839 232;
  • 67) 0.365 725 852 822 961 664 677 354 536 783 839 232 × 2 = 0 + 0.731 451 705 645 923 329 354 709 073 567 678 464;
  • 68) 0.731 451 705 645 923 329 354 709 073 567 678 464 × 2 = 1 + 0.462 903 411 291 846 658 709 418 147 135 356 928;

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:


0.000 020 830 729 321 671 205 134 999 154 509 938(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:

0.000 020 830 729 321 671 205 134 999 154 509 938(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:


0.000 020 830 729 321 671 205 134 999 154 509 938(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 0.000 020 830 729 321 671 205 134 999 154 509 938 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