0.000 020 830 729 321 671 205 134 999 154 509 763 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 763(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
0.000 020 830 729 321 671 205 134 999 154 509 763(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: 0.000 020 830 729 321 671 205 134 999 154 509 763.

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) 0.000 020 830 729 321 671 205 134 999 154 509 763 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 526;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 526 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 039 052;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 039 052 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 078 104;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 078 104 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 156 208;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 156 208 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 312 416;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 312 416 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 624 832;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 624 832 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 249 664;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 249 664 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 499 328;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 499 328 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 998 656;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 998 656 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 997 312;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 997 312 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 994 624;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 994 624 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 989 248;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 989 248 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 978 496;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 978 496 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 487 956 992;
  • 15) 0.341 290 669 206 261 024 931 826 147 487 956 992 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 975 913 984;
  • 16) 0.682 581 338 412 522 049 863 652 294 975 913 984 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 951 827 968;
  • 17) 0.365 162 676 825 044 099 727 304 589 951 827 968 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 903 655 936;
  • 18) 0.730 325 353 650 088 199 454 609 179 903 655 936 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 807 311 872;
  • 19) 0.460 650 707 300 176 398 909 218 359 807 311 872 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 614 623 744;
  • 20) 0.921 301 414 600 352 797 818 436 719 614 623 744 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 229 247 488;
  • 21) 0.842 602 829 200 705 595 636 873 439 229 247 488 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 458 494 976;
  • 22) 0.685 205 658 401 411 191 273 746 878 458 494 976 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 916 989 952;
  • 23) 0.370 411 316 802 822 382 547 493 756 916 989 952 × 2 = 0 + 0.740 822 633 605 644 765 094 987 513 833 979 904;
  • 24) 0.740 822 633 605 644 765 094 987 513 833 979 904 × 2 = 1 + 0.481 645 267 211 289 530 189 975 027 667 959 808;
  • 25) 0.481 645 267 211 289 530 189 975 027 667 959 808 × 2 = 0 + 0.963 290 534 422 579 060 379 950 055 335 919 616;
  • 26) 0.963 290 534 422 579 060 379 950 055 335 919 616 × 2 = 1 + 0.926 581 068 845 158 120 759 900 110 671 839 232;
  • 27) 0.926 581 068 845 158 120 759 900 110 671 839 232 × 2 = 1 + 0.853 162 137 690 316 241 519 800 221 343 678 464;
  • 28) 0.853 162 137 690 316 241 519 800 221 343 678 464 × 2 = 1 + 0.706 324 275 380 632 483 039 600 442 687 356 928;
  • 29) 0.706 324 275 380 632 483 039 600 442 687 356 928 × 2 = 1 + 0.412 648 550 761 264 966 079 200 885 374 713 856;
  • 30) 0.412 648 550 761 264 966 079 200 885 374 713 856 × 2 = 0 + 0.825 297 101 522 529 932 158 401 770 749 427 712;
  • 31) 0.825 297 101 522 529 932 158 401 770 749 427 712 × 2 = 1 + 0.650 594 203 045 059 864 316 803 541 498 855 424;
  • 32) 0.650 594 203 045 059 864 316 803 541 498 855 424 × 2 = 1 + 0.301 188 406 090 119 728 633 607 082 997 710 848;
  • 33) 0.301 188 406 090 119 728 633 607 082 997 710 848 × 2 = 0 + 0.602 376 812 180 239 457 267 214 165 995 421 696;
  • 34) 0.602 376 812 180 239 457 267 214 165 995 421 696 × 2 = 1 + 0.204 753 624 360 478 914 534 428 331 990 843 392;
  • 35) 0.204 753 624 360 478 914 534 428 331 990 843 392 × 2 = 0 + 0.409 507 248 720 957 829 068 856 663 981 686 784;
  • 36) 0.409 507 248 720 957 829 068 856 663 981 686 784 × 2 = 0 + 0.819 014 497 441 915 658 137 713 327 963 373 568;
  • 37) 0.819 014 497 441 915 658 137 713 327 963 373 568 × 2 = 1 + 0.638 028 994 883 831 316 275 426 655 926 747 136;
  • 38) 0.638 028 994 883 831 316 275 426 655 926 747 136 × 2 = 1 + 0.276 057 989 767 662 632 550 853 311 853 494 272;
  • 39) 0.276 057 989 767 662 632 550 853 311 853 494 272 × 2 = 0 + 0.552 115 979 535 325 265 101 706 623 706 988 544;
  • 40) 0.552 115 979 535 325 265 101 706 623 706 988 544 × 2 = 1 + 0.104 231 959 070 650 530 203 413 247 413 977 088;
  • 41) 0.104 231 959 070 650 530 203 413 247 413 977 088 × 2 = 0 + 0.208 463 918 141 301 060 406 826 494 827 954 176;
  • 42) 0.208 463 918 141 301 060 406 826 494 827 954 176 × 2 = 0 + 0.416 927 836 282 602 120 813 652 989 655 908 352;
  • 43) 0.416 927 836 282 602 120 813 652 989 655 908 352 × 2 = 0 + 0.833 855 672 565 204 241 627 305 979 311 816 704;
  • 44) 0.833 855 672 565 204 241 627 305 979 311 816 704 × 2 = 1 + 0.667 711 345 130 408 483 254 611 958 623 633 408;
  • 45) 0.667 711 345 130 408 483 254 611 958 623 633 408 × 2 = 1 + 0.335 422 690 260 816 966 509 223 917 247 266 816;
  • 46) 0.335 422 690 260 816 966 509 223 917 247 266 816 × 2 = 0 + 0.670 845 380 521 633 933 018 447 834 494 533 632;
  • 47) 0.670 845 380 521 633 933 018 447 834 494 533 632 × 2 = 1 + 0.341 690 761 043 267 866 036 895 668 989 067 264;
  • 48) 0.341 690 761 043 267 866 036 895 668 989 067 264 × 2 = 0 + 0.683 381 522 086 535 732 073 791 337 978 134 528;
  • 49) 0.683 381 522 086 535 732 073 791 337 978 134 528 × 2 = 1 + 0.366 763 044 173 071 464 147 582 675 956 269 056;
  • 50) 0.366 763 044 173 071 464 147 582 675 956 269 056 × 2 = 0 + 0.733 526 088 346 142 928 295 165 351 912 538 112;
  • 51) 0.733 526 088 346 142 928 295 165 351 912 538 112 × 2 = 1 + 0.467 052 176 692 285 856 590 330 703 825 076 224;
  • 52) 0.467 052 176 692 285 856 590 330 703 825 076 224 × 2 = 0 + 0.934 104 353 384 571 713 180 661 407 650 152 448;
  • 53) 0.934 104 353 384 571 713 180 661 407 650 152 448 × 2 = 1 + 0.868 208 706 769 143 426 361 322 815 300 304 896;
  • 54) 0.868 208 706 769 143 426 361 322 815 300 304 896 × 2 = 1 + 0.736 417 413 538 286 852 722 645 630 600 609 792;
  • 55) 0.736 417 413 538 286 852 722 645 630 600 609 792 × 2 = 1 + 0.472 834 827 076 573 705 445 291 261 201 219 584;
  • 56) 0.472 834 827 076 573 705 445 291 261 201 219 584 × 2 = 0 + 0.945 669 654 153 147 410 890 582 522 402 439 168;
  • 57) 0.945 669 654 153 147 410 890 582 522 402 439 168 × 2 = 1 + 0.891 339 308 306 294 821 781 165 044 804 878 336;
  • 58) 0.891 339 308 306 294 821 781 165 044 804 878 336 × 2 = 1 + 0.782 678 616 612 589 643 562 330 089 609 756 672;
  • 59) 0.782 678 616 612 589 643 562 330 089 609 756 672 × 2 = 1 + 0.565 357 233 225 179 287 124 660 179 219 513 344;
  • 60) 0.565 357 233 225 179 287 124 660 179 219 513 344 × 2 = 1 + 0.130 714 466 450 358 574 249 320 358 439 026 688;
  • 61) 0.130 714 466 450 358 574 249 320 358 439 026 688 × 2 = 0 + 0.261 428 932 900 717 148 498 640 716 878 053 376;
  • 62) 0.261 428 932 900 717 148 498 640 716 878 053 376 × 2 = 0 + 0.522 857 865 801 434 296 997 281 433 756 106 752;
  • 63) 0.522 857 865 801 434 296 997 281 433 756 106 752 × 2 = 1 + 0.045 715 731 602 868 593 994 562 867 512 213 504;
  • 64) 0.045 715 731 602 868 593 994 562 867 512 213 504 × 2 = 0 + 0.091 431 463 205 737 187 989 125 735 024 427 008;
  • 65) 0.091 431 463 205 737 187 989 125 735 024 427 008 × 2 = 0 + 0.182 862 926 411 474 375 978 251 470 048 854 016;
  • 66) 0.182 862 926 411 474 375 978 251 470 048 854 016 × 2 = 0 + 0.365 725 852 822 948 751 956 502 940 097 708 032;
  • 67) 0.365 725 852 822 948 751 956 502 940 097 708 032 × 2 = 0 + 0.731 451 705 645 897 503 913 005 880 195 416 064;
  • 68) 0.731 451 705 645 897 503 913 005 880 195 416 064 × 2 = 1 + 0.462 903 411 291 795 007 826 011 760 390 832 128;

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:


0.000 020 830 729 321 671 205 134 999 154 509 763(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:

0.000 020 830 729 321 671 205 134 999 154 509 763(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:


0.000 020 830 729 321 671 205 134 999 154 509 763(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 0.000 020 830 729 321 671 205 134 999 154 509 763 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