0.000 020 830 729 321 671 205 134 999 154 506 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 506 2(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
0.000 020 830 729 321 671 205 134 999 154 506 2(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: 0.000 020 830 729 321 671 205 134 999 154 506 2.

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) 0.000 020 830 729 321 671 205 134 999 154 506 2 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 012 4;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 012 4 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 024 8;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 024 8 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 049 6;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 049 6 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 099 2;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 099 2 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 198 4;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 198 4 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 396 8;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 396 8 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 776 793 6;
  • 8) 0.002 666 333 353 173 914 257 279 891 776 793 6 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 553 587 2;
  • 9) 0.005 332 666 706 347 828 514 559 783 553 587 2 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 107 174 4;
  • 10) 0.010 665 333 412 695 657 029 119 567 107 174 4 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 214 348 8;
  • 11) 0.021 330 666 825 391 314 058 239 134 214 348 8 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 428 697 6;
  • 12) 0.042 661 333 650 782 628 116 478 268 428 697 6 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 857 395 2;
  • 13) 0.085 322 667 301 565 256 232 956 536 857 395 2 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 714 790 4;
  • 14) 0.170 645 334 603 130 512 465 913 073 714 790 4 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 429 580 8;
  • 15) 0.341 290 669 206 261 024 931 826 147 429 580 8 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 859 161 6;
  • 16) 0.682 581 338 412 522 049 863 652 294 859 161 6 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 718 323 2;
  • 17) 0.365 162 676 825 044 099 727 304 589 718 323 2 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 436 646 4;
  • 18) 0.730 325 353 650 088 199 454 609 179 436 646 4 × 2 = 1 + 0.460 650 707 300 176 398 909 218 358 873 292 8;
  • 19) 0.460 650 707 300 176 398 909 218 358 873 292 8 × 2 = 0 + 0.921 301 414 600 352 797 818 436 717 746 585 6;
  • 20) 0.921 301 414 600 352 797 818 436 717 746 585 6 × 2 = 1 + 0.842 602 829 200 705 595 636 873 435 493 171 2;
  • 21) 0.842 602 829 200 705 595 636 873 435 493 171 2 × 2 = 1 + 0.685 205 658 401 411 191 273 746 870 986 342 4;
  • 22) 0.685 205 658 401 411 191 273 746 870 986 342 4 × 2 = 1 + 0.370 411 316 802 822 382 547 493 741 972 684 8;
  • 23) 0.370 411 316 802 822 382 547 493 741 972 684 8 × 2 = 0 + 0.740 822 633 605 644 765 094 987 483 945 369 6;
  • 24) 0.740 822 633 605 644 765 094 987 483 945 369 6 × 2 = 1 + 0.481 645 267 211 289 530 189 974 967 890 739 2;
  • 25) 0.481 645 267 211 289 530 189 974 967 890 739 2 × 2 = 0 + 0.963 290 534 422 579 060 379 949 935 781 478 4;
  • 26) 0.963 290 534 422 579 060 379 949 935 781 478 4 × 2 = 1 + 0.926 581 068 845 158 120 759 899 871 562 956 8;
  • 27) 0.926 581 068 845 158 120 759 899 871 562 956 8 × 2 = 1 + 0.853 162 137 690 316 241 519 799 743 125 913 6;
  • 28) 0.853 162 137 690 316 241 519 799 743 125 913 6 × 2 = 1 + 0.706 324 275 380 632 483 039 599 486 251 827 2;
  • 29) 0.706 324 275 380 632 483 039 599 486 251 827 2 × 2 = 1 + 0.412 648 550 761 264 966 079 198 972 503 654 4;
  • 30) 0.412 648 550 761 264 966 079 198 972 503 654 4 × 2 = 0 + 0.825 297 101 522 529 932 158 397 945 007 308 8;
  • 31) 0.825 297 101 522 529 932 158 397 945 007 308 8 × 2 = 1 + 0.650 594 203 045 059 864 316 795 890 014 617 6;
  • 32) 0.650 594 203 045 059 864 316 795 890 014 617 6 × 2 = 1 + 0.301 188 406 090 119 728 633 591 780 029 235 2;
  • 33) 0.301 188 406 090 119 728 633 591 780 029 235 2 × 2 = 0 + 0.602 376 812 180 239 457 267 183 560 058 470 4;
  • 34) 0.602 376 812 180 239 457 267 183 560 058 470 4 × 2 = 1 + 0.204 753 624 360 478 914 534 367 120 116 940 8;
  • 35) 0.204 753 624 360 478 914 534 367 120 116 940 8 × 2 = 0 + 0.409 507 248 720 957 829 068 734 240 233 881 6;
  • 36) 0.409 507 248 720 957 829 068 734 240 233 881 6 × 2 = 0 + 0.819 014 497 441 915 658 137 468 480 467 763 2;
  • 37) 0.819 014 497 441 915 658 137 468 480 467 763 2 × 2 = 1 + 0.638 028 994 883 831 316 274 936 960 935 526 4;
  • 38) 0.638 028 994 883 831 316 274 936 960 935 526 4 × 2 = 1 + 0.276 057 989 767 662 632 549 873 921 871 052 8;
  • 39) 0.276 057 989 767 662 632 549 873 921 871 052 8 × 2 = 0 + 0.552 115 979 535 325 265 099 747 843 742 105 6;
  • 40) 0.552 115 979 535 325 265 099 747 843 742 105 6 × 2 = 1 + 0.104 231 959 070 650 530 199 495 687 484 211 2;
  • 41) 0.104 231 959 070 650 530 199 495 687 484 211 2 × 2 = 0 + 0.208 463 918 141 301 060 398 991 374 968 422 4;
  • 42) 0.208 463 918 141 301 060 398 991 374 968 422 4 × 2 = 0 + 0.416 927 836 282 602 120 797 982 749 936 844 8;
  • 43) 0.416 927 836 282 602 120 797 982 749 936 844 8 × 2 = 0 + 0.833 855 672 565 204 241 595 965 499 873 689 6;
  • 44) 0.833 855 672 565 204 241 595 965 499 873 689 6 × 2 = 1 + 0.667 711 345 130 408 483 191 930 999 747 379 2;
  • 45) 0.667 711 345 130 408 483 191 930 999 747 379 2 × 2 = 1 + 0.335 422 690 260 816 966 383 861 999 494 758 4;
  • 46) 0.335 422 690 260 816 966 383 861 999 494 758 4 × 2 = 0 + 0.670 845 380 521 633 932 767 723 998 989 516 8;
  • 47) 0.670 845 380 521 633 932 767 723 998 989 516 8 × 2 = 1 + 0.341 690 761 043 267 865 535 447 997 979 033 6;
  • 48) 0.341 690 761 043 267 865 535 447 997 979 033 6 × 2 = 0 + 0.683 381 522 086 535 731 070 895 995 958 067 2;
  • 49) 0.683 381 522 086 535 731 070 895 995 958 067 2 × 2 = 1 + 0.366 763 044 173 071 462 141 791 991 916 134 4;
  • 50) 0.366 763 044 173 071 462 141 791 991 916 134 4 × 2 = 0 + 0.733 526 088 346 142 924 283 583 983 832 268 8;
  • 51) 0.733 526 088 346 142 924 283 583 983 832 268 8 × 2 = 1 + 0.467 052 176 692 285 848 567 167 967 664 537 6;
  • 52) 0.467 052 176 692 285 848 567 167 967 664 537 6 × 2 = 0 + 0.934 104 353 384 571 697 134 335 935 329 075 2;
  • 53) 0.934 104 353 384 571 697 134 335 935 329 075 2 × 2 = 1 + 0.868 208 706 769 143 394 268 671 870 658 150 4;
  • 54) 0.868 208 706 769 143 394 268 671 870 658 150 4 × 2 = 1 + 0.736 417 413 538 286 788 537 343 741 316 300 8;
  • 55) 0.736 417 413 538 286 788 537 343 741 316 300 8 × 2 = 1 + 0.472 834 827 076 573 577 074 687 482 632 601 6;
  • 56) 0.472 834 827 076 573 577 074 687 482 632 601 6 × 2 = 0 + 0.945 669 654 153 147 154 149 374 965 265 203 2;
  • 57) 0.945 669 654 153 147 154 149 374 965 265 203 2 × 2 = 1 + 0.891 339 308 306 294 308 298 749 930 530 406 4;
  • 58) 0.891 339 308 306 294 308 298 749 930 530 406 4 × 2 = 1 + 0.782 678 616 612 588 616 597 499 861 060 812 8;
  • 59) 0.782 678 616 612 588 616 597 499 861 060 812 8 × 2 = 1 + 0.565 357 233 225 177 233 194 999 722 121 625 6;
  • 60) 0.565 357 233 225 177 233 194 999 722 121 625 6 × 2 = 1 + 0.130 714 466 450 354 466 389 999 444 243 251 2;
  • 61) 0.130 714 466 450 354 466 389 999 444 243 251 2 × 2 = 0 + 0.261 428 932 900 708 932 779 998 888 486 502 4;
  • 62) 0.261 428 932 900 708 932 779 998 888 486 502 4 × 2 = 0 + 0.522 857 865 801 417 865 559 997 776 973 004 8;
  • 63) 0.522 857 865 801 417 865 559 997 776 973 004 8 × 2 = 1 + 0.045 715 731 602 835 731 119 995 553 946 009 6;
  • 64) 0.045 715 731 602 835 731 119 995 553 946 009 6 × 2 = 0 + 0.091 431 463 205 671 462 239 991 107 892 019 2;
  • 65) 0.091 431 463 205 671 462 239 991 107 892 019 2 × 2 = 0 + 0.182 862 926 411 342 924 479 982 215 784 038 4;
  • 66) 0.182 862 926 411 342 924 479 982 215 784 038 4 × 2 = 0 + 0.365 725 852 822 685 848 959 964 431 568 076 8;
  • 67) 0.365 725 852 822 685 848 959 964 431 568 076 8 × 2 = 0 + 0.731 451 705 645 371 697 919 928 863 136 153 6;
  • 68) 0.731 451 705 645 371 697 919 928 863 136 153 6 × 2 = 1 + 0.462 903 411 290 743 395 839 857 726 272 307 2;

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:


0.000 020 830 729 321 671 205 134 999 154 506 2(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:

0.000 020 830 729 321 671 205 134 999 154 506 2(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:


0.000 020 830 729 321 671 205 134 999 154 506 2(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 0.000 020 830 729 321 671 205 134 999 154 506 2 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

» Convert decimal 0.000 020 830 729 321 671 205 134 999 154 515 3 to 64 bit double precision IEEE 754 binary floating point representation standard

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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