0.000 020 830 729 321 671 205 134 999 154 515 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 515 3(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
0.000 020 830 729 321 671 205 134 999 154 515 3(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: 0.000 020 830 729 321 671 205 134 999 154 515 3.

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) 0.000 020 830 729 321 671 205 134 999 154 515 3 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 030 6;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 030 6 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 061 2;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 061 2 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 122 4;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 122 4 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 244 8;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 244 8 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 489 6;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 489 6 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 979 2;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 979 2 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 958 4;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 958 4 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 555 916 8;
  • 9) 0.005 332 666 706 347 828 514 559 783 555 916 8 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 111 833 6;
  • 10) 0.010 665 333 412 695 657 029 119 567 111 833 6 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 223 667 2;
  • 11) 0.021 330 666 825 391 314 058 239 134 223 667 2 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 447 334 4;
  • 12) 0.042 661 333 650 782 628 116 478 268 447 334 4 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 894 668 8;
  • 13) 0.085 322 667 301 565 256 232 956 536 894 668 8 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 789 337 6;
  • 14) 0.170 645 334 603 130 512 465 913 073 789 337 6 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 578 675 2;
  • 15) 0.341 290 669 206 261 024 931 826 147 578 675 2 × 2 = 0 + 0.682 581 338 412 522 049 863 652 295 157 350 4;
  • 16) 0.682 581 338 412 522 049 863 652 295 157 350 4 × 2 = 1 + 0.365 162 676 825 044 099 727 304 590 314 700 8;
  • 17) 0.365 162 676 825 044 099 727 304 590 314 700 8 × 2 = 0 + 0.730 325 353 650 088 199 454 609 180 629 401 6;
  • 18) 0.730 325 353 650 088 199 454 609 180 629 401 6 × 2 = 1 + 0.460 650 707 300 176 398 909 218 361 258 803 2;
  • 19) 0.460 650 707 300 176 398 909 218 361 258 803 2 × 2 = 0 + 0.921 301 414 600 352 797 818 436 722 517 606 4;
  • 20) 0.921 301 414 600 352 797 818 436 722 517 606 4 × 2 = 1 + 0.842 602 829 200 705 595 636 873 445 035 212 8;
  • 21) 0.842 602 829 200 705 595 636 873 445 035 212 8 × 2 = 1 + 0.685 205 658 401 411 191 273 746 890 070 425 6;
  • 22) 0.685 205 658 401 411 191 273 746 890 070 425 6 × 2 = 1 + 0.370 411 316 802 822 382 547 493 780 140 851 2;
  • 23) 0.370 411 316 802 822 382 547 493 780 140 851 2 × 2 = 0 + 0.740 822 633 605 644 765 094 987 560 281 702 4;
  • 24) 0.740 822 633 605 644 765 094 987 560 281 702 4 × 2 = 1 + 0.481 645 267 211 289 530 189 975 120 563 404 8;
  • 25) 0.481 645 267 211 289 530 189 975 120 563 404 8 × 2 = 0 + 0.963 290 534 422 579 060 379 950 241 126 809 6;
  • 26) 0.963 290 534 422 579 060 379 950 241 126 809 6 × 2 = 1 + 0.926 581 068 845 158 120 759 900 482 253 619 2;
  • 27) 0.926 581 068 845 158 120 759 900 482 253 619 2 × 2 = 1 + 0.853 162 137 690 316 241 519 800 964 507 238 4;
  • 28) 0.853 162 137 690 316 241 519 800 964 507 238 4 × 2 = 1 + 0.706 324 275 380 632 483 039 601 929 014 476 8;
  • 29) 0.706 324 275 380 632 483 039 601 929 014 476 8 × 2 = 1 + 0.412 648 550 761 264 966 079 203 858 028 953 6;
  • 30) 0.412 648 550 761 264 966 079 203 858 028 953 6 × 2 = 0 + 0.825 297 101 522 529 932 158 407 716 057 907 2;
  • 31) 0.825 297 101 522 529 932 158 407 716 057 907 2 × 2 = 1 + 0.650 594 203 045 059 864 316 815 432 115 814 4;
  • 32) 0.650 594 203 045 059 864 316 815 432 115 814 4 × 2 = 1 + 0.301 188 406 090 119 728 633 630 864 231 628 8;
  • 33) 0.301 188 406 090 119 728 633 630 864 231 628 8 × 2 = 0 + 0.602 376 812 180 239 457 267 261 728 463 257 6;
  • 34) 0.602 376 812 180 239 457 267 261 728 463 257 6 × 2 = 1 + 0.204 753 624 360 478 914 534 523 456 926 515 2;
  • 35) 0.204 753 624 360 478 914 534 523 456 926 515 2 × 2 = 0 + 0.409 507 248 720 957 829 069 046 913 853 030 4;
  • 36) 0.409 507 248 720 957 829 069 046 913 853 030 4 × 2 = 0 + 0.819 014 497 441 915 658 138 093 827 706 060 8;
  • 37) 0.819 014 497 441 915 658 138 093 827 706 060 8 × 2 = 1 + 0.638 028 994 883 831 316 276 187 655 412 121 6;
  • 38) 0.638 028 994 883 831 316 276 187 655 412 121 6 × 2 = 1 + 0.276 057 989 767 662 632 552 375 310 824 243 2;
  • 39) 0.276 057 989 767 662 632 552 375 310 824 243 2 × 2 = 0 + 0.552 115 979 535 325 265 104 750 621 648 486 4;
  • 40) 0.552 115 979 535 325 265 104 750 621 648 486 4 × 2 = 1 + 0.104 231 959 070 650 530 209 501 243 296 972 8;
  • 41) 0.104 231 959 070 650 530 209 501 243 296 972 8 × 2 = 0 + 0.208 463 918 141 301 060 419 002 486 593 945 6;
  • 42) 0.208 463 918 141 301 060 419 002 486 593 945 6 × 2 = 0 + 0.416 927 836 282 602 120 838 004 973 187 891 2;
  • 43) 0.416 927 836 282 602 120 838 004 973 187 891 2 × 2 = 0 + 0.833 855 672 565 204 241 676 009 946 375 782 4;
  • 44) 0.833 855 672 565 204 241 676 009 946 375 782 4 × 2 = 1 + 0.667 711 345 130 408 483 352 019 892 751 564 8;
  • 45) 0.667 711 345 130 408 483 352 019 892 751 564 8 × 2 = 1 + 0.335 422 690 260 816 966 704 039 785 503 129 6;
  • 46) 0.335 422 690 260 816 966 704 039 785 503 129 6 × 2 = 0 + 0.670 845 380 521 633 933 408 079 571 006 259 2;
  • 47) 0.670 845 380 521 633 933 408 079 571 006 259 2 × 2 = 1 + 0.341 690 761 043 267 866 816 159 142 012 518 4;
  • 48) 0.341 690 761 043 267 866 816 159 142 012 518 4 × 2 = 0 + 0.683 381 522 086 535 733 632 318 284 025 036 8;
  • 49) 0.683 381 522 086 535 733 632 318 284 025 036 8 × 2 = 1 + 0.366 763 044 173 071 467 264 636 568 050 073 6;
  • 50) 0.366 763 044 173 071 467 264 636 568 050 073 6 × 2 = 0 + 0.733 526 088 346 142 934 529 273 136 100 147 2;
  • 51) 0.733 526 088 346 142 934 529 273 136 100 147 2 × 2 = 1 + 0.467 052 176 692 285 869 058 546 272 200 294 4;
  • 52) 0.467 052 176 692 285 869 058 546 272 200 294 4 × 2 = 0 + 0.934 104 353 384 571 738 117 092 544 400 588 8;
  • 53) 0.934 104 353 384 571 738 117 092 544 400 588 8 × 2 = 1 + 0.868 208 706 769 143 476 234 185 088 801 177 6;
  • 54) 0.868 208 706 769 143 476 234 185 088 801 177 6 × 2 = 1 + 0.736 417 413 538 286 952 468 370 177 602 355 2;
  • 55) 0.736 417 413 538 286 952 468 370 177 602 355 2 × 2 = 1 + 0.472 834 827 076 573 904 936 740 355 204 710 4;
  • 56) 0.472 834 827 076 573 904 936 740 355 204 710 4 × 2 = 0 + 0.945 669 654 153 147 809 873 480 710 409 420 8;
  • 57) 0.945 669 654 153 147 809 873 480 710 409 420 8 × 2 = 1 + 0.891 339 308 306 295 619 746 961 420 818 841 6;
  • 58) 0.891 339 308 306 295 619 746 961 420 818 841 6 × 2 = 1 + 0.782 678 616 612 591 239 493 922 841 637 683 2;
  • 59) 0.782 678 616 612 591 239 493 922 841 637 683 2 × 2 = 1 + 0.565 357 233 225 182 478 987 845 683 275 366 4;
  • 60) 0.565 357 233 225 182 478 987 845 683 275 366 4 × 2 = 1 + 0.130 714 466 450 364 957 975 691 366 550 732 8;
  • 61) 0.130 714 466 450 364 957 975 691 366 550 732 8 × 2 = 0 + 0.261 428 932 900 729 915 951 382 733 101 465 6;
  • 62) 0.261 428 932 900 729 915 951 382 733 101 465 6 × 2 = 0 + 0.522 857 865 801 459 831 902 765 466 202 931 2;
  • 63) 0.522 857 865 801 459 831 902 765 466 202 931 2 × 2 = 1 + 0.045 715 731 602 919 663 805 530 932 405 862 4;
  • 64) 0.045 715 731 602 919 663 805 530 932 405 862 4 × 2 = 0 + 0.091 431 463 205 839 327 611 061 864 811 724 8;
  • 65) 0.091 431 463 205 839 327 611 061 864 811 724 8 × 2 = 0 + 0.182 862 926 411 678 655 222 123 729 623 449 6;
  • 66) 0.182 862 926 411 678 655 222 123 729 623 449 6 × 2 = 0 + 0.365 725 852 823 357 310 444 247 459 246 899 2;
  • 67) 0.365 725 852 823 357 310 444 247 459 246 899 2 × 2 = 0 + 0.731 451 705 646 714 620 888 494 918 493 798 4;
  • 68) 0.731 451 705 646 714 620 888 494 918 493 798 4 × 2 = 1 + 0.462 903 411 293 429 241 776 989 836 987 596 8;

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:


0.000 020 830 729 321 671 205 134 999 154 515 3(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:

0.000 020 830 729 321 671 205 134 999 154 515 3(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:


0.000 020 830 729 321 671 205 134 999 154 515 3(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 0.000 020 830 729 321 671 205 134 999 154 515 3 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 0.000 020 830 729 321 671 205 134 999 154 520 1 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