0.000 020 830 729 321 671 205 134 999 154 53 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 53(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 53(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 53.

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 53 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 06;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 06 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 12;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 12 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 24;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 24 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 48;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 48 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 96;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 96 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 889 92;
  • 7) 0.001 333 166 676 586 957 128 639 945 889 92 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 779 84;
  • 8) 0.002 666 333 353 173 914 257 279 891 779 84 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 559 68;
  • 9) 0.005 332 666 706 347 828 514 559 783 559 68 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 119 36;
  • 10) 0.010 665 333 412 695 657 029 119 567 119 36 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 238 72;
  • 11) 0.021 330 666 825 391 314 058 239 134 238 72 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 477 44;
  • 12) 0.042 661 333 650 782 628 116 478 268 477 44 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 954 88;
  • 13) 0.085 322 667 301 565 256 232 956 536 954 88 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 909 76;
  • 14) 0.170 645 334 603 130 512 465 913 073 909 76 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 819 52;
  • 15) 0.341 290 669 206 261 024 931 826 147 819 52 × 2 = 0 + 0.682 581 338 412 522 049 863 652 295 639 04;
  • 16) 0.682 581 338 412 522 049 863 652 295 639 04 × 2 = 1 + 0.365 162 676 825 044 099 727 304 591 278 08;
  • 17) 0.365 162 676 825 044 099 727 304 591 278 08 × 2 = 0 + 0.730 325 353 650 088 199 454 609 182 556 16;
  • 18) 0.730 325 353 650 088 199 454 609 182 556 16 × 2 = 1 + 0.460 650 707 300 176 398 909 218 365 112 32;
  • 19) 0.460 650 707 300 176 398 909 218 365 112 32 × 2 = 0 + 0.921 301 414 600 352 797 818 436 730 224 64;
  • 20) 0.921 301 414 600 352 797 818 436 730 224 64 × 2 = 1 + 0.842 602 829 200 705 595 636 873 460 449 28;
  • 21) 0.842 602 829 200 705 595 636 873 460 449 28 × 2 = 1 + 0.685 205 658 401 411 191 273 746 920 898 56;
  • 22) 0.685 205 658 401 411 191 273 746 920 898 56 × 2 = 1 + 0.370 411 316 802 822 382 547 493 841 797 12;
  • 23) 0.370 411 316 802 822 382 547 493 841 797 12 × 2 = 0 + 0.740 822 633 605 644 765 094 987 683 594 24;
  • 24) 0.740 822 633 605 644 765 094 987 683 594 24 × 2 = 1 + 0.481 645 267 211 289 530 189 975 367 188 48;
  • 25) 0.481 645 267 211 289 530 189 975 367 188 48 × 2 = 0 + 0.963 290 534 422 579 060 379 950 734 376 96;
  • 26) 0.963 290 534 422 579 060 379 950 734 376 96 × 2 = 1 + 0.926 581 068 845 158 120 759 901 468 753 92;
  • 27) 0.926 581 068 845 158 120 759 901 468 753 92 × 2 = 1 + 0.853 162 137 690 316 241 519 802 937 507 84;
  • 28) 0.853 162 137 690 316 241 519 802 937 507 84 × 2 = 1 + 0.706 324 275 380 632 483 039 605 875 015 68;
  • 29) 0.706 324 275 380 632 483 039 605 875 015 68 × 2 = 1 + 0.412 648 550 761 264 966 079 211 750 031 36;
  • 30) 0.412 648 550 761 264 966 079 211 750 031 36 × 2 = 0 + 0.825 297 101 522 529 932 158 423 500 062 72;
  • 31) 0.825 297 101 522 529 932 158 423 500 062 72 × 2 = 1 + 0.650 594 203 045 059 864 316 847 000 125 44;
  • 32) 0.650 594 203 045 059 864 316 847 000 125 44 × 2 = 1 + 0.301 188 406 090 119 728 633 694 000 250 88;
  • 33) 0.301 188 406 090 119 728 633 694 000 250 88 × 2 = 0 + 0.602 376 812 180 239 457 267 388 000 501 76;
  • 34) 0.602 376 812 180 239 457 267 388 000 501 76 × 2 = 1 + 0.204 753 624 360 478 914 534 776 001 003 52;
  • 35) 0.204 753 624 360 478 914 534 776 001 003 52 × 2 = 0 + 0.409 507 248 720 957 829 069 552 002 007 04;
  • 36) 0.409 507 248 720 957 829 069 552 002 007 04 × 2 = 0 + 0.819 014 497 441 915 658 139 104 004 014 08;
  • 37) 0.819 014 497 441 915 658 139 104 004 014 08 × 2 = 1 + 0.638 028 994 883 831 316 278 208 008 028 16;
  • 38) 0.638 028 994 883 831 316 278 208 008 028 16 × 2 = 1 + 0.276 057 989 767 662 632 556 416 016 056 32;
  • 39) 0.276 057 989 767 662 632 556 416 016 056 32 × 2 = 0 + 0.552 115 979 535 325 265 112 832 032 112 64;
  • 40) 0.552 115 979 535 325 265 112 832 032 112 64 × 2 = 1 + 0.104 231 959 070 650 530 225 664 064 225 28;
  • 41) 0.104 231 959 070 650 530 225 664 064 225 28 × 2 = 0 + 0.208 463 918 141 301 060 451 328 128 450 56;
  • 42) 0.208 463 918 141 301 060 451 328 128 450 56 × 2 = 0 + 0.416 927 836 282 602 120 902 656 256 901 12;
  • 43) 0.416 927 836 282 602 120 902 656 256 901 12 × 2 = 0 + 0.833 855 672 565 204 241 805 312 513 802 24;
  • 44) 0.833 855 672 565 204 241 805 312 513 802 24 × 2 = 1 + 0.667 711 345 130 408 483 610 625 027 604 48;
  • 45) 0.667 711 345 130 408 483 610 625 027 604 48 × 2 = 1 + 0.335 422 690 260 816 967 221 250 055 208 96;
  • 46) 0.335 422 690 260 816 967 221 250 055 208 96 × 2 = 0 + 0.670 845 380 521 633 934 442 500 110 417 92;
  • 47) 0.670 845 380 521 633 934 442 500 110 417 92 × 2 = 1 + 0.341 690 761 043 267 868 885 000 220 835 84;
  • 48) 0.341 690 761 043 267 868 885 000 220 835 84 × 2 = 0 + 0.683 381 522 086 535 737 770 000 441 671 68;
  • 49) 0.683 381 522 086 535 737 770 000 441 671 68 × 2 = 1 + 0.366 763 044 173 071 475 540 000 883 343 36;
  • 50) 0.366 763 044 173 071 475 540 000 883 343 36 × 2 = 0 + 0.733 526 088 346 142 951 080 001 766 686 72;
  • 51) 0.733 526 088 346 142 951 080 001 766 686 72 × 2 = 1 + 0.467 052 176 692 285 902 160 003 533 373 44;
  • 52) 0.467 052 176 692 285 902 160 003 533 373 44 × 2 = 0 + 0.934 104 353 384 571 804 320 007 066 746 88;
  • 53) 0.934 104 353 384 571 804 320 007 066 746 88 × 2 = 1 + 0.868 208 706 769 143 608 640 014 133 493 76;
  • 54) 0.868 208 706 769 143 608 640 014 133 493 76 × 2 = 1 + 0.736 417 413 538 287 217 280 028 266 987 52;
  • 55) 0.736 417 413 538 287 217 280 028 266 987 52 × 2 = 1 + 0.472 834 827 076 574 434 560 056 533 975 04;
  • 56) 0.472 834 827 076 574 434 560 056 533 975 04 × 2 = 0 + 0.945 669 654 153 148 869 120 113 067 950 08;
  • 57) 0.945 669 654 153 148 869 120 113 067 950 08 × 2 = 1 + 0.891 339 308 306 297 738 240 226 135 900 16;
  • 58) 0.891 339 308 306 297 738 240 226 135 900 16 × 2 = 1 + 0.782 678 616 612 595 476 480 452 271 800 32;
  • 59) 0.782 678 616 612 595 476 480 452 271 800 32 × 2 = 1 + 0.565 357 233 225 190 952 960 904 543 600 64;
  • 60) 0.565 357 233 225 190 952 960 904 543 600 64 × 2 = 1 + 0.130 714 466 450 381 905 921 809 087 201 28;
  • 61) 0.130 714 466 450 381 905 921 809 087 201 28 × 2 = 0 + 0.261 428 932 900 763 811 843 618 174 402 56;
  • 62) 0.261 428 932 900 763 811 843 618 174 402 56 × 2 = 0 + 0.522 857 865 801 527 623 687 236 348 805 12;
  • 63) 0.522 857 865 801 527 623 687 236 348 805 12 × 2 = 1 + 0.045 715 731 603 055 247 374 472 697 610 24;
  • 64) 0.045 715 731 603 055 247 374 472 697 610 24 × 2 = 0 + 0.091 431 463 206 110 494 748 945 395 220 48;
  • 65) 0.091 431 463 206 110 494 748 945 395 220 48 × 2 = 0 + 0.182 862 926 412 220 989 497 890 790 440 96;
  • 66) 0.182 862 926 412 220 989 497 890 790 440 96 × 2 = 0 + 0.365 725 852 824 441 978 995 781 580 881 92;
  • 67) 0.365 725 852 824 441 978 995 781 580 881 92 × 2 = 0 + 0.731 451 705 648 883 957 991 563 161 763 84;
  • 68) 0.731 451 705 648 883 957 991 563 161 763 84 × 2 = 1 + 0.462 903 411 297 767 915 983 126 323 527 68;

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 53(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 53(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 53(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 53 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

Share

» Convert decimal 0.000 020 830 729 321 671 205 134 999 154 73 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 08, 2026 [August]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: August - by our visitors

» Improve your social skills: The Art of Strategic Ambiguity. Gemini Notebook YouTube Video of 6.59 minutes

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