0.000 020 830 729 321 671 205 134 999 154 473 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 473(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
0.000 020 830 729 321 671 205 134 999 154 473(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: 0.000 020 830 729 321 671 205 134 999 154 473.

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) 0.000 020 830 729 321 671 205 134 999 154 473 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 308 946;
  • 2) 0.000 041 661 458 643 342 410 269 998 308 946 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 617 892;
  • 3) 0.000 083 322 917 286 684 820 539 996 617 892 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 235 784;
  • 4) 0.000 166 645 834 573 369 641 079 993 235 784 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 471 568;
  • 5) 0.000 333 291 669 146 739 282 159 986 471 568 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 943 136;
  • 6) 0.000 666 583 338 293 478 564 319 972 943 136 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 886 272;
  • 7) 0.001 333 166 676 586 957 128 639 945 886 272 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 772 544;
  • 8) 0.002 666 333 353 173 914 257 279 891 772 544 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 545 088;
  • 9) 0.005 332 666 706 347 828 514 559 783 545 088 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 090 176;
  • 10) 0.010 665 333 412 695 657 029 119 567 090 176 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 180 352;
  • 11) 0.021 330 666 825 391 314 058 239 134 180 352 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 360 704;
  • 12) 0.042 661 333 650 782 628 116 478 268 360 704 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 721 408;
  • 13) 0.085 322 667 301 565 256 232 956 536 721 408 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 442 816;
  • 14) 0.170 645 334 603 130 512 465 913 073 442 816 × 2 = 0 + 0.341 290 669 206 261 024 931 826 146 885 632;
  • 15) 0.341 290 669 206 261 024 931 826 146 885 632 × 2 = 0 + 0.682 581 338 412 522 049 863 652 293 771 264;
  • 16) 0.682 581 338 412 522 049 863 652 293 771 264 × 2 = 1 + 0.365 162 676 825 044 099 727 304 587 542 528;
  • 17) 0.365 162 676 825 044 099 727 304 587 542 528 × 2 = 0 + 0.730 325 353 650 088 199 454 609 175 085 056;
  • 18) 0.730 325 353 650 088 199 454 609 175 085 056 × 2 = 1 + 0.460 650 707 300 176 398 909 218 350 170 112;
  • 19) 0.460 650 707 300 176 398 909 218 350 170 112 × 2 = 0 + 0.921 301 414 600 352 797 818 436 700 340 224;
  • 20) 0.921 301 414 600 352 797 818 436 700 340 224 × 2 = 1 + 0.842 602 829 200 705 595 636 873 400 680 448;
  • 21) 0.842 602 829 200 705 595 636 873 400 680 448 × 2 = 1 + 0.685 205 658 401 411 191 273 746 801 360 896;
  • 22) 0.685 205 658 401 411 191 273 746 801 360 896 × 2 = 1 + 0.370 411 316 802 822 382 547 493 602 721 792;
  • 23) 0.370 411 316 802 822 382 547 493 602 721 792 × 2 = 0 + 0.740 822 633 605 644 765 094 987 205 443 584;
  • 24) 0.740 822 633 605 644 765 094 987 205 443 584 × 2 = 1 + 0.481 645 267 211 289 530 189 974 410 887 168;
  • 25) 0.481 645 267 211 289 530 189 974 410 887 168 × 2 = 0 + 0.963 290 534 422 579 060 379 948 821 774 336;
  • 26) 0.963 290 534 422 579 060 379 948 821 774 336 × 2 = 1 + 0.926 581 068 845 158 120 759 897 643 548 672;
  • 27) 0.926 581 068 845 158 120 759 897 643 548 672 × 2 = 1 + 0.853 162 137 690 316 241 519 795 287 097 344;
  • 28) 0.853 162 137 690 316 241 519 795 287 097 344 × 2 = 1 + 0.706 324 275 380 632 483 039 590 574 194 688;
  • 29) 0.706 324 275 380 632 483 039 590 574 194 688 × 2 = 1 + 0.412 648 550 761 264 966 079 181 148 389 376;
  • 30) 0.412 648 550 761 264 966 079 181 148 389 376 × 2 = 0 + 0.825 297 101 522 529 932 158 362 296 778 752;
  • 31) 0.825 297 101 522 529 932 158 362 296 778 752 × 2 = 1 + 0.650 594 203 045 059 864 316 724 593 557 504;
  • 32) 0.650 594 203 045 059 864 316 724 593 557 504 × 2 = 1 + 0.301 188 406 090 119 728 633 449 187 115 008;
  • 33) 0.301 188 406 090 119 728 633 449 187 115 008 × 2 = 0 + 0.602 376 812 180 239 457 266 898 374 230 016;
  • 34) 0.602 376 812 180 239 457 266 898 374 230 016 × 2 = 1 + 0.204 753 624 360 478 914 533 796 748 460 032;
  • 35) 0.204 753 624 360 478 914 533 796 748 460 032 × 2 = 0 + 0.409 507 248 720 957 829 067 593 496 920 064;
  • 36) 0.409 507 248 720 957 829 067 593 496 920 064 × 2 = 0 + 0.819 014 497 441 915 658 135 186 993 840 128;
  • 37) 0.819 014 497 441 915 658 135 186 993 840 128 × 2 = 1 + 0.638 028 994 883 831 316 270 373 987 680 256;
  • 38) 0.638 028 994 883 831 316 270 373 987 680 256 × 2 = 1 + 0.276 057 989 767 662 632 540 747 975 360 512;
  • 39) 0.276 057 989 767 662 632 540 747 975 360 512 × 2 = 0 + 0.552 115 979 535 325 265 081 495 950 721 024;
  • 40) 0.552 115 979 535 325 265 081 495 950 721 024 × 2 = 1 + 0.104 231 959 070 650 530 162 991 901 442 048;
  • 41) 0.104 231 959 070 650 530 162 991 901 442 048 × 2 = 0 + 0.208 463 918 141 301 060 325 983 802 884 096;
  • 42) 0.208 463 918 141 301 060 325 983 802 884 096 × 2 = 0 + 0.416 927 836 282 602 120 651 967 605 768 192;
  • 43) 0.416 927 836 282 602 120 651 967 605 768 192 × 2 = 0 + 0.833 855 672 565 204 241 303 935 211 536 384;
  • 44) 0.833 855 672 565 204 241 303 935 211 536 384 × 2 = 1 + 0.667 711 345 130 408 482 607 870 423 072 768;
  • 45) 0.667 711 345 130 408 482 607 870 423 072 768 × 2 = 1 + 0.335 422 690 260 816 965 215 740 846 145 536;
  • 46) 0.335 422 690 260 816 965 215 740 846 145 536 × 2 = 0 + 0.670 845 380 521 633 930 431 481 692 291 072;
  • 47) 0.670 845 380 521 633 930 431 481 692 291 072 × 2 = 1 + 0.341 690 761 043 267 860 862 963 384 582 144;
  • 48) 0.341 690 761 043 267 860 862 963 384 582 144 × 2 = 0 + 0.683 381 522 086 535 721 725 926 769 164 288;
  • 49) 0.683 381 522 086 535 721 725 926 769 164 288 × 2 = 1 + 0.366 763 044 173 071 443 451 853 538 328 576;
  • 50) 0.366 763 044 173 071 443 451 853 538 328 576 × 2 = 0 + 0.733 526 088 346 142 886 903 707 076 657 152;
  • 51) 0.733 526 088 346 142 886 903 707 076 657 152 × 2 = 1 + 0.467 052 176 692 285 773 807 414 153 314 304;
  • 52) 0.467 052 176 692 285 773 807 414 153 314 304 × 2 = 0 + 0.934 104 353 384 571 547 614 828 306 628 608;
  • 53) 0.934 104 353 384 571 547 614 828 306 628 608 × 2 = 1 + 0.868 208 706 769 143 095 229 656 613 257 216;
  • 54) 0.868 208 706 769 143 095 229 656 613 257 216 × 2 = 1 + 0.736 417 413 538 286 190 459 313 226 514 432;
  • 55) 0.736 417 413 538 286 190 459 313 226 514 432 × 2 = 1 + 0.472 834 827 076 572 380 918 626 453 028 864;
  • 56) 0.472 834 827 076 572 380 918 626 453 028 864 × 2 = 0 + 0.945 669 654 153 144 761 837 252 906 057 728;
  • 57) 0.945 669 654 153 144 761 837 252 906 057 728 × 2 = 1 + 0.891 339 308 306 289 523 674 505 812 115 456;
  • 58) 0.891 339 308 306 289 523 674 505 812 115 456 × 2 = 1 + 0.782 678 616 612 579 047 349 011 624 230 912;
  • 59) 0.782 678 616 612 579 047 349 011 624 230 912 × 2 = 1 + 0.565 357 233 225 158 094 698 023 248 461 824;
  • 60) 0.565 357 233 225 158 094 698 023 248 461 824 × 2 = 1 + 0.130 714 466 450 316 189 396 046 496 923 648;
  • 61) 0.130 714 466 450 316 189 396 046 496 923 648 × 2 = 0 + 0.261 428 932 900 632 378 792 092 993 847 296;
  • 62) 0.261 428 932 900 632 378 792 092 993 847 296 × 2 = 0 + 0.522 857 865 801 264 757 584 185 987 694 592;
  • 63) 0.522 857 865 801 264 757 584 185 987 694 592 × 2 = 1 + 0.045 715 731 602 529 515 168 371 975 389 184;
  • 64) 0.045 715 731 602 529 515 168 371 975 389 184 × 2 = 0 + 0.091 431 463 205 059 030 336 743 950 778 368;
  • 65) 0.091 431 463 205 059 030 336 743 950 778 368 × 2 = 0 + 0.182 862 926 410 118 060 673 487 901 556 736;
  • 66) 0.182 862 926 410 118 060 673 487 901 556 736 × 2 = 0 + 0.365 725 852 820 236 121 346 975 803 113 472;
  • 67) 0.365 725 852 820 236 121 346 975 803 113 472 × 2 = 0 + 0.731 451 705 640 472 242 693 951 606 226 944;
  • 68) 0.731 451 705 640 472 242 693 951 606 226 944 × 2 = 1 + 0.462 903 411 280 944 485 387 903 212 453 888;

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:


0.000 020 830 729 321 671 205 134 999 154 473(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:

0.000 020 830 729 321 671 205 134 999 154 473(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:


0.000 020 830 729 321 671 205 134 999 154 473(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 0.000 020 830 729 321 671 205 134 999 154 473 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 0.000 020 830 729 321 671 205 134 999 154 552 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: Are You Using Communication by Design, or by Default? Gemini Notebook YouTube Video of 6.33 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