0.000 020 830 729 321 671 205 134 999 154 507 68 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 507 68(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
0.000 020 830 729 321 671 205 134 999 154 507 68(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: 0.000 020 830 729 321 671 205 134 999 154 507 68.

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) 0.000 020 830 729 321 671 205 134 999 154 507 68 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 015 36;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 015 36 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 030 72;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 030 72 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 061 44;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 061 44 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 122 88;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 122 88 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 245 76;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 245 76 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 491 52;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 491 52 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 776 983 04;
  • 8) 0.002 666 333 353 173 914 257 279 891 776 983 04 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 553 966 08;
  • 9) 0.005 332 666 706 347 828 514 559 783 553 966 08 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 107 932 16;
  • 10) 0.010 665 333 412 695 657 029 119 567 107 932 16 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 215 864 32;
  • 11) 0.021 330 666 825 391 314 058 239 134 215 864 32 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 431 728 64;
  • 12) 0.042 661 333 650 782 628 116 478 268 431 728 64 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 863 457 28;
  • 13) 0.085 322 667 301 565 256 232 956 536 863 457 28 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 726 914 56;
  • 14) 0.170 645 334 603 130 512 465 913 073 726 914 56 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 453 829 12;
  • 15) 0.341 290 669 206 261 024 931 826 147 453 829 12 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 907 658 24;
  • 16) 0.682 581 338 412 522 049 863 652 294 907 658 24 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 815 316 48;
  • 17) 0.365 162 676 825 044 099 727 304 589 815 316 48 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 630 632 96;
  • 18) 0.730 325 353 650 088 199 454 609 179 630 632 96 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 261 265 92;
  • 19) 0.460 650 707 300 176 398 909 218 359 261 265 92 × 2 = 0 + 0.921 301 414 600 352 797 818 436 718 522 531 84;
  • 20) 0.921 301 414 600 352 797 818 436 718 522 531 84 × 2 = 1 + 0.842 602 829 200 705 595 636 873 437 045 063 68;
  • 21) 0.842 602 829 200 705 595 636 873 437 045 063 68 × 2 = 1 + 0.685 205 658 401 411 191 273 746 874 090 127 36;
  • 22) 0.685 205 658 401 411 191 273 746 874 090 127 36 × 2 = 1 + 0.370 411 316 802 822 382 547 493 748 180 254 72;
  • 23) 0.370 411 316 802 822 382 547 493 748 180 254 72 × 2 = 0 + 0.740 822 633 605 644 765 094 987 496 360 509 44;
  • 24) 0.740 822 633 605 644 765 094 987 496 360 509 44 × 2 = 1 + 0.481 645 267 211 289 530 189 974 992 721 018 88;
  • 25) 0.481 645 267 211 289 530 189 974 992 721 018 88 × 2 = 0 + 0.963 290 534 422 579 060 379 949 985 442 037 76;
  • 26) 0.963 290 534 422 579 060 379 949 985 442 037 76 × 2 = 1 + 0.926 581 068 845 158 120 759 899 970 884 075 52;
  • 27) 0.926 581 068 845 158 120 759 899 970 884 075 52 × 2 = 1 + 0.853 162 137 690 316 241 519 799 941 768 151 04;
  • 28) 0.853 162 137 690 316 241 519 799 941 768 151 04 × 2 = 1 + 0.706 324 275 380 632 483 039 599 883 536 302 08;
  • 29) 0.706 324 275 380 632 483 039 599 883 536 302 08 × 2 = 1 + 0.412 648 550 761 264 966 079 199 767 072 604 16;
  • 30) 0.412 648 550 761 264 966 079 199 767 072 604 16 × 2 = 0 + 0.825 297 101 522 529 932 158 399 534 145 208 32;
  • 31) 0.825 297 101 522 529 932 158 399 534 145 208 32 × 2 = 1 + 0.650 594 203 045 059 864 316 799 068 290 416 64;
  • 32) 0.650 594 203 045 059 864 316 799 068 290 416 64 × 2 = 1 + 0.301 188 406 090 119 728 633 598 136 580 833 28;
  • 33) 0.301 188 406 090 119 728 633 598 136 580 833 28 × 2 = 0 + 0.602 376 812 180 239 457 267 196 273 161 666 56;
  • 34) 0.602 376 812 180 239 457 267 196 273 161 666 56 × 2 = 1 + 0.204 753 624 360 478 914 534 392 546 323 333 12;
  • 35) 0.204 753 624 360 478 914 534 392 546 323 333 12 × 2 = 0 + 0.409 507 248 720 957 829 068 785 092 646 666 24;
  • 36) 0.409 507 248 720 957 829 068 785 092 646 666 24 × 2 = 0 + 0.819 014 497 441 915 658 137 570 185 293 332 48;
  • 37) 0.819 014 497 441 915 658 137 570 185 293 332 48 × 2 = 1 + 0.638 028 994 883 831 316 275 140 370 586 664 96;
  • 38) 0.638 028 994 883 831 316 275 140 370 586 664 96 × 2 = 1 + 0.276 057 989 767 662 632 550 280 741 173 329 92;
  • 39) 0.276 057 989 767 662 632 550 280 741 173 329 92 × 2 = 0 + 0.552 115 979 535 325 265 100 561 482 346 659 84;
  • 40) 0.552 115 979 535 325 265 100 561 482 346 659 84 × 2 = 1 + 0.104 231 959 070 650 530 201 122 964 693 319 68;
  • 41) 0.104 231 959 070 650 530 201 122 964 693 319 68 × 2 = 0 + 0.208 463 918 141 301 060 402 245 929 386 639 36;
  • 42) 0.208 463 918 141 301 060 402 245 929 386 639 36 × 2 = 0 + 0.416 927 836 282 602 120 804 491 858 773 278 72;
  • 43) 0.416 927 836 282 602 120 804 491 858 773 278 72 × 2 = 0 + 0.833 855 672 565 204 241 608 983 717 546 557 44;
  • 44) 0.833 855 672 565 204 241 608 983 717 546 557 44 × 2 = 1 + 0.667 711 345 130 408 483 217 967 435 093 114 88;
  • 45) 0.667 711 345 130 408 483 217 967 435 093 114 88 × 2 = 1 + 0.335 422 690 260 816 966 435 934 870 186 229 76;
  • 46) 0.335 422 690 260 816 966 435 934 870 186 229 76 × 2 = 0 + 0.670 845 380 521 633 932 871 869 740 372 459 52;
  • 47) 0.670 845 380 521 633 932 871 869 740 372 459 52 × 2 = 1 + 0.341 690 761 043 267 865 743 739 480 744 919 04;
  • 48) 0.341 690 761 043 267 865 743 739 480 744 919 04 × 2 = 0 + 0.683 381 522 086 535 731 487 478 961 489 838 08;
  • 49) 0.683 381 522 086 535 731 487 478 961 489 838 08 × 2 = 1 + 0.366 763 044 173 071 462 974 957 922 979 676 16;
  • 50) 0.366 763 044 173 071 462 974 957 922 979 676 16 × 2 = 0 + 0.733 526 088 346 142 925 949 915 845 959 352 32;
  • 51) 0.733 526 088 346 142 925 949 915 845 959 352 32 × 2 = 1 + 0.467 052 176 692 285 851 899 831 691 918 704 64;
  • 52) 0.467 052 176 692 285 851 899 831 691 918 704 64 × 2 = 0 + 0.934 104 353 384 571 703 799 663 383 837 409 28;
  • 53) 0.934 104 353 384 571 703 799 663 383 837 409 28 × 2 = 1 + 0.868 208 706 769 143 407 599 326 767 674 818 56;
  • 54) 0.868 208 706 769 143 407 599 326 767 674 818 56 × 2 = 1 + 0.736 417 413 538 286 815 198 653 535 349 637 12;
  • 55) 0.736 417 413 538 286 815 198 653 535 349 637 12 × 2 = 1 + 0.472 834 827 076 573 630 397 307 070 699 274 24;
  • 56) 0.472 834 827 076 573 630 397 307 070 699 274 24 × 2 = 0 + 0.945 669 654 153 147 260 794 614 141 398 548 48;
  • 57) 0.945 669 654 153 147 260 794 614 141 398 548 48 × 2 = 1 + 0.891 339 308 306 294 521 589 228 282 797 096 96;
  • 58) 0.891 339 308 306 294 521 589 228 282 797 096 96 × 2 = 1 + 0.782 678 616 612 589 043 178 456 565 594 193 92;
  • 59) 0.782 678 616 612 589 043 178 456 565 594 193 92 × 2 = 1 + 0.565 357 233 225 178 086 356 913 131 188 387 84;
  • 60) 0.565 357 233 225 178 086 356 913 131 188 387 84 × 2 = 1 + 0.130 714 466 450 356 172 713 826 262 376 775 68;
  • 61) 0.130 714 466 450 356 172 713 826 262 376 775 68 × 2 = 0 + 0.261 428 932 900 712 345 427 652 524 753 551 36;
  • 62) 0.261 428 932 900 712 345 427 652 524 753 551 36 × 2 = 0 + 0.522 857 865 801 424 690 855 305 049 507 102 72;
  • 63) 0.522 857 865 801 424 690 855 305 049 507 102 72 × 2 = 1 + 0.045 715 731 602 849 381 710 610 099 014 205 44;
  • 64) 0.045 715 731 602 849 381 710 610 099 014 205 44 × 2 = 0 + 0.091 431 463 205 698 763 421 220 198 028 410 88;
  • 65) 0.091 431 463 205 698 763 421 220 198 028 410 88 × 2 = 0 + 0.182 862 926 411 397 526 842 440 396 056 821 76;
  • 66) 0.182 862 926 411 397 526 842 440 396 056 821 76 × 2 = 0 + 0.365 725 852 822 795 053 684 880 792 113 643 52;
  • 67) 0.365 725 852 822 795 053 684 880 792 113 643 52 × 2 = 0 + 0.731 451 705 645 590 107 369 761 584 227 287 04;
  • 68) 0.731 451 705 645 590 107 369 761 584 227 287 04 × 2 = 1 + 0.462 903 411 291 180 214 739 523 168 454 574 08;

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:


0.000 020 830 729 321 671 205 134 999 154 507 68(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:

0.000 020 830 729 321 671 205 134 999 154 507 68(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:


0.000 020 830 729 321 671 205 134 999 154 507 68(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 0.000 020 830 729 321 671 205 134 999 154 507 68 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