0.000 020 830 729 321 671 205 134 999 154 509 347 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 347(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
0.000 020 830 729 321 671 205 134 999 154 509 347(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: 0.000 020 830 729 321 671 205 134 999 154 509 347.

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) 0.000 020 830 729 321 671 205 134 999 154 509 347 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 018 694;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 018 694 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 037 388;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 037 388 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 074 776;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 074 776 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 149 552;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 149 552 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 299 104;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 299 104 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 598 208;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 598 208 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 196 416;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 196 416 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 392 832;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 392 832 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 785 664;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 785 664 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 571 328;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 571 328 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 142 656;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 142 656 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 870 285 312;
  • 13) 0.085 322 667 301 565 256 232 956 536 870 285 312 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 740 570 624;
  • 14) 0.170 645 334 603 130 512 465 913 073 740 570 624 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 481 141 248;
  • 15) 0.341 290 669 206 261 024 931 826 147 481 141 248 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 962 282 496;
  • 16) 0.682 581 338 412 522 049 863 652 294 962 282 496 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 924 564 992;
  • 17) 0.365 162 676 825 044 099 727 304 589 924 564 992 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 849 129 984;
  • 18) 0.730 325 353 650 088 199 454 609 179 849 129 984 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 698 259 968;
  • 19) 0.460 650 707 300 176 398 909 218 359 698 259 968 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 396 519 936;
  • 20) 0.921 301 414 600 352 797 818 436 719 396 519 936 × 2 = 1 + 0.842 602 829 200 705 595 636 873 438 793 039 872;
  • 21) 0.842 602 829 200 705 595 636 873 438 793 039 872 × 2 = 1 + 0.685 205 658 401 411 191 273 746 877 586 079 744;
  • 22) 0.685 205 658 401 411 191 273 746 877 586 079 744 × 2 = 1 + 0.370 411 316 802 822 382 547 493 755 172 159 488;
  • 23) 0.370 411 316 802 822 382 547 493 755 172 159 488 × 2 = 0 + 0.740 822 633 605 644 765 094 987 510 344 318 976;
  • 24) 0.740 822 633 605 644 765 094 987 510 344 318 976 × 2 = 1 + 0.481 645 267 211 289 530 189 975 020 688 637 952;
  • 25) 0.481 645 267 211 289 530 189 975 020 688 637 952 × 2 = 0 + 0.963 290 534 422 579 060 379 950 041 377 275 904;
  • 26) 0.963 290 534 422 579 060 379 950 041 377 275 904 × 2 = 1 + 0.926 581 068 845 158 120 759 900 082 754 551 808;
  • 27) 0.926 581 068 845 158 120 759 900 082 754 551 808 × 2 = 1 + 0.853 162 137 690 316 241 519 800 165 509 103 616;
  • 28) 0.853 162 137 690 316 241 519 800 165 509 103 616 × 2 = 1 + 0.706 324 275 380 632 483 039 600 331 018 207 232;
  • 29) 0.706 324 275 380 632 483 039 600 331 018 207 232 × 2 = 1 + 0.412 648 550 761 264 966 079 200 662 036 414 464;
  • 30) 0.412 648 550 761 264 966 079 200 662 036 414 464 × 2 = 0 + 0.825 297 101 522 529 932 158 401 324 072 828 928;
  • 31) 0.825 297 101 522 529 932 158 401 324 072 828 928 × 2 = 1 + 0.650 594 203 045 059 864 316 802 648 145 657 856;
  • 32) 0.650 594 203 045 059 864 316 802 648 145 657 856 × 2 = 1 + 0.301 188 406 090 119 728 633 605 296 291 315 712;
  • 33) 0.301 188 406 090 119 728 633 605 296 291 315 712 × 2 = 0 + 0.602 376 812 180 239 457 267 210 592 582 631 424;
  • 34) 0.602 376 812 180 239 457 267 210 592 582 631 424 × 2 = 1 + 0.204 753 624 360 478 914 534 421 185 165 262 848;
  • 35) 0.204 753 624 360 478 914 534 421 185 165 262 848 × 2 = 0 + 0.409 507 248 720 957 829 068 842 370 330 525 696;
  • 36) 0.409 507 248 720 957 829 068 842 370 330 525 696 × 2 = 0 + 0.819 014 497 441 915 658 137 684 740 661 051 392;
  • 37) 0.819 014 497 441 915 658 137 684 740 661 051 392 × 2 = 1 + 0.638 028 994 883 831 316 275 369 481 322 102 784;
  • 38) 0.638 028 994 883 831 316 275 369 481 322 102 784 × 2 = 1 + 0.276 057 989 767 662 632 550 738 962 644 205 568;
  • 39) 0.276 057 989 767 662 632 550 738 962 644 205 568 × 2 = 0 + 0.552 115 979 535 325 265 101 477 925 288 411 136;
  • 40) 0.552 115 979 535 325 265 101 477 925 288 411 136 × 2 = 1 + 0.104 231 959 070 650 530 202 955 850 576 822 272;
  • 41) 0.104 231 959 070 650 530 202 955 850 576 822 272 × 2 = 0 + 0.208 463 918 141 301 060 405 911 701 153 644 544;
  • 42) 0.208 463 918 141 301 060 405 911 701 153 644 544 × 2 = 0 + 0.416 927 836 282 602 120 811 823 402 307 289 088;
  • 43) 0.416 927 836 282 602 120 811 823 402 307 289 088 × 2 = 0 + 0.833 855 672 565 204 241 623 646 804 614 578 176;
  • 44) 0.833 855 672 565 204 241 623 646 804 614 578 176 × 2 = 1 + 0.667 711 345 130 408 483 247 293 609 229 156 352;
  • 45) 0.667 711 345 130 408 483 247 293 609 229 156 352 × 2 = 1 + 0.335 422 690 260 816 966 494 587 218 458 312 704;
  • 46) 0.335 422 690 260 816 966 494 587 218 458 312 704 × 2 = 0 + 0.670 845 380 521 633 932 989 174 436 916 625 408;
  • 47) 0.670 845 380 521 633 932 989 174 436 916 625 408 × 2 = 1 + 0.341 690 761 043 267 865 978 348 873 833 250 816;
  • 48) 0.341 690 761 043 267 865 978 348 873 833 250 816 × 2 = 0 + 0.683 381 522 086 535 731 956 697 747 666 501 632;
  • 49) 0.683 381 522 086 535 731 956 697 747 666 501 632 × 2 = 1 + 0.366 763 044 173 071 463 913 395 495 333 003 264;
  • 50) 0.366 763 044 173 071 463 913 395 495 333 003 264 × 2 = 0 + 0.733 526 088 346 142 927 826 790 990 666 006 528;
  • 51) 0.733 526 088 346 142 927 826 790 990 666 006 528 × 2 = 1 + 0.467 052 176 692 285 855 653 581 981 332 013 056;
  • 52) 0.467 052 176 692 285 855 653 581 981 332 013 056 × 2 = 0 + 0.934 104 353 384 571 711 307 163 962 664 026 112;
  • 53) 0.934 104 353 384 571 711 307 163 962 664 026 112 × 2 = 1 + 0.868 208 706 769 143 422 614 327 925 328 052 224;
  • 54) 0.868 208 706 769 143 422 614 327 925 328 052 224 × 2 = 1 + 0.736 417 413 538 286 845 228 655 850 656 104 448;
  • 55) 0.736 417 413 538 286 845 228 655 850 656 104 448 × 2 = 1 + 0.472 834 827 076 573 690 457 311 701 312 208 896;
  • 56) 0.472 834 827 076 573 690 457 311 701 312 208 896 × 2 = 0 + 0.945 669 654 153 147 380 914 623 402 624 417 792;
  • 57) 0.945 669 654 153 147 380 914 623 402 624 417 792 × 2 = 1 + 0.891 339 308 306 294 761 829 246 805 248 835 584;
  • 58) 0.891 339 308 306 294 761 829 246 805 248 835 584 × 2 = 1 + 0.782 678 616 612 589 523 658 493 610 497 671 168;
  • 59) 0.782 678 616 612 589 523 658 493 610 497 671 168 × 2 = 1 + 0.565 357 233 225 179 047 316 987 220 995 342 336;
  • 60) 0.565 357 233 225 179 047 316 987 220 995 342 336 × 2 = 1 + 0.130 714 466 450 358 094 633 974 441 990 684 672;
  • 61) 0.130 714 466 450 358 094 633 974 441 990 684 672 × 2 = 0 + 0.261 428 932 900 716 189 267 948 883 981 369 344;
  • 62) 0.261 428 932 900 716 189 267 948 883 981 369 344 × 2 = 0 + 0.522 857 865 801 432 378 535 897 767 962 738 688;
  • 63) 0.522 857 865 801 432 378 535 897 767 962 738 688 × 2 = 1 + 0.045 715 731 602 864 757 071 795 535 925 477 376;
  • 64) 0.045 715 731 602 864 757 071 795 535 925 477 376 × 2 = 0 + 0.091 431 463 205 729 514 143 591 071 850 954 752;
  • 65) 0.091 431 463 205 729 514 143 591 071 850 954 752 × 2 = 0 + 0.182 862 926 411 459 028 287 182 143 701 909 504;
  • 66) 0.182 862 926 411 459 028 287 182 143 701 909 504 × 2 = 0 + 0.365 725 852 822 918 056 574 364 287 403 819 008;
  • 67) 0.365 725 852 822 918 056 574 364 287 403 819 008 × 2 = 0 + 0.731 451 705 645 836 113 148 728 574 807 638 016;
  • 68) 0.731 451 705 645 836 113 148 728 574 807 638 016 × 2 = 1 + 0.462 903 411 291 672 226 297 457 149 615 276 032;

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:


0.000 020 830 729 321 671 205 134 999 154 509 347(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:

0.000 020 830 729 321 671 205 134 999 154 509 347(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:


0.000 020 830 729 321 671 205 134 999 154 509 347(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 0.000 020 830 729 321 671 205 134 999 154 509 347 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