0.000 020 830 729 321 671 205 134 999 154 509 636 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 636(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
0.000 020 830 729 321 671 205 134 999 154 509 636(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: 0.000 020 830 729 321 671 205 134 999 154 509 636.

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) 0.000 020 830 729 321 671 205 134 999 154 509 636 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 272;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 272 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 544;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 544 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 088;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 088 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 176;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 176 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 308 352;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 308 352 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 616 704;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 616 704 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 233 408;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 233 408 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 466 816;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 466 816 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 933 632;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 933 632 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 867 264;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 867 264 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 734 528;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 734 528 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 469 056;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 469 056 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 742 938 112;
  • 14) 0.170 645 334 603 130 512 465 913 073 742 938 112 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 485 876 224;
  • 15) 0.341 290 669 206 261 024 931 826 147 485 876 224 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 971 752 448;
  • 16) 0.682 581 338 412 522 049 863 652 294 971 752 448 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 943 504 896;
  • 17) 0.365 162 676 825 044 099 727 304 589 943 504 896 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 887 009 792;
  • 18) 0.730 325 353 650 088 199 454 609 179 887 009 792 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 774 019 584;
  • 19) 0.460 650 707 300 176 398 909 218 359 774 019 584 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 548 039 168;
  • 20) 0.921 301 414 600 352 797 818 436 719 548 039 168 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 096 078 336;
  • 21) 0.842 602 829 200 705 595 636 873 439 096 078 336 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 192 156 672;
  • 22) 0.685 205 658 401 411 191 273 746 878 192 156 672 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 384 313 344;
  • 23) 0.370 411 316 802 822 382 547 493 756 384 313 344 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 768 626 688;
  • 24) 0.740 822 633 605 644 765 094 987 512 768 626 688 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 537 253 376;
  • 25) 0.481 645 267 211 289 530 189 975 025 537 253 376 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 074 506 752;
  • 26) 0.963 290 534 422 579 060 379 950 051 074 506 752 × 2 = 1 + 0.926 581 068 845 158 120 759 900 102 149 013 504;
  • 27) 0.926 581 068 845 158 120 759 900 102 149 013 504 × 2 = 1 + 0.853 162 137 690 316 241 519 800 204 298 027 008;
  • 28) 0.853 162 137 690 316 241 519 800 204 298 027 008 × 2 = 1 + 0.706 324 275 380 632 483 039 600 408 596 054 016;
  • 29) 0.706 324 275 380 632 483 039 600 408 596 054 016 × 2 = 1 + 0.412 648 550 761 264 966 079 200 817 192 108 032;
  • 30) 0.412 648 550 761 264 966 079 200 817 192 108 032 × 2 = 0 + 0.825 297 101 522 529 932 158 401 634 384 216 064;
  • 31) 0.825 297 101 522 529 932 158 401 634 384 216 064 × 2 = 1 + 0.650 594 203 045 059 864 316 803 268 768 432 128;
  • 32) 0.650 594 203 045 059 864 316 803 268 768 432 128 × 2 = 1 + 0.301 188 406 090 119 728 633 606 537 536 864 256;
  • 33) 0.301 188 406 090 119 728 633 606 537 536 864 256 × 2 = 0 + 0.602 376 812 180 239 457 267 213 075 073 728 512;
  • 34) 0.602 376 812 180 239 457 267 213 075 073 728 512 × 2 = 1 + 0.204 753 624 360 478 914 534 426 150 147 457 024;
  • 35) 0.204 753 624 360 478 914 534 426 150 147 457 024 × 2 = 0 + 0.409 507 248 720 957 829 068 852 300 294 914 048;
  • 36) 0.409 507 248 720 957 829 068 852 300 294 914 048 × 2 = 0 + 0.819 014 497 441 915 658 137 704 600 589 828 096;
  • 37) 0.819 014 497 441 915 658 137 704 600 589 828 096 × 2 = 1 + 0.638 028 994 883 831 316 275 409 201 179 656 192;
  • 38) 0.638 028 994 883 831 316 275 409 201 179 656 192 × 2 = 1 + 0.276 057 989 767 662 632 550 818 402 359 312 384;
  • 39) 0.276 057 989 767 662 632 550 818 402 359 312 384 × 2 = 0 + 0.552 115 979 535 325 265 101 636 804 718 624 768;
  • 40) 0.552 115 979 535 325 265 101 636 804 718 624 768 × 2 = 1 + 0.104 231 959 070 650 530 203 273 609 437 249 536;
  • 41) 0.104 231 959 070 650 530 203 273 609 437 249 536 × 2 = 0 + 0.208 463 918 141 301 060 406 547 218 874 499 072;
  • 42) 0.208 463 918 141 301 060 406 547 218 874 499 072 × 2 = 0 + 0.416 927 836 282 602 120 813 094 437 748 998 144;
  • 43) 0.416 927 836 282 602 120 813 094 437 748 998 144 × 2 = 0 + 0.833 855 672 565 204 241 626 188 875 497 996 288;
  • 44) 0.833 855 672 565 204 241 626 188 875 497 996 288 × 2 = 1 + 0.667 711 345 130 408 483 252 377 750 995 992 576;
  • 45) 0.667 711 345 130 408 483 252 377 750 995 992 576 × 2 = 1 + 0.335 422 690 260 816 966 504 755 501 991 985 152;
  • 46) 0.335 422 690 260 816 966 504 755 501 991 985 152 × 2 = 0 + 0.670 845 380 521 633 933 009 511 003 983 970 304;
  • 47) 0.670 845 380 521 633 933 009 511 003 983 970 304 × 2 = 1 + 0.341 690 761 043 267 866 019 022 007 967 940 608;
  • 48) 0.341 690 761 043 267 866 019 022 007 967 940 608 × 2 = 0 + 0.683 381 522 086 535 732 038 044 015 935 881 216;
  • 49) 0.683 381 522 086 535 732 038 044 015 935 881 216 × 2 = 1 + 0.366 763 044 173 071 464 076 088 031 871 762 432;
  • 50) 0.366 763 044 173 071 464 076 088 031 871 762 432 × 2 = 0 + 0.733 526 088 346 142 928 152 176 063 743 524 864;
  • 51) 0.733 526 088 346 142 928 152 176 063 743 524 864 × 2 = 1 + 0.467 052 176 692 285 856 304 352 127 487 049 728;
  • 52) 0.467 052 176 692 285 856 304 352 127 487 049 728 × 2 = 0 + 0.934 104 353 384 571 712 608 704 254 974 099 456;
  • 53) 0.934 104 353 384 571 712 608 704 254 974 099 456 × 2 = 1 + 0.868 208 706 769 143 425 217 408 509 948 198 912;
  • 54) 0.868 208 706 769 143 425 217 408 509 948 198 912 × 2 = 1 + 0.736 417 413 538 286 850 434 817 019 896 397 824;
  • 55) 0.736 417 413 538 286 850 434 817 019 896 397 824 × 2 = 1 + 0.472 834 827 076 573 700 869 634 039 792 795 648;
  • 56) 0.472 834 827 076 573 700 869 634 039 792 795 648 × 2 = 0 + 0.945 669 654 153 147 401 739 268 079 585 591 296;
  • 57) 0.945 669 654 153 147 401 739 268 079 585 591 296 × 2 = 1 + 0.891 339 308 306 294 803 478 536 159 171 182 592;
  • 58) 0.891 339 308 306 294 803 478 536 159 171 182 592 × 2 = 1 + 0.782 678 616 612 589 606 957 072 318 342 365 184;
  • 59) 0.782 678 616 612 589 606 957 072 318 342 365 184 × 2 = 1 + 0.565 357 233 225 179 213 914 144 636 684 730 368;
  • 60) 0.565 357 233 225 179 213 914 144 636 684 730 368 × 2 = 1 + 0.130 714 466 450 358 427 828 289 273 369 460 736;
  • 61) 0.130 714 466 450 358 427 828 289 273 369 460 736 × 2 = 0 + 0.261 428 932 900 716 855 656 578 546 738 921 472;
  • 62) 0.261 428 932 900 716 855 656 578 546 738 921 472 × 2 = 0 + 0.522 857 865 801 433 711 313 157 093 477 842 944;
  • 63) 0.522 857 865 801 433 711 313 157 093 477 842 944 × 2 = 1 + 0.045 715 731 602 867 422 626 314 186 955 685 888;
  • 64) 0.045 715 731 602 867 422 626 314 186 955 685 888 × 2 = 0 + 0.091 431 463 205 734 845 252 628 373 911 371 776;
  • 65) 0.091 431 463 205 734 845 252 628 373 911 371 776 × 2 = 0 + 0.182 862 926 411 469 690 505 256 747 822 743 552;
  • 66) 0.182 862 926 411 469 690 505 256 747 822 743 552 × 2 = 0 + 0.365 725 852 822 939 381 010 513 495 645 487 104;
  • 67) 0.365 725 852 822 939 381 010 513 495 645 487 104 × 2 = 0 + 0.731 451 705 645 878 762 021 026 991 290 974 208;
  • 68) 0.731 451 705 645 878 762 021 026 991 290 974 208 × 2 = 1 + 0.462 903 411 291 757 524 042 053 982 581 948 416;

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:


0.000 020 830 729 321 671 205 134 999 154 509 636(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:

0.000 020 830 729 321 671 205 134 999 154 509 636(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:


0.000 020 830 729 321 671 205 134 999 154 509 636(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 0.000 020 830 729 321 671 205 134 999 154 509 636 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