0.000 020 830 729 321 671 205 134 999 154 509 818 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 818(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
0.000 020 830 729 321 671 205 134 999 154 509 818(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: 0.000 020 830 729 321 671 205 134 999 154 509 818.

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) 0.000 020 830 729 321 671 205 134 999 154 509 818 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 636;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 636 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 039 272;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 039 272 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 078 544;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 078 544 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 157 088;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 157 088 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 314 176;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 314 176 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 628 352;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 628 352 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 256 704;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 256 704 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 513 408;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 513 408 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 109 026 816;
  • 10) 0.010 665 333 412 695 657 029 119 567 109 026 816 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 218 053 632;
  • 11) 0.021 330 666 825 391 314 058 239 134 218 053 632 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 436 107 264;
  • 12) 0.042 661 333 650 782 628 116 478 268 436 107 264 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 872 214 528;
  • 13) 0.085 322 667 301 565 256 232 956 536 872 214 528 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 744 429 056;
  • 14) 0.170 645 334 603 130 512 465 913 073 744 429 056 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 488 858 112;
  • 15) 0.341 290 669 206 261 024 931 826 147 488 858 112 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 977 716 224;
  • 16) 0.682 581 338 412 522 049 863 652 294 977 716 224 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 955 432 448;
  • 17) 0.365 162 676 825 044 099 727 304 589 955 432 448 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 910 864 896;
  • 18) 0.730 325 353 650 088 199 454 609 179 910 864 896 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 821 729 792;
  • 19) 0.460 650 707 300 176 398 909 218 359 821 729 792 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 643 459 584;
  • 20) 0.921 301 414 600 352 797 818 436 719 643 459 584 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 286 919 168;
  • 21) 0.842 602 829 200 705 595 636 873 439 286 919 168 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 573 838 336;
  • 22) 0.685 205 658 401 411 191 273 746 878 573 838 336 × 2 = 1 + 0.370 411 316 802 822 382 547 493 757 147 676 672;
  • 23) 0.370 411 316 802 822 382 547 493 757 147 676 672 × 2 = 0 + 0.740 822 633 605 644 765 094 987 514 295 353 344;
  • 24) 0.740 822 633 605 644 765 094 987 514 295 353 344 × 2 = 1 + 0.481 645 267 211 289 530 189 975 028 590 706 688;
  • 25) 0.481 645 267 211 289 530 189 975 028 590 706 688 × 2 = 0 + 0.963 290 534 422 579 060 379 950 057 181 413 376;
  • 26) 0.963 290 534 422 579 060 379 950 057 181 413 376 × 2 = 1 + 0.926 581 068 845 158 120 759 900 114 362 826 752;
  • 27) 0.926 581 068 845 158 120 759 900 114 362 826 752 × 2 = 1 + 0.853 162 137 690 316 241 519 800 228 725 653 504;
  • 28) 0.853 162 137 690 316 241 519 800 228 725 653 504 × 2 = 1 + 0.706 324 275 380 632 483 039 600 457 451 307 008;
  • 29) 0.706 324 275 380 632 483 039 600 457 451 307 008 × 2 = 1 + 0.412 648 550 761 264 966 079 200 914 902 614 016;
  • 30) 0.412 648 550 761 264 966 079 200 914 902 614 016 × 2 = 0 + 0.825 297 101 522 529 932 158 401 829 805 228 032;
  • 31) 0.825 297 101 522 529 932 158 401 829 805 228 032 × 2 = 1 + 0.650 594 203 045 059 864 316 803 659 610 456 064;
  • 32) 0.650 594 203 045 059 864 316 803 659 610 456 064 × 2 = 1 + 0.301 188 406 090 119 728 633 607 319 220 912 128;
  • 33) 0.301 188 406 090 119 728 633 607 319 220 912 128 × 2 = 0 + 0.602 376 812 180 239 457 267 214 638 441 824 256;
  • 34) 0.602 376 812 180 239 457 267 214 638 441 824 256 × 2 = 1 + 0.204 753 624 360 478 914 534 429 276 883 648 512;
  • 35) 0.204 753 624 360 478 914 534 429 276 883 648 512 × 2 = 0 + 0.409 507 248 720 957 829 068 858 553 767 297 024;
  • 36) 0.409 507 248 720 957 829 068 858 553 767 297 024 × 2 = 0 + 0.819 014 497 441 915 658 137 717 107 534 594 048;
  • 37) 0.819 014 497 441 915 658 137 717 107 534 594 048 × 2 = 1 + 0.638 028 994 883 831 316 275 434 215 069 188 096;
  • 38) 0.638 028 994 883 831 316 275 434 215 069 188 096 × 2 = 1 + 0.276 057 989 767 662 632 550 868 430 138 376 192;
  • 39) 0.276 057 989 767 662 632 550 868 430 138 376 192 × 2 = 0 + 0.552 115 979 535 325 265 101 736 860 276 752 384;
  • 40) 0.552 115 979 535 325 265 101 736 860 276 752 384 × 2 = 1 + 0.104 231 959 070 650 530 203 473 720 553 504 768;
  • 41) 0.104 231 959 070 650 530 203 473 720 553 504 768 × 2 = 0 + 0.208 463 918 141 301 060 406 947 441 107 009 536;
  • 42) 0.208 463 918 141 301 060 406 947 441 107 009 536 × 2 = 0 + 0.416 927 836 282 602 120 813 894 882 214 019 072;
  • 43) 0.416 927 836 282 602 120 813 894 882 214 019 072 × 2 = 0 + 0.833 855 672 565 204 241 627 789 764 428 038 144;
  • 44) 0.833 855 672 565 204 241 627 789 764 428 038 144 × 2 = 1 + 0.667 711 345 130 408 483 255 579 528 856 076 288;
  • 45) 0.667 711 345 130 408 483 255 579 528 856 076 288 × 2 = 1 + 0.335 422 690 260 816 966 511 159 057 712 152 576;
  • 46) 0.335 422 690 260 816 966 511 159 057 712 152 576 × 2 = 0 + 0.670 845 380 521 633 933 022 318 115 424 305 152;
  • 47) 0.670 845 380 521 633 933 022 318 115 424 305 152 × 2 = 1 + 0.341 690 761 043 267 866 044 636 230 848 610 304;
  • 48) 0.341 690 761 043 267 866 044 636 230 848 610 304 × 2 = 0 + 0.683 381 522 086 535 732 089 272 461 697 220 608;
  • 49) 0.683 381 522 086 535 732 089 272 461 697 220 608 × 2 = 1 + 0.366 763 044 173 071 464 178 544 923 394 441 216;
  • 50) 0.366 763 044 173 071 464 178 544 923 394 441 216 × 2 = 0 + 0.733 526 088 346 142 928 357 089 846 788 882 432;
  • 51) 0.733 526 088 346 142 928 357 089 846 788 882 432 × 2 = 1 + 0.467 052 176 692 285 856 714 179 693 577 764 864;
  • 52) 0.467 052 176 692 285 856 714 179 693 577 764 864 × 2 = 0 + 0.934 104 353 384 571 713 428 359 387 155 529 728;
  • 53) 0.934 104 353 384 571 713 428 359 387 155 529 728 × 2 = 1 + 0.868 208 706 769 143 426 856 718 774 311 059 456;
  • 54) 0.868 208 706 769 143 426 856 718 774 311 059 456 × 2 = 1 + 0.736 417 413 538 286 853 713 437 548 622 118 912;
  • 55) 0.736 417 413 538 286 853 713 437 548 622 118 912 × 2 = 1 + 0.472 834 827 076 573 707 426 875 097 244 237 824;
  • 56) 0.472 834 827 076 573 707 426 875 097 244 237 824 × 2 = 0 + 0.945 669 654 153 147 414 853 750 194 488 475 648;
  • 57) 0.945 669 654 153 147 414 853 750 194 488 475 648 × 2 = 1 + 0.891 339 308 306 294 829 707 500 388 976 951 296;
  • 58) 0.891 339 308 306 294 829 707 500 388 976 951 296 × 2 = 1 + 0.782 678 616 612 589 659 415 000 777 953 902 592;
  • 59) 0.782 678 616 612 589 659 415 000 777 953 902 592 × 2 = 1 + 0.565 357 233 225 179 318 830 001 555 907 805 184;
  • 60) 0.565 357 233 225 179 318 830 001 555 907 805 184 × 2 = 1 + 0.130 714 466 450 358 637 660 003 111 815 610 368;
  • 61) 0.130 714 466 450 358 637 660 003 111 815 610 368 × 2 = 0 + 0.261 428 932 900 717 275 320 006 223 631 220 736;
  • 62) 0.261 428 932 900 717 275 320 006 223 631 220 736 × 2 = 0 + 0.522 857 865 801 434 550 640 012 447 262 441 472;
  • 63) 0.522 857 865 801 434 550 640 012 447 262 441 472 × 2 = 1 + 0.045 715 731 602 869 101 280 024 894 524 882 944;
  • 64) 0.045 715 731 602 869 101 280 024 894 524 882 944 × 2 = 0 + 0.091 431 463 205 738 202 560 049 789 049 765 888;
  • 65) 0.091 431 463 205 738 202 560 049 789 049 765 888 × 2 = 0 + 0.182 862 926 411 476 405 120 099 578 099 531 776;
  • 66) 0.182 862 926 411 476 405 120 099 578 099 531 776 × 2 = 0 + 0.365 725 852 822 952 810 240 199 156 199 063 552;
  • 67) 0.365 725 852 822 952 810 240 199 156 199 063 552 × 2 = 0 + 0.731 451 705 645 905 620 480 398 312 398 127 104;
  • 68) 0.731 451 705 645 905 620 480 398 312 398 127 104 × 2 = 1 + 0.462 903 411 291 811 240 960 796 624 796 254 208;

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:


0.000 020 830 729 321 671 205 134 999 154 509 818(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:

0.000 020 830 729 321 671 205 134 999 154 509 818(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:


0.000 020 830 729 321 671 205 134 999 154 509 818(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 0.000 020 830 729 321 671 205 134 999 154 509 818 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