0.000 020 830 729 321 671 205 134 999 154 509 477 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 477(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
0.000 020 830 729 321 671 205 134 999 154 509 477(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: 0.000 020 830 729 321 671 205 134 999 154 509 477.

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) 0.000 020 830 729 321 671 205 134 999 154 509 477 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 018 954;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 018 954 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 037 908;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 037 908 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 075 816;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 075 816 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 151 632;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 151 632 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 303 264;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 303 264 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 606 528;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 606 528 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 213 056;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 213 056 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 426 112;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 426 112 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 852 224;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 852 224 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 704 448;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 704 448 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 408 896;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 408 896 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 870 817 792;
  • 13) 0.085 322 667 301 565 256 232 956 536 870 817 792 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 741 635 584;
  • 14) 0.170 645 334 603 130 512 465 913 073 741 635 584 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 483 271 168;
  • 15) 0.341 290 669 206 261 024 931 826 147 483 271 168 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 966 542 336;
  • 16) 0.682 581 338 412 522 049 863 652 294 966 542 336 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 933 084 672;
  • 17) 0.365 162 676 825 044 099 727 304 589 933 084 672 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 866 169 344;
  • 18) 0.730 325 353 650 088 199 454 609 179 866 169 344 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 732 338 688;
  • 19) 0.460 650 707 300 176 398 909 218 359 732 338 688 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 464 677 376;
  • 20) 0.921 301 414 600 352 797 818 436 719 464 677 376 × 2 = 1 + 0.842 602 829 200 705 595 636 873 438 929 354 752;
  • 21) 0.842 602 829 200 705 595 636 873 438 929 354 752 × 2 = 1 + 0.685 205 658 401 411 191 273 746 877 858 709 504;
  • 22) 0.685 205 658 401 411 191 273 746 877 858 709 504 × 2 = 1 + 0.370 411 316 802 822 382 547 493 755 717 419 008;
  • 23) 0.370 411 316 802 822 382 547 493 755 717 419 008 × 2 = 0 + 0.740 822 633 605 644 765 094 987 511 434 838 016;
  • 24) 0.740 822 633 605 644 765 094 987 511 434 838 016 × 2 = 1 + 0.481 645 267 211 289 530 189 975 022 869 676 032;
  • 25) 0.481 645 267 211 289 530 189 975 022 869 676 032 × 2 = 0 + 0.963 290 534 422 579 060 379 950 045 739 352 064;
  • 26) 0.963 290 534 422 579 060 379 950 045 739 352 064 × 2 = 1 + 0.926 581 068 845 158 120 759 900 091 478 704 128;
  • 27) 0.926 581 068 845 158 120 759 900 091 478 704 128 × 2 = 1 + 0.853 162 137 690 316 241 519 800 182 957 408 256;
  • 28) 0.853 162 137 690 316 241 519 800 182 957 408 256 × 2 = 1 + 0.706 324 275 380 632 483 039 600 365 914 816 512;
  • 29) 0.706 324 275 380 632 483 039 600 365 914 816 512 × 2 = 1 + 0.412 648 550 761 264 966 079 200 731 829 633 024;
  • 30) 0.412 648 550 761 264 966 079 200 731 829 633 024 × 2 = 0 + 0.825 297 101 522 529 932 158 401 463 659 266 048;
  • 31) 0.825 297 101 522 529 932 158 401 463 659 266 048 × 2 = 1 + 0.650 594 203 045 059 864 316 802 927 318 532 096;
  • 32) 0.650 594 203 045 059 864 316 802 927 318 532 096 × 2 = 1 + 0.301 188 406 090 119 728 633 605 854 637 064 192;
  • 33) 0.301 188 406 090 119 728 633 605 854 637 064 192 × 2 = 0 + 0.602 376 812 180 239 457 267 211 709 274 128 384;
  • 34) 0.602 376 812 180 239 457 267 211 709 274 128 384 × 2 = 1 + 0.204 753 624 360 478 914 534 423 418 548 256 768;
  • 35) 0.204 753 624 360 478 914 534 423 418 548 256 768 × 2 = 0 + 0.409 507 248 720 957 829 068 846 837 096 513 536;
  • 36) 0.409 507 248 720 957 829 068 846 837 096 513 536 × 2 = 0 + 0.819 014 497 441 915 658 137 693 674 193 027 072;
  • 37) 0.819 014 497 441 915 658 137 693 674 193 027 072 × 2 = 1 + 0.638 028 994 883 831 316 275 387 348 386 054 144;
  • 38) 0.638 028 994 883 831 316 275 387 348 386 054 144 × 2 = 1 + 0.276 057 989 767 662 632 550 774 696 772 108 288;
  • 39) 0.276 057 989 767 662 632 550 774 696 772 108 288 × 2 = 0 + 0.552 115 979 535 325 265 101 549 393 544 216 576;
  • 40) 0.552 115 979 535 325 265 101 549 393 544 216 576 × 2 = 1 + 0.104 231 959 070 650 530 203 098 787 088 433 152;
  • 41) 0.104 231 959 070 650 530 203 098 787 088 433 152 × 2 = 0 + 0.208 463 918 141 301 060 406 197 574 176 866 304;
  • 42) 0.208 463 918 141 301 060 406 197 574 176 866 304 × 2 = 0 + 0.416 927 836 282 602 120 812 395 148 353 732 608;
  • 43) 0.416 927 836 282 602 120 812 395 148 353 732 608 × 2 = 0 + 0.833 855 672 565 204 241 624 790 296 707 465 216;
  • 44) 0.833 855 672 565 204 241 624 790 296 707 465 216 × 2 = 1 + 0.667 711 345 130 408 483 249 580 593 414 930 432;
  • 45) 0.667 711 345 130 408 483 249 580 593 414 930 432 × 2 = 1 + 0.335 422 690 260 816 966 499 161 186 829 860 864;
  • 46) 0.335 422 690 260 816 966 499 161 186 829 860 864 × 2 = 0 + 0.670 845 380 521 633 932 998 322 373 659 721 728;
  • 47) 0.670 845 380 521 633 932 998 322 373 659 721 728 × 2 = 1 + 0.341 690 761 043 267 865 996 644 747 319 443 456;
  • 48) 0.341 690 761 043 267 865 996 644 747 319 443 456 × 2 = 0 + 0.683 381 522 086 535 731 993 289 494 638 886 912;
  • 49) 0.683 381 522 086 535 731 993 289 494 638 886 912 × 2 = 1 + 0.366 763 044 173 071 463 986 578 989 277 773 824;
  • 50) 0.366 763 044 173 071 463 986 578 989 277 773 824 × 2 = 0 + 0.733 526 088 346 142 927 973 157 978 555 547 648;
  • 51) 0.733 526 088 346 142 927 973 157 978 555 547 648 × 2 = 1 + 0.467 052 176 692 285 855 946 315 957 111 095 296;
  • 52) 0.467 052 176 692 285 855 946 315 957 111 095 296 × 2 = 0 + 0.934 104 353 384 571 711 892 631 914 222 190 592;
  • 53) 0.934 104 353 384 571 711 892 631 914 222 190 592 × 2 = 1 + 0.868 208 706 769 143 423 785 263 828 444 381 184;
  • 54) 0.868 208 706 769 143 423 785 263 828 444 381 184 × 2 = 1 + 0.736 417 413 538 286 847 570 527 656 888 762 368;
  • 55) 0.736 417 413 538 286 847 570 527 656 888 762 368 × 2 = 1 + 0.472 834 827 076 573 695 141 055 313 777 524 736;
  • 56) 0.472 834 827 076 573 695 141 055 313 777 524 736 × 2 = 0 + 0.945 669 654 153 147 390 282 110 627 555 049 472;
  • 57) 0.945 669 654 153 147 390 282 110 627 555 049 472 × 2 = 1 + 0.891 339 308 306 294 780 564 221 255 110 098 944;
  • 58) 0.891 339 308 306 294 780 564 221 255 110 098 944 × 2 = 1 + 0.782 678 616 612 589 561 128 442 510 220 197 888;
  • 59) 0.782 678 616 612 589 561 128 442 510 220 197 888 × 2 = 1 + 0.565 357 233 225 179 122 256 885 020 440 395 776;
  • 60) 0.565 357 233 225 179 122 256 885 020 440 395 776 × 2 = 1 + 0.130 714 466 450 358 244 513 770 040 880 791 552;
  • 61) 0.130 714 466 450 358 244 513 770 040 880 791 552 × 2 = 0 + 0.261 428 932 900 716 489 027 540 081 761 583 104;
  • 62) 0.261 428 932 900 716 489 027 540 081 761 583 104 × 2 = 0 + 0.522 857 865 801 432 978 055 080 163 523 166 208;
  • 63) 0.522 857 865 801 432 978 055 080 163 523 166 208 × 2 = 1 + 0.045 715 731 602 865 956 110 160 327 046 332 416;
  • 64) 0.045 715 731 602 865 956 110 160 327 046 332 416 × 2 = 0 + 0.091 431 463 205 731 912 220 320 654 092 664 832;
  • 65) 0.091 431 463 205 731 912 220 320 654 092 664 832 × 2 = 0 + 0.182 862 926 411 463 824 440 641 308 185 329 664;
  • 66) 0.182 862 926 411 463 824 440 641 308 185 329 664 × 2 = 0 + 0.365 725 852 822 927 648 881 282 616 370 659 328;
  • 67) 0.365 725 852 822 927 648 881 282 616 370 659 328 × 2 = 0 + 0.731 451 705 645 855 297 762 565 232 741 318 656;
  • 68) 0.731 451 705 645 855 297 762 565 232 741 318 656 × 2 = 1 + 0.462 903 411 291 710 595 525 130 465 482 637 312;

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:


0.000 020 830 729 321 671 205 134 999 154 509 477(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:

0.000 020 830 729 321 671 205 134 999 154 509 477(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:


0.000 020 830 729 321 671 205 134 999 154 509 477(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 0.000 020 830 729 321 671 205 134 999 154 509 477 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