0.000 020 830 729 321 671 205 134 999 154 507 55 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 55(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 55(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 55.

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 55 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 015 1;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 015 1 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 030 2;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 030 2 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 060 4;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 060 4 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 120 8;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 120 8 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 241 6;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 241 6 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 483 2;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 483 2 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 776 966 4;
  • 8) 0.002 666 333 353 173 914 257 279 891 776 966 4 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 553 932 8;
  • 9) 0.005 332 666 706 347 828 514 559 783 553 932 8 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 107 865 6;
  • 10) 0.010 665 333 412 695 657 029 119 567 107 865 6 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 215 731 2;
  • 11) 0.021 330 666 825 391 314 058 239 134 215 731 2 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 431 462 4;
  • 12) 0.042 661 333 650 782 628 116 478 268 431 462 4 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 862 924 8;
  • 13) 0.085 322 667 301 565 256 232 956 536 862 924 8 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 725 849 6;
  • 14) 0.170 645 334 603 130 512 465 913 073 725 849 6 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 451 699 2;
  • 15) 0.341 290 669 206 261 024 931 826 147 451 699 2 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 903 398 4;
  • 16) 0.682 581 338 412 522 049 863 652 294 903 398 4 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 806 796 8;
  • 17) 0.365 162 676 825 044 099 727 304 589 806 796 8 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 613 593 6;
  • 18) 0.730 325 353 650 088 199 454 609 179 613 593 6 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 227 187 2;
  • 19) 0.460 650 707 300 176 398 909 218 359 227 187 2 × 2 = 0 + 0.921 301 414 600 352 797 818 436 718 454 374 4;
  • 20) 0.921 301 414 600 352 797 818 436 718 454 374 4 × 2 = 1 + 0.842 602 829 200 705 595 636 873 436 908 748 8;
  • 21) 0.842 602 829 200 705 595 636 873 436 908 748 8 × 2 = 1 + 0.685 205 658 401 411 191 273 746 873 817 497 6;
  • 22) 0.685 205 658 401 411 191 273 746 873 817 497 6 × 2 = 1 + 0.370 411 316 802 822 382 547 493 747 634 995 2;
  • 23) 0.370 411 316 802 822 382 547 493 747 634 995 2 × 2 = 0 + 0.740 822 633 605 644 765 094 987 495 269 990 4;
  • 24) 0.740 822 633 605 644 765 094 987 495 269 990 4 × 2 = 1 + 0.481 645 267 211 289 530 189 974 990 539 980 8;
  • 25) 0.481 645 267 211 289 530 189 974 990 539 980 8 × 2 = 0 + 0.963 290 534 422 579 060 379 949 981 079 961 6;
  • 26) 0.963 290 534 422 579 060 379 949 981 079 961 6 × 2 = 1 + 0.926 581 068 845 158 120 759 899 962 159 923 2;
  • 27) 0.926 581 068 845 158 120 759 899 962 159 923 2 × 2 = 1 + 0.853 162 137 690 316 241 519 799 924 319 846 4;
  • 28) 0.853 162 137 690 316 241 519 799 924 319 846 4 × 2 = 1 + 0.706 324 275 380 632 483 039 599 848 639 692 8;
  • 29) 0.706 324 275 380 632 483 039 599 848 639 692 8 × 2 = 1 + 0.412 648 550 761 264 966 079 199 697 279 385 6;
  • 30) 0.412 648 550 761 264 966 079 199 697 279 385 6 × 2 = 0 + 0.825 297 101 522 529 932 158 399 394 558 771 2;
  • 31) 0.825 297 101 522 529 932 158 399 394 558 771 2 × 2 = 1 + 0.650 594 203 045 059 864 316 798 789 117 542 4;
  • 32) 0.650 594 203 045 059 864 316 798 789 117 542 4 × 2 = 1 + 0.301 188 406 090 119 728 633 597 578 235 084 8;
  • 33) 0.301 188 406 090 119 728 633 597 578 235 084 8 × 2 = 0 + 0.602 376 812 180 239 457 267 195 156 470 169 6;
  • 34) 0.602 376 812 180 239 457 267 195 156 470 169 6 × 2 = 1 + 0.204 753 624 360 478 914 534 390 312 940 339 2;
  • 35) 0.204 753 624 360 478 914 534 390 312 940 339 2 × 2 = 0 + 0.409 507 248 720 957 829 068 780 625 880 678 4;
  • 36) 0.409 507 248 720 957 829 068 780 625 880 678 4 × 2 = 0 + 0.819 014 497 441 915 658 137 561 251 761 356 8;
  • 37) 0.819 014 497 441 915 658 137 561 251 761 356 8 × 2 = 1 + 0.638 028 994 883 831 316 275 122 503 522 713 6;
  • 38) 0.638 028 994 883 831 316 275 122 503 522 713 6 × 2 = 1 + 0.276 057 989 767 662 632 550 245 007 045 427 2;
  • 39) 0.276 057 989 767 662 632 550 245 007 045 427 2 × 2 = 0 + 0.552 115 979 535 325 265 100 490 014 090 854 4;
  • 40) 0.552 115 979 535 325 265 100 490 014 090 854 4 × 2 = 1 + 0.104 231 959 070 650 530 200 980 028 181 708 8;
  • 41) 0.104 231 959 070 650 530 200 980 028 181 708 8 × 2 = 0 + 0.208 463 918 141 301 060 401 960 056 363 417 6;
  • 42) 0.208 463 918 141 301 060 401 960 056 363 417 6 × 2 = 0 + 0.416 927 836 282 602 120 803 920 112 726 835 2;
  • 43) 0.416 927 836 282 602 120 803 920 112 726 835 2 × 2 = 0 + 0.833 855 672 565 204 241 607 840 225 453 670 4;
  • 44) 0.833 855 672 565 204 241 607 840 225 453 670 4 × 2 = 1 + 0.667 711 345 130 408 483 215 680 450 907 340 8;
  • 45) 0.667 711 345 130 408 483 215 680 450 907 340 8 × 2 = 1 + 0.335 422 690 260 816 966 431 360 901 814 681 6;
  • 46) 0.335 422 690 260 816 966 431 360 901 814 681 6 × 2 = 0 + 0.670 845 380 521 633 932 862 721 803 629 363 2;
  • 47) 0.670 845 380 521 633 932 862 721 803 629 363 2 × 2 = 1 + 0.341 690 761 043 267 865 725 443 607 258 726 4;
  • 48) 0.341 690 761 043 267 865 725 443 607 258 726 4 × 2 = 0 + 0.683 381 522 086 535 731 450 887 214 517 452 8;
  • 49) 0.683 381 522 086 535 731 450 887 214 517 452 8 × 2 = 1 + 0.366 763 044 173 071 462 901 774 429 034 905 6;
  • 50) 0.366 763 044 173 071 462 901 774 429 034 905 6 × 2 = 0 + 0.733 526 088 346 142 925 803 548 858 069 811 2;
  • 51) 0.733 526 088 346 142 925 803 548 858 069 811 2 × 2 = 1 + 0.467 052 176 692 285 851 607 097 716 139 622 4;
  • 52) 0.467 052 176 692 285 851 607 097 716 139 622 4 × 2 = 0 + 0.934 104 353 384 571 703 214 195 432 279 244 8;
  • 53) 0.934 104 353 384 571 703 214 195 432 279 244 8 × 2 = 1 + 0.868 208 706 769 143 406 428 390 864 558 489 6;
  • 54) 0.868 208 706 769 143 406 428 390 864 558 489 6 × 2 = 1 + 0.736 417 413 538 286 812 856 781 729 116 979 2;
  • 55) 0.736 417 413 538 286 812 856 781 729 116 979 2 × 2 = 1 + 0.472 834 827 076 573 625 713 563 458 233 958 4;
  • 56) 0.472 834 827 076 573 625 713 563 458 233 958 4 × 2 = 0 + 0.945 669 654 153 147 251 427 126 916 467 916 8;
  • 57) 0.945 669 654 153 147 251 427 126 916 467 916 8 × 2 = 1 + 0.891 339 308 306 294 502 854 253 832 935 833 6;
  • 58) 0.891 339 308 306 294 502 854 253 832 935 833 6 × 2 = 1 + 0.782 678 616 612 589 005 708 507 665 871 667 2;
  • 59) 0.782 678 616 612 589 005 708 507 665 871 667 2 × 2 = 1 + 0.565 357 233 225 178 011 417 015 331 743 334 4;
  • 60) 0.565 357 233 225 178 011 417 015 331 743 334 4 × 2 = 1 + 0.130 714 466 450 356 022 834 030 663 486 668 8;
  • 61) 0.130 714 466 450 356 022 834 030 663 486 668 8 × 2 = 0 + 0.261 428 932 900 712 045 668 061 326 973 337 6;
  • 62) 0.261 428 932 900 712 045 668 061 326 973 337 6 × 2 = 0 + 0.522 857 865 801 424 091 336 122 653 946 675 2;
  • 63) 0.522 857 865 801 424 091 336 122 653 946 675 2 × 2 = 1 + 0.045 715 731 602 848 182 672 245 307 893 350 4;
  • 64) 0.045 715 731 602 848 182 672 245 307 893 350 4 × 2 = 0 + 0.091 431 463 205 696 365 344 490 615 786 700 8;
  • 65) 0.091 431 463 205 696 365 344 490 615 786 700 8 × 2 = 0 + 0.182 862 926 411 392 730 688 981 231 573 401 6;
  • 66) 0.182 862 926 411 392 730 688 981 231 573 401 6 × 2 = 0 + 0.365 725 852 822 785 461 377 962 463 146 803 2;
  • 67) 0.365 725 852 822 785 461 377 962 463 146 803 2 × 2 = 0 + 0.731 451 705 645 570 922 755 924 926 293 606 4;
  • 68) 0.731 451 705 645 570 922 755 924 926 293 606 4 × 2 = 1 + 0.462 903 411 291 141 845 511 849 852 587 212 8;

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 55(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 55(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 55(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 55 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