90.899 999 999 999 991 473 487 170 878 797 769 546 508 758 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 90.899 999 999 999 991 473 487 170 878 797 769 546 508 758 2(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
90.899 999 999 999 991 473 487 170 878 797 769 546 508 758 2(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: 90.
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;
  • 90 ÷ 2 = 45 + 0;
  • 45 ÷ 2 = 22 + 1;
  • 22 ÷ 2 = 11 + 0;
  • 11 ÷ 2 = 5 + 1;
  • 5 ÷ 2 = 2 + 1;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

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.

90(10) =

101 1010(2)


3. Convert to binary (base 2) the fractional part: 0.899 999 999 999 991 473 487 170 878 797 769 546 508 758 2.

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) 0.899 999 999 999 991 473 487 170 878 797 769 546 508 758 2 × 2 = 1 + 0.799 999 999 999 982 946 974 341 757 595 539 093 017 516 4;
  • 2) 0.799 999 999 999 982 946 974 341 757 595 539 093 017 516 4 × 2 = 1 + 0.599 999 999 999 965 893 948 683 515 191 078 186 035 032 8;
  • 3) 0.599 999 999 999 965 893 948 683 515 191 078 186 035 032 8 × 2 = 1 + 0.199 999 999 999 931 787 897 367 030 382 156 372 070 065 6;
  • 4) 0.199 999 999 999 931 787 897 367 030 382 156 372 070 065 6 × 2 = 0 + 0.399 999 999 999 863 575 794 734 060 764 312 744 140 131 2;
  • 5) 0.399 999 999 999 863 575 794 734 060 764 312 744 140 131 2 × 2 = 0 + 0.799 999 999 999 727 151 589 468 121 528 625 488 280 262 4;
  • 6) 0.799 999 999 999 727 151 589 468 121 528 625 488 280 262 4 × 2 = 1 + 0.599 999 999 999 454 303 178 936 243 057 250 976 560 524 8;
  • 7) 0.599 999 999 999 454 303 178 936 243 057 250 976 560 524 8 × 2 = 1 + 0.199 999 999 998 908 606 357 872 486 114 501 953 121 049 6;
  • 8) 0.199 999 999 998 908 606 357 872 486 114 501 953 121 049 6 × 2 = 0 + 0.399 999 999 997 817 212 715 744 972 229 003 906 242 099 2;
  • 9) 0.399 999 999 997 817 212 715 744 972 229 003 906 242 099 2 × 2 = 0 + 0.799 999 999 995 634 425 431 489 944 458 007 812 484 198 4;
  • 10) 0.799 999 999 995 634 425 431 489 944 458 007 812 484 198 4 × 2 = 1 + 0.599 999 999 991 268 850 862 979 888 916 015 624 968 396 8;
  • 11) 0.599 999 999 991 268 850 862 979 888 916 015 624 968 396 8 × 2 = 1 + 0.199 999 999 982 537 701 725 959 777 832 031 249 936 793 6;
  • 12) 0.199 999 999 982 537 701 725 959 777 832 031 249 936 793 6 × 2 = 0 + 0.399 999 999 965 075 403 451 919 555 664 062 499 873 587 2;
  • 13) 0.399 999 999 965 075 403 451 919 555 664 062 499 873 587 2 × 2 = 0 + 0.799 999 999 930 150 806 903 839 111 328 124 999 747 174 4;
  • 14) 0.799 999 999 930 150 806 903 839 111 328 124 999 747 174 4 × 2 = 1 + 0.599 999 999 860 301 613 807 678 222 656 249 999 494 348 8;
  • 15) 0.599 999 999 860 301 613 807 678 222 656 249 999 494 348 8 × 2 = 1 + 0.199 999 999 720 603 227 615 356 445 312 499 998 988 697 6;
  • 16) 0.199 999 999 720 603 227 615 356 445 312 499 998 988 697 6 × 2 = 0 + 0.399 999 999 441 206 455 230 712 890 624 999 997 977 395 2;
  • 17) 0.399 999 999 441 206 455 230 712 890 624 999 997 977 395 2 × 2 = 0 + 0.799 999 998 882 412 910 461 425 781 249 999 995 954 790 4;
  • 18) 0.799 999 998 882 412 910 461 425 781 249 999 995 954 790 4 × 2 = 1 + 0.599 999 997 764 825 820 922 851 562 499 999 991 909 580 8;
  • 19) 0.599 999 997 764 825 820 922 851 562 499 999 991 909 580 8 × 2 = 1 + 0.199 999 995 529 651 641 845 703 124 999 999 983 819 161 6;
  • 20) 0.199 999 995 529 651 641 845 703 124 999 999 983 819 161 6 × 2 = 0 + 0.399 999 991 059 303 283 691 406 249 999 999 967 638 323 2;
  • 21) 0.399 999 991 059 303 283 691 406 249 999 999 967 638 323 2 × 2 = 0 + 0.799 999 982 118 606 567 382 812 499 999 999 935 276 646 4;
  • 22) 0.799 999 982 118 606 567 382 812 499 999 999 935 276 646 4 × 2 = 1 + 0.599 999 964 237 213 134 765 624 999 999 999 870 553 292 8;
  • 23) 0.599 999 964 237 213 134 765 624 999 999 999 870 553 292 8 × 2 = 1 + 0.199 999 928 474 426 269 531 249 999 999 999 741 106 585 6;
  • 24) 0.199 999 928 474 426 269 531 249 999 999 999 741 106 585 6 × 2 = 0 + 0.399 999 856 948 852 539 062 499 999 999 999 482 213 171 2;
  • 25) 0.399 999 856 948 852 539 062 499 999 999 999 482 213 171 2 × 2 = 0 + 0.799 999 713 897 705 078 124 999 999 999 998 964 426 342 4;
  • 26) 0.799 999 713 897 705 078 124 999 999 999 998 964 426 342 4 × 2 = 1 + 0.599 999 427 795 410 156 249 999 999 999 997 928 852 684 8;
  • 27) 0.599 999 427 795 410 156 249 999 999 999 997 928 852 684 8 × 2 = 1 + 0.199 998 855 590 820 312 499 999 999 999 995 857 705 369 6;
  • 28) 0.199 998 855 590 820 312 499 999 999 999 995 857 705 369 6 × 2 = 0 + 0.399 997 711 181 640 624 999 999 999 999 991 715 410 739 2;
  • 29) 0.399 997 711 181 640 624 999 999 999 999 991 715 410 739 2 × 2 = 0 + 0.799 995 422 363 281 249 999 999 999 999 983 430 821 478 4;
  • 30) 0.799 995 422 363 281 249 999 999 999 999 983 430 821 478 4 × 2 = 1 + 0.599 990 844 726 562 499 999 999 999 999 966 861 642 956 8;
  • 31) 0.599 990 844 726 562 499 999 999 999 999 966 861 642 956 8 × 2 = 1 + 0.199 981 689 453 124 999 999 999 999 999 933 723 285 913 6;
  • 32) 0.199 981 689 453 124 999 999 999 999 999 933 723 285 913 6 × 2 = 0 + 0.399 963 378 906 249 999 999 999 999 999 867 446 571 827 2;
  • 33) 0.399 963 378 906 249 999 999 999 999 999 867 446 571 827 2 × 2 = 0 + 0.799 926 757 812 499 999 999 999 999 999 734 893 143 654 4;
  • 34) 0.799 926 757 812 499 999 999 999 999 999 734 893 143 654 4 × 2 = 1 + 0.599 853 515 624 999 999 999 999 999 999 469 786 287 308 8;
  • 35) 0.599 853 515 624 999 999 999 999 999 999 469 786 287 308 8 × 2 = 1 + 0.199 707 031 249 999 999 999 999 999 998 939 572 574 617 6;
  • 36) 0.199 707 031 249 999 999 999 999 999 998 939 572 574 617 6 × 2 = 0 + 0.399 414 062 499 999 999 999 999 999 997 879 145 149 235 2;
  • 37) 0.399 414 062 499 999 999 999 999 999 997 879 145 149 235 2 × 2 = 0 + 0.798 828 124 999 999 999 999 999 999 995 758 290 298 470 4;
  • 38) 0.798 828 124 999 999 999 999 999 999 995 758 290 298 470 4 × 2 = 1 + 0.597 656 249 999 999 999 999 999 999 991 516 580 596 940 8;
  • 39) 0.597 656 249 999 999 999 999 999 999 991 516 580 596 940 8 × 2 = 1 + 0.195 312 499 999 999 999 999 999 999 983 033 161 193 881 6;
  • 40) 0.195 312 499 999 999 999 999 999 999 983 033 161 193 881 6 × 2 = 0 + 0.390 624 999 999 999 999 999 999 999 966 066 322 387 763 2;
  • 41) 0.390 624 999 999 999 999 999 999 999 966 066 322 387 763 2 × 2 = 0 + 0.781 249 999 999 999 999 999 999 999 932 132 644 775 526 4;
  • 42) 0.781 249 999 999 999 999 999 999 999 932 132 644 775 526 4 × 2 = 1 + 0.562 499 999 999 999 999 999 999 999 864 265 289 551 052 8;
  • 43) 0.562 499 999 999 999 999 999 999 999 864 265 289 551 052 8 × 2 = 1 + 0.124 999 999 999 999 999 999 999 999 728 530 579 102 105 6;
  • 44) 0.124 999 999 999 999 999 999 999 999 728 530 579 102 105 6 × 2 = 0 + 0.249 999 999 999 999 999 999 999 999 457 061 158 204 211 2;
  • 45) 0.249 999 999 999 999 999 999 999 999 457 061 158 204 211 2 × 2 = 0 + 0.499 999 999 999 999 999 999 999 998 914 122 316 408 422 4;
  • 46) 0.499 999 999 999 999 999 999 999 998 914 122 316 408 422 4 × 2 = 0 + 0.999 999 999 999 999 999 999 999 997 828 244 632 816 844 8;
  • 47) 0.999 999 999 999 999 999 999 999 997 828 244 632 816 844 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 995 656 489 265 633 689 6;
  • 48) 0.999 999 999 999 999 999 999 999 995 656 489 265 633 689 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 991 312 978 531 267 379 2;
  • 49) 0.999 999 999 999 999 999 999 999 991 312 978 531 267 379 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 982 625 957 062 534 758 4;
  • 50) 0.999 999 999 999 999 999 999 999 982 625 957 062 534 758 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 965 251 914 125 069 516 8;
  • 51) 0.999 999 999 999 999 999 999 999 965 251 914 125 069 516 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 930 503 828 250 139 033 6;
  • 52) 0.999 999 999 999 999 999 999 999 930 503 828 250 139 033 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 861 007 656 500 278 067 2;
  • 53) 0.999 999 999 999 999 999 999 999 861 007 656 500 278 067 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 722 015 313 000 556 134 4;

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:


0.899 999 999 999 991 473 487 170 878 797 769 546 508 758 2(10) =

0.1110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0011 1111 1(2)

5. Positive number before normalization:

90.899 999 999 999 991 473 487 170 878 797 769 546 508 758 2(10) =

101 1010.1110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0011 1111 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 6 positions to the left, so that only one non zero digit remains to the left of it:


90.899 999 999 999 991 473 487 170 878 797 769 546 508 758 2(10) =

101 1010.1110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0011 1111 1(2) =

101 1010.1110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0011 1111 1(2) × 20 =

1.0110 1011 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1000 1111 111(2) × 26


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): 6

Mantissa (not normalized):
1.0110 1011 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1000 1111 111


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

6 + 2(11-1) - 1 =

(6 + 1 023)(10) =

1 029(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;
  • 1 029 ÷ 2 = 514 + 1;
  • 514 ÷ 2 = 257 + 0;
  • 257 ÷ 2 = 128 + 1;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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) =

1029(10) =

100 0000 0101(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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 0110 1011 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1000 111 1111 =

0110 1011 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1000


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) =
100 0000 0101

Mantissa (52 bits) =
0110 1011 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1000


Decimal number 90.899 999 999 999 991 473 487 170 878 797 769 546 508 758 2 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0101 - 0110 1011 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1000

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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