87.549 999 999 999 997 157 829 056 959 599 256 515 502 929 27 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 87.549 999 999 999 997 157 829 056 959 599 256 515 502 929 27(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
87.549 999 999 999 997 157 829 056 959 599 256 515 502 929 27(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: 87.
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;
  • 87 ÷ 2 = 43 + 1;
  • 43 ÷ 2 = 21 + 1;
  • 21 ÷ 2 = 10 + 1;
  • 10 ÷ 2 = 5 + 0;
  • 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.

87(10) =

101 0111(2)


3. Convert to binary (base 2) the fractional part: 0.549 999 999 999 997 157 829 056 959 599 256 515 502 929 27.

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) 0.549 999 999 999 997 157 829 056 959 599 256 515 502 929 27 × 2 = 1 + 0.099 999 999 999 994 315 658 113 919 198 513 031 005 858 54;
  • 2) 0.099 999 999 999 994 315 658 113 919 198 513 031 005 858 54 × 2 = 0 + 0.199 999 999 999 988 631 316 227 838 397 026 062 011 717 08;
  • 3) 0.199 999 999 999 988 631 316 227 838 397 026 062 011 717 08 × 2 = 0 + 0.399 999 999 999 977 262 632 455 676 794 052 124 023 434 16;
  • 4) 0.399 999 999 999 977 262 632 455 676 794 052 124 023 434 16 × 2 = 0 + 0.799 999 999 999 954 525 264 911 353 588 104 248 046 868 32;
  • 5) 0.799 999 999 999 954 525 264 911 353 588 104 248 046 868 32 × 2 = 1 + 0.599 999 999 999 909 050 529 822 707 176 208 496 093 736 64;
  • 6) 0.599 999 999 999 909 050 529 822 707 176 208 496 093 736 64 × 2 = 1 + 0.199 999 999 999 818 101 059 645 414 352 416 992 187 473 28;
  • 7) 0.199 999 999 999 818 101 059 645 414 352 416 992 187 473 28 × 2 = 0 + 0.399 999 999 999 636 202 119 290 828 704 833 984 374 946 56;
  • 8) 0.399 999 999 999 636 202 119 290 828 704 833 984 374 946 56 × 2 = 0 + 0.799 999 999 999 272 404 238 581 657 409 667 968 749 893 12;
  • 9) 0.799 999 999 999 272 404 238 581 657 409 667 968 749 893 12 × 2 = 1 + 0.599 999 999 998 544 808 477 163 314 819 335 937 499 786 24;
  • 10) 0.599 999 999 998 544 808 477 163 314 819 335 937 499 786 24 × 2 = 1 + 0.199 999 999 997 089 616 954 326 629 638 671 874 999 572 48;
  • 11) 0.199 999 999 997 089 616 954 326 629 638 671 874 999 572 48 × 2 = 0 + 0.399 999 999 994 179 233 908 653 259 277 343 749 999 144 96;
  • 12) 0.399 999 999 994 179 233 908 653 259 277 343 749 999 144 96 × 2 = 0 + 0.799 999 999 988 358 467 817 306 518 554 687 499 998 289 92;
  • 13) 0.799 999 999 988 358 467 817 306 518 554 687 499 998 289 92 × 2 = 1 + 0.599 999 999 976 716 935 634 613 037 109 374 999 996 579 84;
  • 14) 0.599 999 999 976 716 935 634 613 037 109 374 999 996 579 84 × 2 = 1 + 0.199 999 999 953 433 871 269 226 074 218 749 999 993 159 68;
  • 15) 0.199 999 999 953 433 871 269 226 074 218 749 999 993 159 68 × 2 = 0 + 0.399 999 999 906 867 742 538 452 148 437 499 999 986 319 36;
  • 16) 0.399 999 999 906 867 742 538 452 148 437 499 999 986 319 36 × 2 = 0 + 0.799 999 999 813 735 485 076 904 296 874 999 999 972 638 72;
  • 17) 0.799 999 999 813 735 485 076 904 296 874 999 999 972 638 72 × 2 = 1 + 0.599 999 999 627 470 970 153 808 593 749 999 999 945 277 44;
  • 18) 0.599 999 999 627 470 970 153 808 593 749 999 999 945 277 44 × 2 = 1 + 0.199 999 999 254 941 940 307 617 187 499 999 999 890 554 88;
  • 19) 0.199 999 999 254 941 940 307 617 187 499 999 999 890 554 88 × 2 = 0 + 0.399 999 998 509 883 880 615 234 374 999 999 999 781 109 76;
  • 20) 0.399 999 998 509 883 880 615 234 374 999 999 999 781 109 76 × 2 = 0 + 0.799 999 997 019 767 761 230 468 749 999 999 999 562 219 52;
  • 21) 0.799 999 997 019 767 761 230 468 749 999 999 999 562 219 52 × 2 = 1 + 0.599 999 994 039 535 522 460 937 499 999 999 999 124 439 04;
  • 22) 0.599 999 994 039 535 522 460 937 499 999 999 999 124 439 04 × 2 = 1 + 0.199 999 988 079 071 044 921 874 999 999 999 998 248 878 08;
  • 23) 0.199 999 988 079 071 044 921 874 999 999 999 998 248 878 08 × 2 = 0 + 0.399 999 976 158 142 089 843 749 999 999 999 996 497 756 16;
  • 24) 0.399 999 976 158 142 089 843 749 999 999 999 996 497 756 16 × 2 = 0 + 0.799 999 952 316 284 179 687 499 999 999 999 992 995 512 32;
  • 25) 0.799 999 952 316 284 179 687 499 999 999 999 992 995 512 32 × 2 = 1 + 0.599 999 904 632 568 359 374 999 999 999 999 985 991 024 64;
  • 26) 0.599 999 904 632 568 359 374 999 999 999 999 985 991 024 64 × 2 = 1 + 0.199 999 809 265 136 718 749 999 999 999 999 971 982 049 28;
  • 27) 0.199 999 809 265 136 718 749 999 999 999 999 971 982 049 28 × 2 = 0 + 0.399 999 618 530 273 437 499 999 999 999 999 943 964 098 56;
  • 28) 0.399 999 618 530 273 437 499 999 999 999 999 943 964 098 56 × 2 = 0 + 0.799 999 237 060 546 874 999 999 999 999 999 887 928 197 12;
  • 29) 0.799 999 237 060 546 874 999 999 999 999 999 887 928 197 12 × 2 = 1 + 0.599 998 474 121 093 749 999 999 999 999 999 775 856 394 24;
  • 30) 0.599 998 474 121 093 749 999 999 999 999 999 775 856 394 24 × 2 = 1 + 0.199 996 948 242 187 499 999 999 999 999 999 551 712 788 48;
  • 31) 0.199 996 948 242 187 499 999 999 999 999 999 551 712 788 48 × 2 = 0 + 0.399 993 896 484 374 999 999 999 999 999 999 103 425 576 96;
  • 32) 0.399 993 896 484 374 999 999 999 999 999 999 103 425 576 96 × 2 = 0 + 0.799 987 792 968 749 999 999 999 999 999 998 206 851 153 92;
  • 33) 0.799 987 792 968 749 999 999 999 999 999 998 206 851 153 92 × 2 = 1 + 0.599 975 585 937 499 999 999 999 999 999 996 413 702 307 84;
  • 34) 0.599 975 585 937 499 999 999 999 999 999 996 413 702 307 84 × 2 = 1 + 0.199 951 171 874 999 999 999 999 999 999 992 827 404 615 68;
  • 35) 0.199 951 171 874 999 999 999 999 999 999 992 827 404 615 68 × 2 = 0 + 0.399 902 343 749 999 999 999 999 999 999 985 654 809 231 36;
  • 36) 0.399 902 343 749 999 999 999 999 999 999 985 654 809 231 36 × 2 = 0 + 0.799 804 687 499 999 999 999 999 999 999 971 309 618 462 72;
  • 37) 0.799 804 687 499 999 999 999 999 999 999 971 309 618 462 72 × 2 = 1 + 0.599 609 374 999 999 999 999 999 999 999 942 619 236 925 44;
  • 38) 0.599 609 374 999 999 999 999 999 999 999 942 619 236 925 44 × 2 = 1 + 0.199 218 749 999 999 999 999 999 999 999 885 238 473 850 88;
  • 39) 0.199 218 749 999 999 999 999 999 999 999 885 238 473 850 88 × 2 = 0 + 0.398 437 499 999 999 999 999 999 999 999 770 476 947 701 76;
  • 40) 0.398 437 499 999 999 999 999 999 999 999 770 476 947 701 76 × 2 = 0 + 0.796 874 999 999 999 999 999 999 999 999 540 953 895 403 52;
  • 41) 0.796 874 999 999 999 999 999 999 999 999 540 953 895 403 52 × 2 = 1 + 0.593 749 999 999 999 999 999 999 999 999 081 907 790 807 04;
  • 42) 0.593 749 999 999 999 999 999 999 999 999 081 907 790 807 04 × 2 = 1 + 0.187 499 999 999 999 999 999 999 999 998 163 815 581 614 08;
  • 43) 0.187 499 999 999 999 999 999 999 999 998 163 815 581 614 08 × 2 = 0 + 0.374 999 999 999 999 999 999 999 999 996 327 631 163 228 16;
  • 44) 0.374 999 999 999 999 999 999 999 999 996 327 631 163 228 16 × 2 = 0 + 0.749 999 999 999 999 999 999 999 999 992 655 262 326 456 32;
  • 45) 0.749 999 999 999 999 999 999 999 999 992 655 262 326 456 32 × 2 = 1 + 0.499 999 999 999 999 999 999 999 999 985 310 524 652 912 64;
  • 46) 0.499 999 999 999 999 999 999 999 999 985 310 524 652 912 64 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 970 621 049 305 825 28;
  • 47) 0.999 999 999 999 999 999 999 999 999 970 621 049 305 825 28 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 941 242 098 611 650 56;
  • 48) 0.999 999 999 999 999 999 999 999 999 941 242 098 611 650 56 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 882 484 197 223 301 12;
  • 49) 0.999 999 999 999 999 999 999 999 999 882 484 197 223 301 12 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 764 968 394 446 602 24;
  • 50) 0.999 999 999 999 999 999 999 999 999 764 968 394 446 602 24 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 529 936 788 893 204 48;
  • 51) 0.999 999 999 999 999 999 999 999 999 529 936 788 893 204 48 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 059 873 577 786 408 96;
  • 52) 0.999 999 999 999 999 999 999 999 999 059 873 577 786 408 96 × 2 = 1 + 0.999 999 999 999 999 999 999 999 998 119 747 155 572 817 92;
  • 53) 0.999 999 999 999 999 999 999 999 998 119 747 155 572 817 92 × 2 = 1 + 0.999 999 999 999 999 999 999 999 996 239 494 311 145 635 84;

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:


0.549 999 999 999 997 157 829 056 959 599 256 515 502 929 27(10) =

0.1000 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1011 1111 1(2)

5. Positive number before normalization:

87.549 999 999 999 997 157 829 056 959 599 256 515 502 929 27(10) =

101 0111.1000 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1011 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:


87.549 999 999 999 997 157 829 056 959 599 256 515 502 929 27(10) =

101 0111.1000 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1011 1111 1(2) =

101 0111.1000 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1011 1111 1(2) × 20 =

1.0101 1110 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0010 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):
1.0101 1110 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0010 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. 0101 1110 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0010 111 1111 =

0101 1110 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0010


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) =
0101 1110 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0010


Decimal number 87.549 999 999 999 997 157 829 056 959 599 256 515 502 929 27 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0101 - 0101 1110 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0010

» Convert decimal 87.549 999 999 999 997 157 829 056 959 599 256 515 502 928 36 to 64 bit double precision IEEE 754 binary floating point representation standard

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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