0.000 999 999 999 999 999 803 976 247 214 620 797 57 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 999 999 999 999 999 803 976 247 214 620 797 57(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
0.000 999 999 999 999 999 803 976 247 214 620 797 57(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: 0.000 999 999 999 999 999 803 976 247 214 620 797 57.

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) 0.000 999 999 999 999 999 803 976 247 214 620 797 57 × 2 = 0 + 0.001 999 999 999 999 999 607 952 494 429 241 595 14;
  • 2) 0.001 999 999 999 999 999 607 952 494 429 241 595 14 × 2 = 0 + 0.003 999 999 999 999 999 215 904 988 858 483 190 28;
  • 3) 0.003 999 999 999 999 999 215 904 988 858 483 190 28 × 2 = 0 + 0.007 999 999 999 999 998 431 809 977 716 966 380 56;
  • 4) 0.007 999 999 999 999 998 431 809 977 716 966 380 56 × 2 = 0 + 0.015 999 999 999 999 996 863 619 955 433 932 761 12;
  • 5) 0.015 999 999 999 999 996 863 619 955 433 932 761 12 × 2 = 0 + 0.031 999 999 999 999 993 727 239 910 867 865 522 24;
  • 6) 0.031 999 999 999 999 993 727 239 910 867 865 522 24 × 2 = 0 + 0.063 999 999 999 999 987 454 479 821 735 731 044 48;
  • 7) 0.063 999 999 999 999 987 454 479 821 735 731 044 48 × 2 = 0 + 0.127 999 999 999 999 974 908 959 643 471 462 088 96;
  • 8) 0.127 999 999 999 999 974 908 959 643 471 462 088 96 × 2 = 0 + 0.255 999 999 999 999 949 817 919 286 942 924 177 92;
  • 9) 0.255 999 999 999 999 949 817 919 286 942 924 177 92 × 2 = 0 + 0.511 999 999 999 999 899 635 838 573 885 848 355 84;
  • 10) 0.511 999 999 999 999 899 635 838 573 885 848 355 84 × 2 = 1 + 0.023 999 999 999 999 799 271 677 147 771 696 711 68;
  • 11) 0.023 999 999 999 999 799 271 677 147 771 696 711 68 × 2 = 0 + 0.047 999 999 999 999 598 543 354 295 543 393 423 36;
  • 12) 0.047 999 999 999 999 598 543 354 295 543 393 423 36 × 2 = 0 + 0.095 999 999 999 999 197 086 708 591 086 786 846 72;
  • 13) 0.095 999 999 999 999 197 086 708 591 086 786 846 72 × 2 = 0 + 0.191 999 999 999 998 394 173 417 182 173 573 693 44;
  • 14) 0.191 999 999 999 998 394 173 417 182 173 573 693 44 × 2 = 0 + 0.383 999 999 999 996 788 346 834 364 347 147 386 88;
  • 15) 0.383 999 999 999 996 788 346 834 364 347 147 386 88 × 2 = 0 + 0.767 999 999 999 993 576 693 668 728 694 294 773 76;
  • 16) 0.767 999 999 999 993 576 693 668 728 694 294 773 76 × 2 = 1 + 0.535 999 999 999 987 153 387 337 457 388 589 547 52;
  • 17) 0.535 999 999 999 987 153 387 337 457 388 589 547 52 × 2 = 1 + 0.071 999 999 999 974 306 774 674 914 777 179 095 04;
  • 18) 0.071 999 999 999 974 306 774 674 914 777 179 095 04 × 2 = 0 + 0.143 999 999 999 948 613 549 349 829 554 358 190 08;
  • 19) 0.143 999 999 999 948 613 549 349 829 554 358 190 08 × 2 = 0 + 0.287 999 999 999 897 227 098 699 659 108 716 380 16;
  • 20) 0.287 999 999 999 897 227 098 699 659 108 716 380 16 × 2 = 0 + 0.575 999 999 999 794 454 197 399 318 217 432 760 32;
  • 21) 0.575 999 999 999 794 454 197 399 318 217 432 760 32 × 2 = 1 + 0.151 999 999 999 588 908 394 798 636 434 865 520 64;
  • 22) 0.151 999 999 999 588 908 394 798 636 434 865 520 64 × 2 = 0 + 0.303 999 999 999 177 816 789 597 272 869 731 041 28;
  • 23) 0.303 999 999 999 177 816 789 597 272 869 731 041 28 × 2 = 0 + 0.607 999 999 998 355 633 579 194 545 739 462 082 56;
  • 24) 0.607 999 999 998 355 633 579 194 545 739 462 082 56 × 2 = 1 + 0.215 999 999 996 711 267 158 389 091 478 924 165 12;
  • 25) 0.215 999 999 996 711 267 158 389 091 478 924 165 12 × 2 = 0 + 0.431 999 999 993 422 534 316 778 182 957 848 330 24;
  • 26) 0.431 999 999 993 422 534 316 778 182 957 848 330 24 × 2 = 0 + 0.863 999 999 986 845 068 633 556 365 915 696 660 48;
  • 27) 0.863 999 999 986 845 068 633 556 365 915 696 660 48 × 2 = 1 + 0.727 999 999 973 690 137 267 112 731 831 393 320 96;
  • 28) 0.727 999 999 973 690 137 267 112 731 831 393 320 96 × 2 = 1 + 0.455 999 999 947 380 274 534 225 463 662 786 641 92;
  • 29) 0.455 999 999 947 380 274 534 225 463 662 786 641 92 × 2 = 0 + 0.911 999 999 894 760 549 068 450 927 325 573 283 84;
  • 30) 0.911 999 999 894 760 549 068 450 927 325 573 283 84 × 2 = 1 + 0.823 999 999 789 521 098 136 901 854 651 146 567 68;
  • 31) 0.823 999 999 789 521 098 136 901 854 651 146 567 68 × 2 = 1 + 0.647 999 999 579 042 196 273 803 709 302 293 135 36;
  • 32) 0.647 999 999 579 042 196 273 803 709 302 293 135 36 × 2 = 1 + 0.295 999 999 158 084 392 547 607 418 604 586 270 72;
  • 33) 0.295 999 999 158 084 392 547 607 418 604 586 270 72 × 2 = 0 + 0.591 999 998 316 168 785 095 214 837 209 172 541 44;
  • 34) 0.591 999 998 316 168 785 095 214 837 209 172 541 44 × 2 = 1 + 0.183 999 996 632 337 570 190 429 674 418 345 082 88;
  • 35) 0.183 999 996 632 337 570 190 429 674 418 345 082 88 × 2 = 0 + 0.367 999 993 264 675 140 380 859 348 836 690 165 76;
  • 36) 0.367 999 993 264 675 140 380 859 348 836 690 165 76 × 2 = 0 + 0.735 999 986 529 350 280 761 718 697 673 380 331 52;
  • 37) 0.735 999 986 529 350 280 761 718 697 673 380 331 52 × 2 = 1 + 0.471 999 973 058 700 561 523 437 395 346 760 663 04;
  • 38) 0.471 999 973 058 700 561 523 437 395 346 760 663 04 × 2 = 0 + 0.943 999 946 117 401 123 046 874 790 693 521 326 08;
  • 39) 0.943 999 946 117 401 123 046 874 790 693 521 326 08 × 2 = 1 + 0.887 999 892 234 802 246 093 749 581 387 042 652 16;
  • 40) 0.887 999 892 234 802 246 093 749 581 387 042 652 16 × 2 = 1 + 0.775 999 784 469 604 492 187 499 162 774 085 304 32;
  • 41) 0.775 999 784 469 604 492 187 499 162 774 085 304 32 × 2 = 1 + 0.551 999 568 939 208 984 374 998 325 548 170 608 64;
  • 42) 0.551 999 568 939 208 984 374 998 325 548 170 608 64 × 2 = 1 + 0.103 999 137 878 417 968 749 996 651 096 341 217 28;
  • 43) 0.103 999 137 878 417 968 749 996 651 096 341 217 28 × 2 = 0 + 0.207 998 275 756 835 937 499 993 302 192 682 434 56;
  • 44) 0.207 998 275 756 835 937 499 993 302 192 682 434 56 × 2 = 0 + 0.415 996 551 513 671 874 999 986 604 385 364 869 12;
  • 45) 0.415 996 551 513 671 874 999 986 604 385 364 869 12 × 2 = 0 + 0.831 993 103 027 343 749 999 973 208 770 729 738 24;
  • 46) 0.831 993 103 027 343 749 999 973 208 770 729 738 24 × 2 = 1 + 0.663 986 206 054 687 499 999 946 417 541 459 476 48;
  • 47) 0.663 986 206 054 687 499 999 946 417 541 459 476 48 × 2 = 1 + 0.327 972 412 109 374 999 999 892 835 082 918 952 96;
  • 48) 0.327 972 412 109 374 999 999 892 835 082 918 952 96 × 2 = 0 + 0.655 944 824 218 749 999 999 785 670 165 837 905 92;
  • 49) 0.655 944 824 218 749 999 999 785 670 165 837 905 92 × 2 = 1 + 0.311 889 648 437 499 999 999 571 340 331 675 811 84;
  • 50) 0.311 889 648 437 499 999 999 571 340 331 675 811 84 × 2 = 0 + 0.623 779 296 874 999 999 999 142 680 663 351 623 68;
  • 51) 0.623 779 296 874 999 999 999 142 680 663 351 623 68 × 2 = 1 + 0.247 558 593 749 999 999 998 285 361 326 703 247 36;
  • 52) 0.247 558 593 749 999 999 998 285 361 326 703 247 36 × 2 = 0 + 0.495 117 187 499 999 999 996 570 722 653 406 494 72;
  • 53) 0.495 117 187 499 999 999 996 570 722 653 406 494 72 × 2 = 0 + 0.990 234 374 999 999 999 993 141 445 306 812 989 44;
  • 54) 0.990 234 374 999 999 999 993 141 445 306 812 989 44 × 2 = 1 + 0.980 468 749 999 999 999 986 282 890 613 625 978 88;
  • 55) 0.980 468 749 999 999 999 986 282 890 613 625 978 88 × 2 = 1 + 0.960 937 499 999 999 999 972 565 781 227 251 957 76;
  • 56) 0.960 937 499 999 999 999 972 565 781 227 251 957 76 × 2 = 1 + 0.921 874 999 999 999 999 945 131 562 454 503 915 52;
  • 57) 0.921 874 999 999 999 999 945 131 562 454 503 915 52 × 2 = 1 + 0.843 749 999 999 999 999 890 263 124 909 007 831 04;
  • 58) 0.843 749 999 999 999 999 890 263 124 909 007 831 04 × 2 = 1 + 0.687 499 999 999 999 999 780 526 249 818 015 662 08;
  • 59) 0.687 499 999 999 999 999 780 526 249 818 015 662 08 × 2 = 1 + 0.374 999 999 999 999 999 561 052 499 636 031 324 16;
  • 60) 0.374 999 999 999 999 999 561 052 499 636 031 324 16 × 2 = 0 + 0.749 999 999 999 999 999 122 104 999 272 062 648 32;
  • 61) 0.749 999 999 999 999 999 122 104 999 272 062 648 32 × 2 = 1 + 0.499 999 999 999 999 998 244 209 998 544 125 296 64;
  • 62) 0.499 999 999 999 999 998 244 209 998 544 125 296 64 × 2 = 0 + 0.999 999 999 999 999 996 488 419 997 088 250 593 28;

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:


0.000 999 999 999 999 999 803 976 247 214 620 797 57(10) =

0.0000 0000 0100 0001 1000 1001 0011 0111 0100 1011 1100 0110 1010 0111 1110 10(2)

5. Positive number before normalization:

0.000 999 999 999 999 999 803 976 247 214 620 797 57(10) =

0.0000 0000 0100 0001 1000 1001 0011 0111 0100 1011 1100 0110 1010 0111 1110 10(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 10 positions to the right, so that only one non zero digit remains to the left of it:


0.000 999 999 999 999 999 803 976 247 214 620 797 57(10) =

0.0000 0000 0100 0001 1000 1001 0011 0111 0100 1011 1100 0110 1010 0111 1110 10(2) =

0.0000 0000 0100 0001 1000 1001 0011 0111 0100 1011 1100 0110 1010 0111 1110 10(2) × 20 =

1.0000 0110 0010 0100 1101 1101 0010 1111 0001 1010 1001 1111 1010(2) × 2-10


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): -10

Mantissa (not normalized):
1.0000 0110 0010 0100 1101 1101 0010 1111 0001 1010 1001 1111 1010


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-10 + 2(11-1) - 1 =

(-10 + 1 023)(10) =

1 013(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;
  • 1 013 ÷ 2 = 506 + 1;
  • 506 ÷ 2 = 253 + 0;
  • 253 ÷ 2 = 126 + 1;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 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) =

1013(10) =

011 1111 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, only if necessary (not the case here).


Mantissa (normalized) =

1. 0000 0110 0010 0100 1101 1101 0010 1111 0001 1010 1001 1111 1010 =

0000 0110 0010 0100 1101 1101 0010 1111 0001 1010 1001 1111 1010


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 1111 0101

Mantissa (52 bits) =
0000 0110 0010 0100 1101 1101 0010 1111 0001 1010 1001 1111 1010


Decimal number 0.000 999 999 999 999 999 803 976 247 214 620 797 57 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 0101 - 0000 0110 0010 0100 1101 1101 0010 1111 0001 1010 1001 1111 1010

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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