Decimal to 64 Bit IEEE 754 Binary: Convert Number 0.000 999 999 999 999 999 803 976 247 214 620 799 01 to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From Base Ten Decimal System

Number 0.000 999 999 999 999 999 803 976 247 214 620 799 01(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation standard (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: 0.000 999 999 999 999 999 803 976 247 214 620 799 01.

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) 0.000 999 999 999 999 999 803 976 247 214 620 799 01 × 2 = 0 + 0.001 999 999 999 999 999 607 952 494 429 241 598 02;
  • 2) 0.001 999 999 999 999 999 607 952 494 429 241 598 02 × 2 = 0 + 0.003 999 999 999 999 999 215 904 988 858 483 196 04;
  • 3) 0.003 999 999 999 999 999 215 904 988 858 483 196 04 × 2 = 0 + 0.007 999 999 999 999 998 431 809 977 716 966 392 08;
  • 4) 0.007 999 999 999 999 998 431 809 977 716 966 392 08 × 2 = 0 + 0.015 999 999 999 999 996 863 619 955 433 932 784 16;
  • 5) 0.015 999 999 999 999 996 863 619 955 433 932 784 16 × 2 = 0 + 0.031 999 999 999 999 993 727 239 910 867 865 568 32;
  • 6) 0.031 999 999 999 999 993 727 239 910 867 865 568 32 × 2 = 0 + 0.063 999 999 999 999 987 454 479 821 735 731 136 64;
  • 7) 0.063 999 999 999 999 987 454 479 821 735 731 136 64 × 2 = 0 + 0.127 999 999 999 999 974 908 959 643 471 462 273 28;
  • 8) 0.127 999 999 999 999 974 908 959 643 471 462 273 28 × 2 = 0 + 0.255 999 999 999 999 949 817 919 286 942 924 546 56;
  • 9) 0.255 999 999 999 999 949 817 919 286 942 924 546 56 × 2 = 0 + 0.511 999 999 999 999 899 635 838 573 885 849 093 12;
  • 10) 0.511 999 999 999 999 899 635 838 573 885 849 093 12 × 2 = 1 + 0.023 999 999 999 999 799 271 677 147 771 698 186 24;
  • 11) 0.023 999 999 999 999 799 271 677 147 771 698 186 24 × 2 = 0 + 0.047 999 999 999 999 598 543 354 295 543 396 372 48;
  • 12) 0.047 999 999 999 999 598 543 354 295 543 396 372 48 × 2 = 0 + 0.095 999 999 999 999 197 086 708 591 086 792 744 96;
  • 13) 0.095 999 999 999 999 197 086 708 591 086 792 744 96 × 2 = 0 + 0.191 999 999 999 998 394 173 417 182 173 585 489 92;
  • 14) 0.191 999 999 999 998 394 173 417 182 173 585 489 92 × 2 = 0 + 0.383 999 999 999 996 788 346 834 364 347 170 979 84;
  • 15) 0.383 999 999 999 996 788 346 834 364 347 170 979 84 × 2 = 0 + 0.767 999 999 999 993 576 693 668 728 694 341 959 68;
  • 16) 0.767 999 999 999 993 576 693 668 728 694 341 959 68 × 2 = 1 + 0.535 999 999 999 987 153 387 337 457 388 683 919 36;
  • 17) 0.535 999 999 999 987 153 387 337 457 388 683 919 36 × 2 = 1 + 0.071 999 999 999 974 306 774 674 914 777 367 838 72;
  • 18) 0.071 999 999 999 974 306 774 674 914 777 367 838 72 × 2 = 0 + 0.143 999 999 999 948 613 549 349 829 554 735 677 44;
  • 19) 0.143 999 999 999 948 613 549 349 829 554 735 677 44 × 2 = 0 + 0.287 999 999 999 897 227 098 699 659 109 471 354 88;
  • 20) 0.287 999 999 999 897 227 098 699 659 109 471 354 88 × 2 = 0 + 0.575 999 999 999 794 454 197 399 318 218 942 709 76;
  • 21) 0.575 999 999 999 794 454 197 399 318 218 942 709 76 × 2 = 1 + 0.151 999 999 999 588 908 394 798 636 437 885 419 52;
  • 22) 0.151 999 999 999 588 908 394 798 636 437 885 419 52 × 2 = 0 + 0.303 999 999 999 177 816 789 597 272 875 770 839 04;
  • 23) 0.303 999 999 999 177 816 789 597 272 875 770 839 04 × 2 = 0 + 0.607 999 999 998 355 633 579 194 545 751 541 678 08;
  • 24) 0.607 999 999 998 355 633 579 194 545 751 541 678 08 × 2 = 1 + 0.215 999 999 996 711 267 158 389 091 503 083 356 16;
  • 25) 0.215 999 999 996 711 267 158 389 091 503 083 356 16 × 2 = 0 + 0.431 999 999 993 422 534 316 778 183 006 166 712 32;
  • 26) 0.431 999 999 993 422 534 316 778 183 006 166 712 32 × 2 = 0 + 0.863 999 999 986 845 068 633 556 366 012 333 424 64;
  • 27) 0.863 999 999 986 845 068 633 556 366 012 333 424 64 × 2 = 1 + 0.727 999 999 973 690 137 267 112 732 024 666 849 28;
  • 28) 0.727 999 999 973 690 137 267 112 732 024 666 849 28 × 2 = 1 + 0.455 999 999 947 380 274 534 225 464 049 333 698 56;
  • 29) 0.455 999 999 947 380 274 534 225 464 049 333 698 56 × 2 = 0 + 0.911 999 999 894 760 549 068 450 928 098 667 397 12;
  • 30) 0.911 999 999 894 760 549 068 450 928 098 667 397 12 × 2 = 1 + 0.823 999 999 789 521 098 136 901 856 197 334 794 24;
  • 31) 0.823 999 999 789 521 098 136 901 856 197 334 794 24 × 2 = 1 + 0.647 999 999 579 042 196 273 803 712 394 669 588 48;
  • 32) 0.647 999 999 579 042 196 273 803 712 394 669 588 48 × 2 = 1 + 0.295 999 999 158 084 392 547 607 424 789 339 176 96;
  • 33) 0.295 999 999 158 084 392 547 607 424 789 339 176 96 × 2 = 0 + 0.591 999 998 316 168 785 095 214 849 578 678 353 92;
  • 34) 0.591 999 998 316 168 785 095 214 849 578 678 353 92 × 2 = 1 + 0.183 999 996 632 337 570 190 429 699 157 356 707 84;
  • 35) 0.183 999 996 632 337 570 190 429 699 157 356 707 84 × 2 = 0 + 0.367 999 993 264 675 140 380 859 398 314 713 415 68;
  • 36) 0.367 999 993 264 675 140 380 859 398 314 713 415 68 × 2 = 0 + 0.735 999 986 529 350 280 761 718 796 629 426 831 36;
  • 37) 0.735 999 986 529 350 280 761 718 796 629 426 831 36 × 2 = 1 + 0.471 999 973 058 700 561 523 437 593 258 853 662 72;
  • 38) 0.471 999 973 058 700 561 523 437 593 258 853 662 72 × 2 = 0 + 0.943 999 946 117 401 123 046 875 186 517 707 325 44;
  • 39) 0.943 999 946 117 401 123 046 875 186 517 707 325 44 × 2 = 1 + 0.887 999 892 234 802 246 093 750 373 035 414 650 88;
  • 40) 0.887 999 892 234 802 246 093 750 373 035 414 650 88 × 2 = 1 + 0.775 999 784 469 604 492 187 500 746 070 829 301 76;
  • 41) 0.775 999 784 469 604 492 187 500 746 070 829 301 76 × 2 = 1 + 0.551 999 568 939 208 984 375 001 492 141 658 603 52;
  • 42) 0.551 999 568 939 208 984 375 001 492 141 658 603 52 × 2 = 1 + 0.103 999 137 878 417 968 750 002 984 283 317 207 04;
  • 43) 0.103 999 137 878 417 968 750 002 984 283 317 207 04 × 2 = 0 + 0.207 998 275 756 835 937 500 005 968 566 634 414 08;
  • 44) 0.207 998 275 756 835 937 500 005 968 566 634 414 08 × 2 = 0 + 0.415 996 551 513 671 875 000 011 937 133 268 828 16;
  • 45) 0.415 996 551 513 671 875 000 011 937 133 268 828 16 × 2 = 0 + 0.831 993 103 027 343 750 000 023 874 266 537 656 32;
  • 46) 0.831 993 103 027 343 750 000 023 874 266 537 656 32 × 2 = 1 + 0.663 986 206 054 687 500 000 047 748 533 075 312 64;
  • 47) 0.663 986 206 054 687 500 000 047 748 533 075 312 64 × 2 = 1 + 0.327 972 412 109 375 000 000 095 497 066 150 625 28;
  • 48) 0.327 972 412 109 375 000 000 095 497 066 150 625 28 × 2 = 0 + 0.655 944 824 218 750 000 000 190 994 132 301 250 56;
  • 49) 0.655 944 824 218 750 000 000 190 994 132 301 250 56 × 2 = 1 + 0.311 889 648 437 500 000 000 381 988 264 602 501 12;
  • 50) 0.311 889 648 437 500 000 000 381 988 264 602 501 12 × 2 = 0 + 0.623 779 296 875 000 000 000 763 976 529 205 002 24;
  • 51) 0.623 779 296 875 000 000 000 763 976 529 205 002 24 × 2 = 1 + 0.247 558 593 750 000 000 001 527 953 058 410 004 48;
  • 52) 0.247 558 593 750 000 000 001 527 953 058 410 004 48 × 2 = 0 + 0.495 117 187 500 000 000 003 055 906 116 820 008 96;
  • 53) 0.495 117 187 500 000 000 003 055 906 116 820 008 96 × 2 = 0 + 0.990 234 375 000 000 000 006 111 812 233 640 017 92;
  • 54) 0.990 234 375 000 000 000 006 111 812 233 640 017 92 × 2 = 1 + 0.980 468 750 000 000 000 012 223 624 467 280 035 84;
  • 55) 0.980 468 750 000 000 000 012 223 624 467 280 035 84 × 2 = 1 + 0.960 937 500 000 000 000 024 447 248 934 560 071 68;
  • 56) 0.960 937 500 000 000 000 024 447 248 934 560 071 68 × 2 = 1 + 0.921 875 000 000 000 000 048 894 497 869 120 143 36;
  • 57) 0.921 875 000 000 000 000 048 894 497 869 120 143 36 × 2 = 1 + 0.843 750 000 000 000 000 097 788 995 738 240 286 72;
  • 58) 0.843 750 000 000 000 000 097 788 995 738 240 286 72 × 2 = 1 + 0.687 500 000 000 000 000 195 577 991 476 480 573 44;
  • 59) 0.687 500 000 000 000 000 195 577 991 476 480 573 44 × 2 = 1 + 0.375 000 000 000 000 000 391 155 982 952 961 146 88;
  • 60) 0.375 000 000 000 000 000 391 155 982 952 961 146 88 × 2 = 0 + 0.750 000 000 000 000 000 782 311 965 905 922 293 76;
  • 61) 0.750 000 000 000 000 000 782 311 965 905 922 293 76 × 2 = 1 + 0.500 000 000 000 000 001 564 623 931 811 844 587 52;
  • 62) 0.500 000 000 000 000 001 564 623 931 811 844 587 52 × 2 = 1 + 0.000 000 000 000 000 003 129 247 863 623 689 175 04;

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:


0.000 999 999 999 999 999 803 976 247 214 620 799 01(10) =

0.0000 0000 0100 0001 1000 1001 0011 0111 0100 1011 1100 0110 1010 0111 1110 11(2)

5. Positive number before normalization:

0.000 999 999 999 999 999 803 976 247 214 620 799 01(10) =

0.0000 0000 0100 0001 1000 1001 0011 0111 0100 1011 1100 0110 1010 0111 1110 11(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:


0.000 999 999 999 999 999 803 976 247 214 620 799 01(10) =

0.0000 0000 0100 0001 1000 1001 0011 0111 0100 1011 1100 0110 1010 0111 1110 11(2) =

0.0000 0000 0100 0001 1000 1001 0011 0111 0100 1011 1100 0110 1010 0111 1110 11(2) × 20 =

1.0000 0110 0010 0100 1101 1101 0010 1111 0001 1010 1001 1111 1011(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 1011


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 1011 =

0000 0110 0010 0100 1101 1101 0010 1111 0001 1010 1001 1111 1011


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 1011


The base ten decimal number 0.000 999 999 999 999 999 803 976 247 214 620 799 01 converted and written in 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 1011

» Decimal number in base ten 0.000 999 999 999 999 999 803 976 247 214 620 798 55 converted and written as 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