-10 064.349 999 999 998 544 808 477 163 314 819 221 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -10 064.349 999 999 998 544 808 477 163 314 819 221(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
-10 064.349 999 999 998 544 808 477 163 314 819 221(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. Start with the positive version of the number:

|-10 064.349 999 999 998 544 808 477 163 314 819 221| = 10 064.349 999 999 998 544 808 477 163 314 819 221


2. First, convert to binary (in base 2) the integer part: 10 064.
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;
  • 10 064 ÷ 2 = 5 032 + 0;
  • 5 032 ÷ 2 = 2 516 + 0;
  • 2 516 ÷ 2 = 1 258 + 0;
  • 1 258 ÷ 2 = 629 + 0;
  • 629 ÷ 2 = 314 + 1;
  • 314 ÷ 2 = 157 + 0;
  • 157 ÷ 2 = 78 + 1;
  • 78 ÷ 2 = 39 + 0;
  • 39 ÷ 2 = 19 + 1;
  • 19 ÷ 2 = 9 + 1;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

3. 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.

10 064(10) =

10 0111 0101 0000(2)


4. Convert to binary (base 2) the fractional part: 0.349 999 999 998 544 808 477 163 314 819 221.

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) 0.349 999 999 998 544 808 477 163 314 819 221 × 2 = 0 + 0.699 999 999 997 089 616 954 326 629 638 442;
  • 2) 0.699 999 999 997 089 616 954 326 629 638 442 × 2 = 1 + 0.399 999 999 994 179 233 908 653 259 276 884;
  • 3) 0.399 999 999 994 179 233 908 653 259 276 884 × 2 = 0 + 0.799 999 999 988 358 467 817 306 518 553 768;
  • 4) 0.799 999 999 988 358 467 817 306 518 553 768 × 2 = 1 + 0.599 999 999 976 716 935 634 613 037 107 536;
  • 5) 0.599 999 999 976 716 935 634 613 037 107 536 × 2 = 1 + 0.199 999 999 953 433 871 269 226 074 215 072;
  • 6) 0.199 999 999 953 433 871 269 226 074 215 072 × 2 = 0 + 0.399 999 999 906 867 742 538 452 148 430 144;
  • 7) 0.399 999 999 906 867 742 538 452 148 430 144 × 2 = 0 + 0.799 999 999 813 735 485 076 904 296 860 288;
  • 8) 0.799 999 999 813 735 485 076 904 296 860 288 × 2 = 1 + 0.599 999 999 627 470 970 153 808 593 720 576;
  • 9) 0.599 999 999 627 470 970 153 808 593 720 576 × 2 = 1 + 0.199 999 999 254 941 940 307 617 187 441 152;
  • 10) 0.199 999 999 254 941 940 307 617 187 441 152 × 2 = 0 + 0.399 999 998 509 883 880 615 234 374 882 304;
  • 11) 0.399 999 998 509 883 880 615 234 374 882 304 × 2 = 0 + 0.799 999 997 019 767 761 230 468 749 764 608;
  • 12) 0.799 999 997 019 767 761 230 468 749 764 608 × 2 = 1 + 0.599 999 994 039 535 522 460 937 499 529 216;
  • 13) 0.599 999 994 039 535 522 460 937 499 529 216 × 2 = 1 + 0.199 999 988 079 071 044 921 874 999 058 432;
  • 14) 0.199 999 988 079 071 044 921 874 999 058 432 × 2 = 0 + 0.399 999 976 158 142 089 843 749 998 116 864;
  • 15) 0.399 999 976 158 142 089 843 749 998 116 864 × 2 = 0 + 0.799 999 952 316 284 179 687 499 996 233 728;
  • 16) 0.799 999 952 316 284 179 687 499 996 233 728 × 2 = 1 + 0.599 999 904 632 568 359 374 999 992 467 456;
  • 17) 0.599 999 904 632 568 359 374 999 992 467 456 × 2 = 1 + 0.199 999 809 265 136 718 749 999 984 934 912;
  • 18) 0.199 999 809 265 136 718 749 999 984 934 912 × 2 = 0 + 0.399 999 618 530 273 437 499 999 969 869 824;
  • 19) 0.399 999 618 530 273 437 499 999 969 869 824 × 2 = 0 + 0.799 999 237 060 546 874 999 999 939 739 648;
  • 20) 0.799 999 237 060 546 874 999 999 939 739 648 × 2 = 1 + 0.599 998 474 121 093 749 999 999 879 479 296;
  • 21) 0.599 998 474 121 093 749 999 999 879 479 296 × 2 = 1 + 0.199 996 948 242 187 499 999 999 758 958 592;
  • 22) 0.199 996 948 242 187 499 999 999 758 958 592 × 2 = 0 + 0.399 993 896 484 374 999 999 999 517 917 184;
  • 23) 0.399 993 896 484 374 999 999 999 517 917 184 × 2 = 0 + 0.799 987 792 968 749 999 999 999 035 834 368;
  • 24) 0.799 987 792 968 749 999 999 999 035 834 368 × 2 = 1 + 0.599 975 585 937 499 999 999 998 071 668 736;
  • 25) 0.599 975 585 937 499 999 999 998 071 668 736 × 2 = 1 + 0.199 951 171 874 999 999 999 996 143 337 472;
  • 26) 0.199 951 171 874 999 999 999 996 143 337 472 × 2 = 0 + 0.399 902 343 749 999 999 999 992 286 674 944;
  • 27) 0.399 902 343 749 999 999 999 992 286 674 944 × 2 = 0 + 0.799 804 687 499 999 999 999 984 573 349 888;
  • 28) 0.799 804 687 499 999 999 999 984 573 349 888 × 2 = 1 + 0.599 609 374 999 999 999 999 969 146 699 776;
  • 29) 0.599 609 374 999 999 999 999 969 146 699 776 × 2 = 1 + 0.199 218 749 999 999 999 999 938 293 399 552;
  • 30) 0.199 218 749 999 999 999 999 938 293 399 552 × 2 = 0 + 0.398 437 499 999 999 999 999 876 586 799 104;
  • 31) 0.398 437 499 999 999 999 999 876 586 799 104 × 2 = 0 + 0.796 874 999 999 999 999 999 753 173 598 208;
  • 32) 0.796 874 999 999 999 999 999 753 173 598 208 × 2 = 1 + 0.593 749 999 999 999 999 999 506 347 196 416;
  • 33) 0.593 749 999 999 999 999 999 506 347 196 416 × 2 = 1 + 0.187 499 999 999 999 999 999 012 694 392 832;
  • 34) 0.187 499 999 999 999 999 999 012 694 392 832 × 2 = 0 + 0.374 999 999 999 999 999 998 025 388 785 664;
  • 35) 0.374 999 999 999 999 999 998 025 388 785 664 × 2 = 0 + 0.749 999 999 999 999 999 996 050 777 571 328;
  • 36) 0.749 999 999 999 999 999 996 050 777 571 328 × 2 = 1 + 0.499 999 999 999 999 999 992 101 555 142 656;
  • 37) 0.499 999 999 999 999 999 992 101 555 142 656 × 2 = 0 + 0.999 999 999 999 999 999 984 203 110 285 312;
  • 38) 0.999 999 999 999 999 999 984 203 110 285 312 × 2 = 1 + 0.999 999 999 999 999 999 968 406 220 570 624;
  • 39) 0.999 999 999 999 999 999 968 406 220 570 624 × 2 = 1 + 0.999 999 999 999 999 999 936 812 441 141 248;
  • 40) 0.999 999 999 999 999 999 936 812 441 141 248 × 2 = 1 + 0.999 999 999 999 999 999 873 624 882 282 496;
  • 41) 0.999 999 999 999 999 999 873 624 882 282 496 × 2 = 1 + 0.999 999 999 999 999 999 747 249 764 564 992;
  • 42) 0.999 999 999 999 999 999 747 249 764 564 992 × 2 = 1 + 0.999 999 999 999 999 999 494 499 529 129 984;
  • 43) 0.999 999 999 999 999 999 494 499 529 129 984 × 2 = 1 + 0.999 999 999 999 999 998 988 999 058 259 968;
  • 44) 0.999 999 999 999 999 998 988 999 058 259 968 × 2 = 1 + 0.999 999 999 999 999 997 977 998 116 519 936;
  • 45) 0.999 999 999 999 999 997 977 998 116 519 936 × 2 = 1 + 0.999 999 999 999 999 995 955 996 233 039 872;
  • 46) 0.999 999 999 999 999 995 955 996 233 039 872 × 2 = 1 + 0.999 999 999 999 999 991 911 992 466 079 744;
  • 47) 0.999 999 999 999 999 991 911 992 466 079 744 × 2 = 1 + 0.999 999 999 999 999 983 823 984 932 159 488;
  • 48) 0.999 999 999 999 999 983 823 984 932 159 488 × 2 = 1 + 0.999 999 999 999 999 967 647 969 864 318 976;
  • 49) 0.999 999 999 999 999 967 647 969 864 318 976 × 2 = 1 + 0.999 999 999 999 999 935 295 939 728 637 952;
  • 50) 0.999 999 999 999 999 935 295 939 728 637 952 × 2 = 1 + 0.999 999 999 999 999 870 591 879 457 275 904;
  • 51) 0.999 999 999 999 999 870 591 879 457 275 904 × 2 = 1 + 0.999 999 999 999 999 741 183 758 914 551 808;
  • 52) 0.999 999 999 999 999 741 183 758 914 551 808 × 2 = 1 + 0.999 999 999 999 999 482 367 517 829 103 616;
  • 53) 0.999 999 999 999 999 482 367 517 829 103 616 × 2 = 1 + 0.999 999 999 999 998 964 735 035 658 207 232;

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).


5. 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:


0.349 999 999 998 544 808 477 163 314 819 221(10) =

0.0101 1001 1001 1001 1001 1001 1001 1001 1001 0111 1111 1111 1111 1(2)

6. Positive number before normalization:

10 064.349 999 999 998 544 808 477 163 314 819 221(10) =

10 0111 0101 0000.0101 1001 1001 1001 1001 1001 1001 1001 1001 0111 1111 1111 1111 1(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 13 positions to the left, so that only one non zero digit remains to the left of it:


10 064.349 999 999 998 544 808 477 163 314 819 221(10) =

10 0111 0101 0000.0101 1001 1001 1001 1001 1001 1001 1001 1001 0111 1111 1111 1111 1(2) =

10 0111 0101 0000.0101 1001 1001 1001 1001 1001 1001 1001 1001 0111 1111 1111 1111 1(2) × 20 =

1.0011 1010 1000 0010 1100 1100 1100 1100 1100 1100 1100 1100 1011 1111 1111 1111 11(2) × 213


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): 13

Mantissa (not normalized):
1.0011 1010 1000 0010 1100 1100 1100 1100 1100 1100 1100 1100 1011 1111 1111 1111 11


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

13 + 2(11-1) - 1 =

(13 + 1 023)(10) =

1 036(10)


10. 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;
  • 1 036 ÷ 2 = 518 + 0;
  • 518 ÷ 2 = 259 + 0;
  • 259 ÷ 2 = 129 + 1;
  • 129 ÷ 2 = 64 + 1;
  • 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;

11. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

1036(10) =

100 0000 1100(2)


12. 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. 0011 1010 1000 0010 1100 1100 1100 1100 1100 1100 1100 1100 1011 11 1111 1111 1111 =

0011 1010 1000 0010 1100 1100 1100 1100 1100 1100 1100 1100 1011


13. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
1 (a negative number)

Exponent (11 bits) =
100 0000 1100

Mantissa (52 bits) =
0011 1010 1000 0010 1100 1100 1100 1100 1100 1100 1100 1100 1011


Decimal number -10 064.349 999 999 998 544 808 477 163 314 819 221 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 100 0000 1100 - 0011 1010 1000 0010 1100 1100 1100 1100 1100 1100 1100 1100 1011

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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