-10 064.349 999 999 998 544 808 477 163 314 819 208 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -10 064.349 999 999 998 544 808 477 163 314 819 208(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
-10 064.349 999 999 998 544 808 477 163 314 819 208(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:

|-10 064.349 999 999 998 544 808 477 163 314 819 208| = 10 064.349 999 999 998 544 808 477 163 314 819 208


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: 0.349 999 999 998 544 808 477 163 314 819 208.

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) 0.349 999 999 998 544 808 477 163 314 819 208 × 2 = 0 + 0.699 999 999 997 089 616 954 326 629 638 416;
  • 2) 0.699 999 999 997 089 616 954 326 629 638 416 × 2 = 1 + 0.399 999 999 994 179 233 908 653 259 276 832;
  • 3) 0.399 999 999 994 179 233 908 653 259 276 832 × 2 = 0 + 0.799 999 999 988 358 467 817 306 518 553 664;
  • 4) 0.799 999 999 988 358 467 817 306 518 553 664 × 2 = 1 + 0.599 999 999 976 716 935 634 613 037 107 328;
  • 5) 0.599 999 999 976 716 935 634 613 037 107 328 × 2 = 1 + 0.199 999 999 953 433 871 269 226 074 214 656;
  • 6) 0.199 999 999 953 433 871 269 226 074 214 656 × 2 = 0 + 0.399 999 999 906 867 742 538 452 148 429 312;
  • 7) 0.399 999 999 906 867 742 538 452 148 429 312 × 2 = 0 + 0.799 999 999 813 735 485 076 904 296 858 624;
  • 8) 0.799 999 999 813 735 485 076 904 296 858 624 × 2 = 1 + 0.599 999 999 627 470 970 153 808 593 717 248;
  • 9) 0.599 999 999 627 470 970 153 808 593 717 248 × 2 = 1 + 0.199 999 999 254 941 940 307 617 187 434 496;
  • 10) 0.199 999 999 254 941 940 307 617 187 434 496 × 2 = 0 + 0.399 999 998 509 883 880 615 234 374 868 992;
  • 11) 0.399 999 998 509 883 880 615 234 374 868 992 × 2 = 0 + 0.799 999 997 019 767 761 230 468 749 737 984;
  • 12) 0.799 999 997 019 767 761 230 468 749 737 984 × 2 = 1 + 0.599 999 994 039 535 522 460 937 499 475 968;
  • 13) 0.599 999 994 039 535 522 460 937 499 475 968 × 2 = 1 + 0.199 999 988 079 071 044 921 874 998 951 936;
  • 14) 0.199 999 988 079 071 044 921 874 998 951 936 × 2 = 0 + 0.399 999 976 158 142 089 843 749 997 903 872;
  • 15) 0.399 999 976 158 142 089 843 749 997 903 872 × 2 = 0 + 0.799 999 952 316 284 179 687 499 995 807 744;
  • 16) 0.799 999 952 316 284 179 687 499 995 807 744 × 2 = 1 + 0.599 999 904 632 568 359 374 999 991 615 488;
  • 17) 0.599 999 904 632 568 359 374 999 991 615 488 × 2 = 1 + 0.199 999 809 265 136 718 749 999 983 230 976;
  • 18) 0.199 999 809 265 136 718 749 999 983 230 976 × 2 = 0 + 0.399 999 618 530 273 437 499 999 966 461 952;
  • 19) 0.399 999 618 530 273 437 499 999 966 461 952 × 2 = 0 + 0.799 999 237 060 546 874 999 999 932 923 904;
  • 20) 0.799 999 237 060 546 874 999 999 932 923 904 × 2 = 1 + 0.599 998 474 121 093 749 999 999 865 847 808;
  • 21) 0.599 998 474 121 093 749 999 999 865 847 808 × 2 = 1 + 0.199 996 948 242 187 499 999 999 731 695 616;
  • 22) 0.199 996 948 242 187 499 999 999 731 695 616 × 2 = 0 + 0.399 993 896 484 374 999 999 999 463 391 232;
  • 23) 0.399 993 896 484 374 999 999 999 463 391 232 × 2 = 0 + 0.799 987 792 968 749 999 999 998 926 782 464;
  • 24) 0.799 987 792 968 749 999 999 998 926 782 464 × 2 = 1 + 0.599 975 585 937 499 999 999 997 853 564 928;
  • 25) 0.599 975 585 937 499 999 999 997 853 564 928 × 2 = 1 + 0.199 951 171 874 999 999 999 995 707 129 856;
  • 26) 0.199 951 171 874 999 999 999 995 707 129 856 × 2 = 0 + 0.399 902 343 749 999 999 999 991 414 259 712;
  • 27) 0.399 902 343 749 999 999 999 991 414 259 712 × 2 = 0 + 0.799 804 687 499 999 999 999 982 828 519 424;
  • 28) 0.799 804 687 499 999 999 999 982 828 519 424 × 2 = 1 + 0.599 609 374 999 999 999 999 965 657 038 848;
  • 29) 0.599 609 374 999 999 999 999 965 657 038 848 × 2 = 1 + 0.199 218 749 999 999 999 999 931 314 077 696;
  • 30) 0.199 218 749 999 999 999 999 931 314 077 696 × 2 = 0 + 0.398 437 499 999 999 999 999 862 628 155 392;
  • 31) 0.398 437 499 999 999 999 999 862 628 155 392 × 2 = 0 + 0.796 874 999 999 999 999 999 725 256 310 784;
  • 32) 0.796 874 999 999 999 999 999 725 256 310 784 × 2 = 1 + 0.593 749 999 999 999 999 999 450 512 621 568;
  • 33) 0.593 749 999 999 999 999 999 450 512 621 568 × 2 = 1 + 0.187 499 999 999 999 999 998 901 025 243 136;
  • 34) 0.187 499 999 999 999 999 998 901 025 243 136 × 2 = 0 + 0.374 999 999 999 999 999 997 802 050 486 272;
  • 35) 0.374 999 999 999 999 999 997 802 050 486 272 × 2 = 0 + 0.749 999 999 999 999 999 995 604 100 972 544;
  • 36) 0.749 999 999 999 999 999 995 604 100 972 544 × 2 = 1 + 0.499 999 999 999 999 999 991 208 201 945 088;
  • 37) 0.499 999 999 999 999 999 991 208 201 945 088 × 2 = 0 + 0.999 999 999 999 999 999 982 416 403 890 176;
  • 38) 0.999 999 999 999 999 999 982 416 403 890 176 × 2 = 1 + 0.999 999 999 999 999 999 964 832 807 780 352;
  • 39) 0.999 999 999 999 999 999 964 832 807 780 352 × 2 = 1 + 0.999 999 999 999 999 999 929 665 615 560 704;
  • 40) 0.999 999 999 999 999 999 929 665 615 560 704 × 2 = 1 + 0.999 999 999 999 999 999 859 331 231 121 408;
  • 41) 0.999 999 999 999 999 999 859 331 231 121 408 × 2 = 1 + 0.999 999 999 999 999 999 718 662 462 242 816;
  • 42) 0.999 999 999 999 999 999 718 662 462 242 816 × 2 = 1 + 0.999 999 999 999 999 999 437 324 924 485 632;
  • 43) 0.999 999 999 999 999 999 437 324 924 485 632 × 2 = 1 + 0.999 999 999 999 999 998 874 649 848 971 264;
  • 44) 0.999 999 999 999 999 998 874 649 848 971 264 × 2 = 1 + 0.999 999 999 999 999 997 749 299 697 942 528;
  • 45) 0.999 999 999 999 999 997 749 299 697 942 528 × 2 = 1 + 0.999 999 999 999 999 995 498 599 395 885 056;
  • 46) 0.999 999 999 999 999 995 498 599 395 885 056 × 2 = 1 + 0.999 999 999 999 999 990 997 198 791 770 112;
  • 47) 0.999 999 999 999 999 990 997 198 791 770 112 × 2 = 1 + 0.999 999 999 999 999 981 994 397 583 540 224;
  • 48) 0.999 999 999 999 999 981 994 397 583 540 224 × 2 = 1 + 0.999 999 999 999 999 963 988 795 167 080 448;
  • 49) 0.999 999 999 999 999 963 988 795 167 080 448 × 2 = 1 + 0.999 999 999 999 999 927 977 590 334 160 896;
  • 50) 0.999 999 999 999 999 927 977 590 334 160 896 × 2 = 1 + 0.999 999 999 999 999 855 955 180 668 321 792;
  • 51) 0.999 999 999 999 999 855 955 180 668 321 792 × 2 = 1 + 0.999 999 999 999 999 711 910 361 336 643 584;
  • 52) 0.999 999 999 999 999 711 910 361 336 643 584 × 2 = 1 + 0.999 999 999 999 999 423 820 722 673 287 168;
  • 53) 0.999 999 999 999 999 423 820 722 673 287 168 × 2 = 1 + 0.999 999 999 999 998 847 641 445 346 574 336;

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:


0.349 999 999 998 544 808 477 163 314 819 208(10) =

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

6. Positive number before normalization:

10 064.349 999 999 998 544 808 477 163 314 819 208(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:


10 064.349 999 999 998 544 808 477 163 314 819 208(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 -10 064.349 999 999 998 544 808 477 163 314 819 208 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