654.599 999 999 999 909 050 529 822 707 171 6 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 654.599 999 999 999 909 050 529 822 707 171 6(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
654.599 999 999 999 909 050 529 822 707 171 6(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: 654.
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;
  • 654 ÷ 2 = 327 + 0;
  • 327 ÷ 2 = 163 + 1;
  • 163 ÷ 2 = 81 + 1;
  • 81 ÷ 2 = 40 + 1;
  • 40 ÷ 2 = 20 + 0;
  • 20 ÷ 2 = 10 + 0;
  • 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.

654(10) =

10 1000 1110(2)


3. Convert to binary (base 2) the fractional part: 0.599 999 999 999 909 050 529 822 707 171 6.

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) 0.599 999 999 999 909 050 529 822 707 171 6 × 2 = 1 + 0.199 999 999 999 818 101 059 645 414 343 2;
  • 2) 0.199 999 999 999 818 101 059 645 414 343 2 × 2 = 0 + 0.399 999 999 999 636 202 119 290 828 686 4;
  • 3) 0.399 999 999 999 636 202 119 290 828 686 4 × 2 = 0 + 0.799 999 999 999 272 404 238 581 657 372 8;
  • 4) 0.799 999 999 999 272 404 238 581 657 372 8 × 2 = 1 + 0.599 999 999 998 544 808 477 163 314 745 6;
  • 5) 0.599 999 999 998 544 808 477 163 314 745 6 × 2 = 1 + 0.199 999 999 997 089 616 954 326 629 491 2;
  • 6) 0.199 999 999 997 089 616 954 326 629 491 2 × 2 = 0 + 0.399 999 999 994 179 233 908 653 258 982 4;
  • 7) 0.399 999 999 994 179 233 908 653 258 982 4 × 2 = 0 + 0.799 999 999 988 358 467 817 306 517 964 8;
  • 8) 0.799 999 999 988 358 467 817 306 517 964 8 × 2 = 1 + 0.599 999 999 976 716 935 634 613 035 929 6;
  • 9) 0.599 999 999 976 716 935 634 613 035 929 6 × 2 = 1 + 0.199 999 999 953 433 871 269 226 071 859 2;
  • 10) 0.199 999 999 953 433 871 269 226 071 859 2 × 2 = 0 + 0.399 999 999 906 867 742 538 452 143 718 4;
  • 11) 0.399 999 999 906 867 742 538 452 143 718 4 × 2 = 0 + 0.799 999 999 813 735 485 076 904 287 436 8;
  • 12) 0.799 999 999 813 735 485 076 904 287 436 8 × 2 = 1 + 0.599 999 999 627 470 970 153 808 574 873 6;
  • 13) 0.599 999 999 627 470 970 153 808 574 873 6 × 2 = 1 + 0.199 999 999 254 941 940 307 617 149 747 2;
  • 14) 0.199 999 999 254 941 940 307 617 149 747 2 × 2 = 0 + 0.399 999 998 509 883 880 615 234 299 494 4;
  • 15) 0.399 999 998 509 883 880 615 234 299 494 4 × 2 = 0 + 0.799 999 997 019 767 761 230 468 598 988 8;
  • 16) 0.799 999 997 019 767 761 230 468 598 988 8 × 2 = 1 + 0.599 999 994 039 535 522 460 937 197 977 6;
  • 17) 0.599 999 994 039 535 522 460 937 197 977 6 × 2 = 1 + 0.199 999 988 079 071 044 921 874 395 955 2;
  • 18) 0.199 999 988 079 071 044 921 874 395 955 2 × 2 = 0 + 0.399 999 976 158 142 089 843 748 791 910 4;
  • 19) 0.399 999 976 158 142 089 843 748 791 910 4 × 2 = 0 + 0.799 999 952 316 284 179 687 497 583 820 8;
  • 20) 0.799 999 952 316 284 179 687 497 583 820 8 × 2 = 1 + 0.599 999 904 632 568 359 374 995 167 641 6;
  • 21) 0.599 999 904 632 568 359 374 995 167 641 6 × 2 = 1 + 0.199 999 809 265 136 718 749 990 335 283 2;
  • 22) 0.199 999 809 265 136 718 749 990 335 283 2 × 2 = 0 + 0.399 999 618 530 273 437 499 980 670 566 4;
  • 23) 0.399 999 618 530 273 437 499 980 670 566 4 × 2 = 0 + 0.799 999 237 060 546 874 999 961 341 132 8;
  • 24) 0.799 999 237 060 546 874 999 961 341 132 8 × 2 = 1 + 0.599 998 474 121 093 749 999 922 682 265 6;
  • 25) 0.599 998 474 121 093 749 999 922 682 265 6 × 2 = 1 + 0.199 996 948 242 187 499 999 845 364 531 2;
  • 26) 0.199 996 948 242 187 499 999 845 364 531 2 × 2 = 0 + 0.399 993 896 484 374 999 999 690 729 062 4;
  • 27) 0.399 993 896 484 374 999 999 690 729 062 4 × 2 = 0 + 0.799 987 792 968 749 999 999 381 458 124 8;
  • 28) 0.799 987 792 968 749 999 999 381 458 124 8 × 2 = 1 + 0.599 975 585 937 499 999 998 762 916 249 6;
  • 29) 0.599 975 585 937 499 999 998 762 916 249 6 × 2 = 1 + 0.199 951 171 874 999 999 997 525 832 499 2;
  • 30) 0.199 951 171 874 999 999 997 525 832 499 2 × 2 = 0 + 0.399 902 343 749 999 999 995 051 664 998 4;
  • 31) 0.399 902 343 749 999 999 995 051 664 998 4 × 2 = 0 + 0.799 804 687 499 999 999 990 103 329 996 8;
  • 32) 0.799 804 687 499 999 999 990 103 329 996 8 × 2 = 1 + 0.599 609 374 999 999 999 980 206 659 993 6;
  • 33) 0.599 609 374 999 999 999 980 206 659 993 6 × 2 = 1 + 0.199 218 749 999 999 999 960 413 319 987 2;
  • 34) 0.199 218 749 999 999 999 960 413 319 987 2 × 2 = 0 + 0.398 437 499 999 999 999 920 826 639 974 4;
  • 35) 0.398 437 499 999 999 999 920 826 639 974 4 × 2 = 0 + 0.796 874 999 999 999 999 841 653 279 948 8;
  • 36) 0.796 874 999 999 999 999 841 653 279 948 8 × 2 = 1 + 0.593 749 999 999 999 999 683 306 559 897 6;
  • 37) 0.593 749 999 999 999 999 683 306 559 897 6 × 2 = 1 + 0.187 499 999 999 999 999 366 613 119 795 2;
  • 38) 0.187 499 999 999 999 999 366 613 119 795 2 × 2 = 0 + 0.374 999 999 999 999 998 733 226 239 590 4;
  • 39) 0.374 999 999 999 999 998 733 226 239 590 4 × 2 = 0 + 0.749 999 999 999 999 997 466 452 479 180 8;
  • 40) 0.749 999 999 999 999 997 466 452 479 180 8 × 2 = 1 + 0.499 999 999 999 999 994 932 904 958 361 6;
  • 41) 0.499 999 999 999 999 994 932 904 958 361 6 × 2 = 0 + 0.999 999 999 999 999 989 865 809 916 723 2;
  • 42) 0.999 999 999 999 999 989 865 809 916 723 2 × 2 = 1 + 0.999 999 999 999 999 979 731 619 833 446 4;
  • 43) 0.999 999 999 999 999 979 731 619 833 446 4 × 2 = 1 + 0.999 999 999 999 999 959 463 239 666 892 8;
  • 44) 0.999 999 999 999 999 959 463 239 666 892 8 × 2 = 1 + 0.999 999 999 999 999 918 926 479 333 785 6;
  • 45) 0.999 999 999 999 999 918 926 479 333 785 6 × 2 = 1 + 0.999 999 999 999 999 837 852 958 667 571 2;
  • 46) 0.999 999 999 999 999 837 852 958 667 571 2 × 2 = 1 + 0.999 999 999 999 999 675 705 917 335 142 4;
  • 47) 0.999 999 999 999 999 675 705 917 335 142 4 × 2 = 1 + 0.999 999 999 999 999 351 411 834 670 284 8;
  • 48) 0.999 999 999 999 999 351 411 834 670 284 8 × 2 = 1 + 0.999 999 999 999 998 702 823 669 340 569 6;
  • 49) 0.999 999 999 999 998 702 823 669 340 569 6 × 2 = 1 + 0.999 999 999 999 997 405 647 338 681 139 2;
  • 50) 0.999 999 999 999 997 405 647 338 681 139 2 × 2 = 1 + 0.999 999 999 999 994 811 294 677 362 278 4;
  • 51) 0.999 999 999 999 994 811 294 677 362 278 4 × 2 = 1 + 0.999 999 999 999 989 622 589 354 724 556 8;
  • 52) 0.999 999 999 999 989 622 589 354 724 556 8 × 2 = 1 + 0.999 999 999 999 979 245 178 709 449 113 6;
  • 53) 0.999 999 999 999 979 245 178 709 449 113 6 × 2 = 1 + 0.999 999 999 999 958 490 357 418 898 227 2;

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:


0.599 999 999 999 909 050 529 822 707 171 6(10) =

0.1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0111 1111 1111 1(2)

5. Positive number before normalization:

654.599 999 999 999 909 050 529 822 707 171 6(10) =

10 1000 1110.1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0111 1111 1111 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 9 positions to the left, so that only one non zero digit remains to the left of it:


654.599 999 999 999 909 050 529 822 707 171 6(10) =

10 1000 1110.1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0111 1111 1111 1(2) =

10 1000 1110.1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0111 1111 1111 1(2) × 20 =

1.0100 0111 0100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1011 1111 1111 11(2) × 29


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): 9

Mantissa (not normalized):
1.0100 0111 0100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1011 1111 1111 11


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

9 + 2(11-1) - 1 =

(9 + 1 023)(10) =

1 032(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;
  • 1 032 ÷ 2 = 516 + 0;
  • 516 ÷ 2 = 258 + 0;
  • 258 ÷ 2 = 129 + 0;
  • 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;

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) =

1032(10) =

100 0000 1000(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. 0100 0111 0100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1011 11 1111 1111 =

0100 0111 0100 1100 1100 1100 1100 1100 1100 1100 1100 1100 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) =
100 0000 1000

Mantissa (52 bits) =
0100 0111 0100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1011


Decimal number 654.599 999 999 999 909 050 529 822 707 171 6 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 1000 - 0100 0111 0100 1100 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