-1 036.699 999 999 999 818 101 059 645 414 352 416 99 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -1 036.699 999 999 999 818 101 059 645 414 352 416 99(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
-1 036.699 999 999 999 818 101 059 645 414 352 416 99(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:

|-1 036.699 999 999 999 818 101 059 645 414 352 416 99| = 1 036.699 999 999 999 818 101 059 645 414 352 416 99


2. First, convert to binary (in base 2) the integer part: 1 036.
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;
  • 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;

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.

1 036(10) =

100 0000 1100(2)


4. Convert to binary (base 2) the fractional part: 0.699 999 999 999 818 101 059 645 414 352 416 99.

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) 0.699 999 999 999 818 101 059 645 414 352 416 99 × 2 = 1 + 0.399 999 999 999 636 202 119 290 828 704 833 98;
  • 2) 0.399 999 999 999 636 202 119 290 828 704 833 98 × 2 = 0 + 0.799 999 999 999 272 404 238 581 657 409 667 96;
  • 3) 0.799 999 999 999 272 404 238 581 657 409 667 96 × 2 = 1 + 0.599 999 999 998 544 808 477 163 314 819 335 92;
  • 4) 0.599 999 999 998 544 808 477 163 314 819 335 92 × 2 = 1 + 0.199 999 999 997 089 616 954 326 629 638 671 84;
  • 5) 0.199 999 999 997 089 616 954 326 629 638 671 84 × 2 = 0 + 0.399 999 999 994 179 233 908 653 259 277 343 68;
  • 6) 0.399 999 999 994 179 233 908 653 259 277 343 68 × 2 = 0 + 0.799 999 999 988 358 467 817 306 518 554 687 36;
  • 7) 0.799 999 999 988 358 467 817 306 518 554 687 36 × 2 = 1 + 0.599 999 999 976 716 935 634 613 037 109 374 72;
  • 8) 0.599 999 999 976 716 935 634 613 037 109 374 72 × 2 = 1 + 0.199 999 999 953 433 871 269 226 074 218 749 44;
  • 9) 0.199 999 999 953 433 871 269 226 074 218 749 44 × 2 = 0 + 0.399 999 999 906 867 742 538 452 148 437 498 88;
  • 10) 0.399 999 999 906 867 742 538 452 148 437 498 88 × 2 = 0 + 0.799 999 999 813 735 485 076 904 296 874 997 76;
  • 11) 0.799 999 999 813 735 485 076 904 296 874 997 76 × 2 = 1 + 0.599 999 999 627 470 970 153 808 593 749 995 52;
  • 12) 0.599 999 999 627 470 970 153 808 593 749 995 52 × 2 = 1 + 0.199 999 999 254 941 940 307 617 187 499 991 04;
  • 13) 0.199 999 999 254 941 940 307 617 187 499 991 04 × 2 = 0 + 0.399 999 998 509 883 880 615 234 374 999 982 08;
  • 14) 0.399 999 998 509 883 880 615 234 374 999 982 08 × 2 = 0 + 0.799 999 997 019 767 761 230 468 749 999 964 16;
  • 15) 0.799 999 997 019 767 761 230 468 749 999 964 16 × 2 = 1 + 0.599 999 994 039 535 522 460 937 499 999 928 32;
  • 16) 0.599 999 994 039 535 522 460 937 499 999 928 32 × 2 = 1 + 0.199 999 988 079 071 044 921 874 999 999 856 64;
  • 17) 0.199 999 988 079 071 044 921 874 999 999 856 64 × 2 = 0 + 0.399 999 976 158 142 089 843 749 999 999 713 28;
  • 18) 0.399 999 976 158 142 089 843 749 999 999 713 28 × 2 = 0 + 0.799 999 952 316 284 179 687 499 999 999 426 56;
  • 19) 0.799 999 952 316 284 179 687 499 999 999 426 56 × 2 = 1 + 0.599 999 904 632 568 359 374 999 999 998 853 12;
  • 20) 0.599 999 904 632 568 359 374 999 999 998 853 12 × 2 = 1 + 0.199 999 809 265 136 718 749 999 999 997 706 24;
  • 21) 0.199 999 809 265 136 718 749 999 999 997 706 24 × 2 = 0 + 0.399 999 618 530 273 437 499 999 999 995 412 48;
  • 22) 0.399 999 618 530 273 437 499 999 999 995 412 48 × 2 = 0 + 0.799 999 237 060 546 874 999 999 999 990 824 96;
  • 23) 0.799 999 237 060 546 874 999 999 999 990 824 96 × 2 = 1 + 0.599 998 474 121 093 749 999 999 999 981 649 92;
  • 24) 0.599 998 474 121 093 749 999 999 999 981 649 92 × 2 = 1 + 0.199 996 948 242 187 499 999 999 999 963 299 84;
  • 25) 0.199 996 948 242 187 499 999 999 999 963 299 84 × 2 = 0 + 0.399 993 896 484 374 999 999 999 999 926 599 68;
  • 26) 0.399 993 896 484 374 999 999 999 999 926 599 68 × 2 = 0 + 0.799 987 792 968 749 999 999 999 999 853 199 36;
  • 27) 0.799 987 792 968 749 999 999 999 999 853 199 36 × 2 = 1 + 0.599 975 585 937 499 999 999 999 999 706 398 72;
  • 28) 0.599 975 585 937 499 999 999 999 999 706 398 72 × 2 = 1 + 0.199 951 171 874 999 999 999 999 999 412 797 44;
  • 29) 0.199 951 171 874 999 999 999 999 999 412 797 44 × 2 = 0 + 0.399 902 343 749 999 999 999 999 998 825 594 88;
  • 30) 0.399 902 343 749 999 999 999 999 998 825 594 88 × 2 = 0 + 0.799 804 687 499 999 999 999 999 997 651 189 76;
  • 31) 0.799 804 687 499 999 999 999 999 997 651 189 76 × 2 = 1 + 0.599 609 374 999 999 999 999 999 995 302 379 52;
  • 32) 0.599 609 374 999 999 999 999 999 995 302 379 52 × 2 = 1 + 0.199 218 749 999 999 999 999 999 990 604 759 04;
  • 33) 0.199 218 749 999 999 999 999 999 990 604 759 04 × 2 = 0 + 0.398 437 499 999 999 999 999 999 981 209 518 08;
  • 34) 0.398 437 499 999 999 999 999 999 981 209 518 08 × 2 = 0 + 0.796 874 999 999 999 999 999 999 962 419 036 16;
  • 35) 0.796 874 999 999 999 999 999 999 962 419 036 16 × 2 = 1 + 0.593 749 999 999 999 999 999 999 924 838 072 32;
  • 36) 0.593 749 999 999 999 999 999 999 924 838 072 32 × 2 = 1 + 0.187 499 999 999 999 999 999 999 849 676 144 64;
  • 37) 0.187 499 999 999 999 999 999 999 849 676 144 64 × 2 = 0 + 0.374 999 999 999 999 999 999 999 699 352 289 28;
  • 38) 0.374 999 999 999 999 999 999 999 699 352 289 28 × 2 = 0 + 0.749 999 999 999 999 999 999 999 398 704 578 56;
  • 39) 0.749 999 999 999 999 999 999 999 398 704 578 56 × 2 = 1 + 0.499 999 999 999 999 999 999 998 797 409 157 12;
  • 40) 0.499 999 999 999 999 999 999 998 797 409 157 12 × 2 = 0 + 0.999 999 999 999 999 999 999 997 594 818 314 24;
  • 41) 0.999 999 999 999 999 999 999 997 594 818 314 24 × 2 = 1 + 0.999 999 999 999 999 999 999 995 189 636 628 48;
  • 42) 0.999 999 999 999 999 999 999 995 189 636 628 48 × 2 = 1 + 0.999 999 999 999 999 999 999 990 379 273 256 96;
  • 43) 0.999 999 999 999 999 999 999 990 379 273 256 96 × 2 = 1 + 0.999 999 999 999 999 999 999 980 758 546 513 92;
  • 44) 0.999 999 999 999 999 999 999 980 758 546 513 92 × 2 = 1 + 0.999 999 999 999 999 999 999 961 517 093 027 84;
  • 45) 0.999 999 999 999 999 999 999 961 517 093 027 84 × 2 = 1 + 0.999 999 999 999 999 999 999 923 034 186 055 68;
  • 46) 0.999 999 999 999 999 999 999 923 034 186 055 68 × 2 = 1 + 0.999 999 999 999 999 999 999 846 068 372 111 36;
  • 47) 0.999 999 999 999 999 999 999 846 068 372 111 36 × 2 = 1 + 0.999 999 999 999 999 999 999 692 136 744 222 72;
  • 48) 0.999 999 999 999 999 999 999 692 136 744 222 72 × 2 = 1 + 0.999 999 999 999 999 999 999 384 273 488 445 44;
  • 49) 0.999 999 999 999 999 999 999 384 273 488 445 44 × 2 = 1 + 0.999 999 999 999 999 999 998 768 546 976 890 88;
  • 50) 0.999 999 999 999 999 999 998 768 546 976 890 88 × 2 = 1 + 0.999 999 999 999 999 999 997 537 093 953 781 76;
  • 51) 0.999 999 999 999 999 999 997 537 093 953 781 76 × 2 = 1 + 0.999 999 999 999 999 999 995 074 187 907 563 52;
  • 52) 0.999 999 999 999 999 999 995 074 187 907 563 52 × 2 = 1 + 0.999 999 999 999 999 999 990 148 375 815 127 04;
  • 53) 0.999 999 999 999 999 999 990 148 375 815 127 04 × 2 = 1 + 0.999 999 999 999 999 999 980 296 751 630 254 08;

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:


0.699 999 999 999 818 101 059 645 414 352 416 99(10) =

0.1011 0011 0011 0011 0011 0011 0011 0011 0011 0010 1111 1111 1111 1(2)

6. Positive number before normalization:

1 036.699 999 999 999 818 101 059 645 414 352 416 99(10) =

100 0000 1100.1011 0011 0011 0011 0011 0011 0011 0011 0011 0010 1111 1111 1111 1(2)

7. Normalize the binary representation of the number.

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


1 036.699 999 999 999 818 101 059 645 414 352 416 99(10) =

100 0000 1100.1011 0011 0011 0011 0011 0011 0011 0011 0011 0010 1111 1111 1111 1(2) =

100 0000 1100.1011 0011 0011 0011 0011 0011 0011 0011 0011 0010 1111 1111 1111 1(2) × 20 =

1.0000 0011 0010 1100 1100 1100 1100 1100 1100 1100 1100 1100 1011 1111 1111 111(2) × 210


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): 10

Mantissa (not normalized):
1.0000 0011 0010 1100 1100 1100 1100 1100 1100 1100 1100 1100 1011 1111 1111 111


9. 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) =

1 033(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;
  • 1 033 ÷ 2 = 516 + 1;
  • 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;

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) =

1033(10) =

100 0000 1001(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. 0000 0011 0010 1100 1100 1100 1100 1100 1100 1100 1100 1100 1011 111 1111 1111 =

0000 0011 0010 1100 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 1001

Mantissa (52 bits) =
0000 0011 0010 1100 1100 1100 1100 1100 1100 1100 1100 1100 1011


Decimal number -1 036.699 999 999 999 818 101 059 645 414 352 416 99 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 100 0000 1001 - 0000 0011 0010 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