-0.000 282 039 4 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 282 039 4(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
-0.000 282 039 4(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:

|-0.000 282 039 4| = 0.000 282 039 4


2. 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;

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.

0(10) =

0(2)


4. Convert to binary (base 2) the fractional part: 0.000 282 039 4.

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) 0.000 282 039 4 × 2 = 0 + 0.000 564 078 8;
  • 2) 0.000 564 078 8 × 2 = 0 + 0.001 128 157 6;
  • 3) 0.001 128 157 6 × 2 = 0 + 0.002 256 315 2;
  • 4) 0.002 256 315 2 × 2 = 0 + 0.004 512 630 4;
  • 5) 0.004 512 630 4 × 2 = 0 + 0.009 025 260 8;
  • 6) 0.009 025 260 8 × 2 = 0 + 0.018 050 521 6;
  • 7) 0.018 050 521 6 × 2 = 0 + 0.036 101 043 2;
  • 8) 0.036 101 043 2 × 2 = 0 + 0.072 202 086 4;
  • 9) 0.072 202 086 4 × 2 = 0 + 0.144 404 172 8;
  • 10) 0.144 404 172 8 × 2 = 0 + 0.288 808 345 6;
  • 11) 0.288 808 345 6 × 2 = 0 + 0.577 616 691 2;
  • 12) 0.577 616 691 2 × 2 = 1 + 0.155 233 382 4;
  • 13) 0.155 233 382 4 × 2 = 0 + 0.310 466 764 8;
  • 14) 0.310 466 764 8 × 2 = 0 + 0.620 933 529 6;
  • 15) 0.620 933 529 6 × 2 = 1 + 0.241 867 059 2;
  • 16) 0.241 867 059 2 × 2 = 0 + 0.483 734 118 4;
  • 17) 0.483 734 118 4 × 2 = 0 + 0.967 468 236 8;
  • 18) 0.967 468 236 8 × 2 = 1 + 0.934 936 473 6;
  • 19) 0.934 936 473 6 × 2 = 1 + 0.869 872 947 2;
  • 20) 0.869 872 947 2 × 2 = 1 + 0.739 745 894 4;
  • 21) 0.739 745 894 4 × 2 = 1 + 0.479 491 788 8;
  • 22) 0.479 491 788 8 × 2 = 0 + 0.958 983 577 6;
  • 23) 0.958 983 577 6 × 2 = 1 + 0.917 967 155 2;
  • 24) 0.917 967 155 2 × 2 = 1 + 0.835 934 310 4;
  • 25) 0.835 934 310 4 × 2 = 1 + 0.671 868 620 8;
  • 26) 0.671 868 620 8 × 2 = 1 + 0.343 737 241 6;
  • 27) 0.343 737 241 6 × 2 = 0 + 0.687 474 483 2;
  • 28) 0.687 474 483 2 × 2 = 1 + 0.374 948 966 4;
  • 29) 0.374 948 966 4 × 2 = 0 + 0.749 897 932 8;
  • 30) 0.749 897 932 8 × 2 = 1 + 0.499 795 865 6;
  • 31) 0.499 795 865 6 × 2 = 0 + 0.999 591 731 2;
  • 32) 0.999 591 731 2 × 2 = 1 + 0.999 183 462 4;
  • 33) 0.999 183 462 4 × 2 = 1 + 0.998 366 924 8;
  • 34) 0.998 366 924 8 × 2 = 1 + 0.996 733 849 6;
  • 35) 0.996 733 849 6 × 2 = 1 + 0.993 467 699 2;
  • 36) 0.993 467 699 2 × 2 = 1 + 0.986 935 398 4;
  • 37) 0.986 935 398 4 × 2 = 1 + 0.973 870 796 8;
  • 38) 0.973 870 796 8 × 2 = 1 + 0.947 741 593 6;
  • 39) 0.947 741 593 6 × 2 = 1 + 0.895 483 187 2;
  • 40) 0.895 483 187 2 × 2 = 1 + 0.790 966 374 4;
  • 41) 0.790 966 374 4 × 2 = 1 + 0.581 932 748 8;
  • 42) 0.581 932 748 8 × 2 = 1 + 0.163 865 497 6;
  • 43) 0.163 865 497 6 × 2 = 0 + 0.327 730 995 2;
  • 44) 0.327 730 995 2 × 2 = 0 + 0.655 461 990 4;
  • 45) 0.655 461 990 4 × 2 = 1 + 0.310 923 980 8;
  • 46) 0.310 923 980 8 × 2 = 0 + 0.621 847 961 6;
  • 47) 0.621 847 961 6 × 2 = 1 + 0.243 695 923 2;
  • 48) 0.243 695 923 2 × 2 = 0 + 0.487 391 846 4;
  • 49) 0.487 391 846 4 × 2 = 0 + 0.974 783 692 8;
  • 50) 0.974 783 692 8 × 2 = 1 + 0.949 567 385 6;
  • 51) 0.949 567 385 6 × 2 = 1 + 0.899 134 771 2;
  • 52) 0.899 134 771 2 × 2 = 1 + 0.798 269 542 4;
  • 53) 0.798 269 542 4 × 2 = 1 + 0.596 539 084 8;
  • 54) 0.596 539 084 8 × 2 = 1 + 0.193 078 169 6;
  • 55) 0.193 078 169 6 × 2 = 0 + 0.386 156 339 2;
  • 56) 0.386 156 339 2 × 2 = 0 + 0.772 312 678 4;
  • 57) 0.772 312 678 4 × 2 = 1 + 0.544 625 356 8;
  • 58) 0.544 625 356 8 × 2 = 1 + 0.089 250 713 6;
  • 59) 0.089 250 713 6 × 2 = 0 + 0.178 501 427 2;
  • 60) 0.178 501 427 2 × 2 = 0 + 0.357 002 854 4;
  • 61) 0.357 002 854 4 × 2 = 0 + 0.714 005 708 8;
  • 62) 0.714 005 708 8 × 2 = 1 + 0.428 011 417 6;
  • 63) 0.428 011 417 6 × 2 = 0 + 0.856 022 835 2;
  • 64) 0.856 022 835 2 × 2 = 1 + 0.712 045 670 4;

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:


0.000 282 039 4(10) =

0.0000 0000 0001 0010 0111 1011 1101 0101 1111 1111 1100 1010 0111 1100 1100 0101(2)

6. Positive number before normalization:

0.000 282 039 4(10) =

0.0000 0000 0001 0010 0111 1011 1101 0101 1111 1111 1100 1010 0111 1100 1100 0101(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 12 positions to the right, so that only one non zero digit remains to the left of it:


0.000 282 039 4(10) =

0.0000 0000 0001 0010 0111 1011 1101 0101 1111 1111 1100 1010 0111 1100 1100 0101(2) =

0.0000 0000 0001 0010 0111 1011 1101 0101 1111 1111 1100 1010 0111 1100 1100 0101(2) × 20 =

1.0010 0111 1011 1101 0101 1111 1111 1100 1010 0111 1100 1100 0101(2) × 2-12


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): -12

Mantissa (not normalized):
1.0010 0111 1011 1101 0101 1111 1111 1100 1010 0111 1100 1100 0101


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-12 + 2(11-1) - 1 =

(-12 + 1 023)(10) =

1 011(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;
  • 1 011 ÷ 2 = 505 + 1;
  • 505 ÷ 2 = 252 + 1;
  • 252 ÷ 2 = 126 + 0;
  • 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;

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) =

1011(10) =

011 1111 0011(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, only if necessary (not the case here).


Mantissa (normalized) =

1. 0010 0111 1011 1101 0101 1111 1111 1100 1010 0111 1100 1100 0101 =

0010 0111 1011 1101 0101 1111 1111 1100 1010 0111 1100 1100 0101


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) =
011 1111 0011

Mantissa (52 bits) =
0010 0111 1011 1101 0101 1111 1111 1100 1010 0111 1100 1100 0101


Decimal number -0.000 282 039 4 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0011 - 0010 0111 1011 1101 0101 1111 1111 1100 1010 0111 1100 1100 0101

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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