-0.000 281 01 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 281 01(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
-0.000 281 01(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:

|-0.000 281 01| = 0.000 281 01


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: 0.000 281 01.

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) 0.000 281 01 × 2 = 0 + 0.000 562 02;
  • 2) 0.000 562 02 × 2 = 0 + 0.001 124 04;
  • 3) 0.001 124 04 × 2 = 0 + 0.002 248 08;
  • 4) 0.002 248 08 × 2 = 0 + 0.004 496 16;
  • 5) 0.004 496 16 × 2 = 0 + 0.008 992 32;
  • 6) 0.008 992 32 × 2 = 0 + 0.017 984 64;
  • 7) 0.017 984 64 × 2 = 0 + 0.035 969 28;
  • 8) 0.035 969 28 × 2 = 0 + 0.071 938 56;
  • 9) 0.071 938 56 × 2 = 0 + 0.143 877 12;
  • 10) 0.143 877 12 × 2 = 0 + 0.287 754 24;
  • 11) 0.287 754 24 × 2 = 0 + 0.575 508 48;
  • 12) 0.575 508 48 × 2 = 1 + 0.151 016 96;
  • 13) 0.151 016 96 × 2 = 0 + 0.302 033 92;
  • 14) 0.302 033 92 × 2 = 0 + 0.604 067 84;
  • 15) 0.604 067 84 × 2 = 1 + 0.208 135 68;
  • 16) 0.208 135 68 × 2 = 0 + 0.416 271 36;
  • 17) 0.416 271 36 × 2 = 0 + 0.832 542 72;
  • 18) 0.832 542 72 × 2 = 1 + 0.665 085 44;
  • 19) 0.665 085 44 × 2 = 1 + 0.330 170 88;
  • 20) 0.330 170 88 × 2 = 0 + 0.660 341 76;
  • 21) 0.660 341 76 × 2 = 1 + 0.320 683 52;
  • 22) 0.320 683 52 × 2 = 0 + 0.641 367 04;
  • 23) 0.641 367 04 × 2 = 1 + 0.282 734 08;
  • 24) 0.282 734 08 × 2 = 0 + 0.565 468 16;
  • 25) 0.565 468 16 × 2 = 1 + 0.130 936 32;
  • 26) 0.130 936 32 × 2 = 0 + 0.261 872 64;
  • 27) 0.261 872 64 × 2 = 0 + 0.523 745 28;
  • 28) 0.523 745 28 × 2 = 1 + 0.047 490 56;
  • 29) 0.047 490 56 × 2 = 0 + 0.094 981 12;
  • 30) 0.094 981 12 × 2 = 0 + 0.189 962 24;
  • 31) 0.189 962 24 × 2 = 0 + 0.379 924 48;
  • 32) 0.379 924 48 × 2 = 0 + 0.759 848 96;
  • 33) 0.759 848 96 × 2 = 1 + 0.519 697 92;
  • 34) 0.519 697 92 × 2 = 1 + 0.039 395 84;
  • 35) 0.039 395 84 × 2 = 0 + 0.078 791 68;
  • 36) 0.078 791 68 × 2 = 0 + 0.157 583 36;
  • 37) 0.157 583 36 × 2 = 0 + 0.315 166 72;
  • 38) 0.315 166 72 × 2 = 0 + 0.630 333 44;
  • 39) 0.630 333 44 × 2 = 1 + 0.260 666 88;
  • 40) 0.260 666 88 × 2 = 0 + 0.521 333 76;
  • 41) 0.521 333 76 × 2 = 1 + 0.042 667 52;
  • 42) 0.042 667 52 × 2 = 0 + 0.085 335 04;
  • 43) 0.085 335 04 × 2 = 0 + 0.170 670 08;
  • 44) 0.170 670 08 × 2 = 0 + 0.341 340 16;
  • 45) 0.341 340 16 × 2 = 0 + 0.682 680 32;
  • 46) 0.682 680 32 × 2 = 1 + 0.365 360 64;
  • 47) 0.365 360 64 × 2 = 0 + 0.730 721 28;
  • 48) 0.730 721 28 × 2 = 1 + 0.461 442 56;
  • 49) 0.461 442 56 × 2 = 0 + 0.922 885 12;
  • 50) 0.922 885 12 × 2 = 1 + 0.845 770 24;
  • 51) 0.845 770 24 × 2 = 1 + 0.691 540 48;
  • 52) 0.691 540 48 × 2 = 1 + 0.383 080 96;
  • 53) 0.383 080 96 × 2 = 0 + 0.766 161 92;
  • 54) 0.766 161 92 × 2 = 1 + 0.532 323 84;
  • 55) 0.532 323 84 × 2 = 1 + 0.064 647 68;
  • 56) 0.064 647 68 × 2 = 0 + 0.129 295 36;
  • 57) 0.129 295 36 × 2 = 0 + 0.258 590 72;
  • 58) 0.258 590 72 × 2 = 0 + 0.517 181 44;
  • 59) 0.517 181 44 × 2 = 1 + 0.034 362 88;
  • 60) 0.034 362 88 × 2 = 0 + 0.068 725 76;
  • 61) 0.068 725 76 × 2 = 0 + 0.137 451 52;
  • 62) 0.137 451 52 × 2 = 0 + 0.274 903 04;
  • 63) 0.274 903 04 × 2 = 0 + 0.549 806 08;
  • 64) 0.549 806 08 × 2 = 1 + 0.099 612 16;

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:


0.000 281 01(10) =

0.0000 0000 0001 0010 0110 1010 1001 0000 1100 0010 1000 0101 0111 0110 0010 0001(2)

6. Positive number before normalization:

0.000 281 01(10) =

0.0000 0000 0001 0010 0110 1010 1001 0000 1100 0010 1000 0101 0111 0110 0010 0001(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:


0.000 281 01(10) =

0.0000 0000 0001 0010 0110 1010 1001 0000 1100 0010 1000 0101 0111 0110 0010 0001(2) =

0.0000 0000 0001 0010 0110 1010 1001 0000 1100 0010 1000 0101 0111 0110 0010 0001(2) × 20 =

1.0010 0110 1010 1001 0000 1100 0010 1000 0101 0111 0110 0010 0001(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 0110 1010 1001 0000 1100 0010 1000 0101 0111 0110 0010 0001


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 0110 1010 1001 0000 1100 0010 1000 0101 0111 0110 0010 0001 =

0010 0110 1010 1001 0000 1100 0010 1000 0101 0111 0110 0010 0001


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 0110 1010 1001 0000 1100 0010 1000 0101 0111 0110 0010 0001


Decimal number -0.000 281 01 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0011 - 0010 0110 1010 1001 0000 1100 0010 1000 0101 0111 0110 0010 0001

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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