-0.000 283 83 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 283 83(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
-0.000 283 83(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:

|-0.000 283 83| = 0.000 283 83


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: 0.000 283 83.

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) 0.000 283 83 × 2 = 0 + 0.000 567 66;
  • 2) 0.000 567 66 × 2 = 0 + 0.001 135 32;
  • 3) 0.001 135 32 × 2 = 0 + 0.002 270 64;
  • 4) 0.002 270 64 × 2 = 0 + 0.004 541 28;
  • 5) 0.004 541 28 × 2 = 0 + 0.009 082 56;
  • 6) 0.009 082 56 × 2 = 0 + 0.018 165 12;
  • 7) 0.018 165 12 × 2 = 0 + 0.036 330 24;
  • 8) 0.036 330 24 × 2 = 0 + 0.072 660 48;
  • 9) 0.072 660 48 × 2 = 0 + 0.145 320 96;
  • 10) 0.145 320 96 × 2 = 0 + 0.290 641 92;
  • 11) 0.290 641 92 × 2 = 0 + 0.581 283 84;
  • 12) 0.581 283 84 × 2 = 1 + 0.162 567 68;
  • 13) 0.162 567 68 × 2 = 0 + 0.325 135 36;
  • 14) 0.325 135 36 × 2 = 0 + 0.650 270 72;
  • 15) 0.650 270 72 × 2 = 1 + 0.300 541 44;
  • 16) 0.300 541 44 × 2 = 0 + 0.601 082 88;
  • 17) 0.601 082 88 × 2 = 1 + 0.202 165 76;
  • 18) 0.202 165 76 × 2 = 0 + 0.404 331 52;
  • 19) 0.404 331 52 × 2 = 0 + 0.808 663 04;
  • 20) 0.808 663 04 × 2 = 1 + 0.617 326 08;
  • 21) 0.617 326 08 × 2 = 1 + 0.234 652 16;
  • 22) 0.234 652 16 × 2 = 0 + 0.469 304 32;
  • 23) 0.469 304 32 × 2 = 0 + 0.938 608 64;
  • 24) 0.938 608 64 × 2 = 1 + 0.877 217 28;
  • 25) 0.877 217 28 × 2 = 1 + 0.754 434 56;
  • 26) 0.754 434 56 × 2 = 1 + 0.508 869 12;
  • 27) 0.508 869 12 × 2 = 1 + 0.017 738 24;
  • 28) 0.017 738 24 × 2 = 0 + 0.035 476 48;
  • 29) 0.035 476 48 × 2 = 0 + 0.070 952 96;
  • 30) 0.070 952 96 × 2 = 0 + 0.141 905 92;
  • 31) 0.141 905 92 × 2 = 0 + 0.283 811 84;
  • 32) 0.283 811 84 × 2 = 0 + 0.567 623 68;
  • 33) 0.567 623 68 × 2 = 1 + 0.135 247 36;
  • 34) 0.135 247 36 × 2 = 0 + 0.270 494 72;
  • 35) 0.270 494 72 × 2 = 0 + 0.540 989 44;
  • 36) 0.540 989 44 × 2 = 1 + 0.081 978 88;
  • 37) 0.081 978 88 × 2 = 0 + 0.163 957 76;
  • 38) 0.163 957 76 × 2 = 0 + 0.327 915 52;
  • 39) 0.327 915 52 × 2 = 0 + 0.655 831 04;
  • 40) 0.655 831 04 × 2 = 1 + 0.311 662 08;
  • 41) 0.311 662 08 × 2 = 0 + 0.623 324 16;
  • 42) 0.623 324 16 × 2 = 1 + 0.246 648 32;
  • 43) 0.246 648 32 × 2 = 0 + 0.493 296 64;
  • 44) 0.493 296 64 × 2 = 0 + 0.986 593 28;
  • 45) 0.986 593 28 × 2 = 1 + 0.973 186 56;
  • 46) 0.973 186 56 × 2 = 1 + 0.946 373 12;
  • 47) 0.946 373 12 × 2 = 1 + 0.892 746 24;
  • 48) 0.892 746 24 × 2 = 1 + 0.785 492 48;
  • 49) 0.785 492 48 × 2 = 1 + 0.570 984 96;
  • 50) 0.570 984 96 × 2 = 1 + 0.141 969 92;
  • 51) 0.141 969 92 × 2 = 0 + 0.283 939 84;
  • 52) 0.283 939 84 × 2 = 0 + 0.567 879 68;
  • 53) 0.567 879 68 × 2 = 1 + 0.135 759 36;
  • 54) 0.135 759 36 × 2 = 0 + 0.271 518 72;
  • 55) 0.271 518 72 × 2 = 0 + 0.543 037 44;
  • 56) 0.543 037 44 × 2 = 1 + 0.086 074 88;
  • 57) 0.086 074 88 × 2 = 0 + 0.172 149 76;
  • 58) 0.172 149 76 × 2 = 0 + 0.344 299 52;
  • 59) 0.344 299 52 × 2 = 0 + 0.688 599 04;
  • 60) 0.688 599 04 × 2 = 1 + 0.377 198 08;
  • 61) 0.377 198 08 × 2 = 0 + 0.754 396 16;
  • 62) 0.754 396 16 × 2 = 1 + 0.508 792 32;
  • 63) 0.508 792 32 × 2 = 1 + 0.017 584 64;
  • 64) 0.017 584 64 × 2 = 0 + 0.035 169 28;

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:


0.000 283 83(10) =

0.0000 0000 0001 0010 1001 1001 1110 0000 1001 0001 0100 1111 1100 1001 0001 0110(2)

6. Positive number before normalization:

0.000 283 83(10) =

0.0000 0000 0001 0010 1001 1001 1110 0000 1001 0001 0100 1111 1100 1001 0001 0110(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:


0.000 283 83(10) =

0.0000 0000 0001 0010 1001 1001 1110 0000 1001 0001 0100 1111 1100 1001 0001 0110(2) =

0.0000 0000 0001 0010 1001 1001 1110 0000 1001 0001 0100 1111 1100 1001 0001 0110(2) × 20 =

1.0010 1001 1001 1110 0000 1001 0001 0100 1111 1100 1001 0001 0110(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 1001 1001 1110 0000 1001 0001 0100 1111 1100 1001 0001 0110


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 1001 1001 1110 0000 1001 0001 0100 1111 1100 1001 0001 0110 =

0010 1001 1001 1110 0000 1001 0001 0100 1111 1100 1001 0001 0110


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 1001 1001 1110 0000 1001 0001 0100 1111 1100 1001 0001 0110


Decimal number -0.000 283 83 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0011 - 0010 1001 1001 1110 0000 1001 0001 0100 1111 1100 1001 0001 0110

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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