-0.000 282 495 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 282 495(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
-0.000 282 495(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:

|-0.000 282 495| = 0.000 282 495


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: 0.000 282 495.

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) 0.000 282 495 × 2 = 0 + 0.000 564 99;
  • 2) 0.000 564 99 × 2 = 0 + 0.001 129 98;
  • 3) 0.001 129 98 × 2 = 0 + 0.002 259 96;
  • 4) 0.002 259 96 × 2 = 0 + 0.004 519 92;
  • 5) 0.004 519 92 × 2 = 0 + 0.009 039 84;
  • 6) 0.009 039 84 × 2 = 0 + 0.018 079 68;
  • 7) 0.018 079 68 × 2 = 0 + 0.036 159 36;
  • 8) 0.036 159 36 × 2 = 0 + 0.072 318 72;
  • 9) 0.072 318 72 × 2 = 0 + 0.144 637 44;
  • 10) 0.144 637 44 × 2 = 0 + 0.289 274 88;
  • 11) 0.289 274 88 × 2 = 0 + 0.578 549 76;
  • 12) 0.578 549 76 × 2 = 1 + 0.157 099 52;
  • 13) 0.157 099 52 × 2 = 0 + 0.314 199 04;
  • 14) 0.314 199 04 × 2 = 0 + 0.628 398 08;
  • 15) 0.628 398 08 × 2 = 1 + 0.256 796 16;
  • 16) 0.256 796 16 × 2 = 0 + 0.513 592 32;
  • 17) 0.513 592 32 × 2 = 1 + 0.027 184 64;
  • 18) 0.027 184 64 × 2 = 0 + 0.054 369 28;
  • 19) 0.054 369 28 × 2 = 0 + 0.108 738 56;
  • 20) 0.108 738 56 × 2 = 0 + 0.217 477 12;
  • 21) 0.217 477 12 × 2 = 0 + 0.434 954 24;
  • 22) 0.434 954 24 × 2 = 0 + 0.869 908 48;
  • 23) 0.869 908 48 × 2 = 1 + 0.739 816 96;
  • 24) 0.739 816 96 × 2 = 1 + 0.479 633 92;
  • 25) 0.479 633 92 × 2 = 0 + 0.959 267 84;
  • 26) 0.959 267 84 × 2 = 1 + 0.918 535 68;
  • 27) 0.918 535 68 × 2 = 1 + 0.837 071 36;
  • 28) 0.837 071 36 × 2 = 1 + 0.674 142 72;
  • 29) 0.674 142 72 × 2 = 1 + 0.348 285 44;
  • 30) 0.348 285 44 × 2 = 0 + 0.696 570 88;
  • 31) 0.696 570 88 × 2 = 1 + 0.393 141 76;
  • 32) 0.393 141 76 × 2 = 0 + 0.786 283 52;
  • 33) 0.786 283 52 × 2 = 1 + 0.572 567 04;
  • 34) 0.572 567 04 × 2 = 1 + 0.145 134 08;
  • 35) 0.145 134 08 × 2 = 0 + 0.290 268 16;
  • 36) 0.290 268 16 × 2 = 0 + 0.580 536 32;
  • 37) 0.580 536 32 × 2 = 1 + 0.161 072 64;
  • 38) 0.161 072 64 × 2 = 0 + 0.322 145 28;
  • 39) 0.322 145 28 × 2 = 0 + 0.644 290 56;
  • 40) 0.644 290 56 × 2 = 1 + 0.288 581 12;
  • 41) 0.288 581 12 × 2 = 0 + 0.577 162 24;
  • 42) 0.577 162 24 × 2 = 1 + 0.154 324 48;
  • 43) 0.154 324 48 × 2 = 0 + 0.308 648 96;
  • 44) 0.308 648 96 × 2 = 0 + 0.617 297 92;
  • 45) 0.617 297 92 × 2 = 1 + 0.234 595 84;
  • 46) 0.234 595 84 × 2 = 0 + 0.469 191 68;
  • 47) 0.469 191 68 × 2 = 0 + 0.938 383 36;
  • 48) 0.938 383 36 × 2 = 1 + 0.876 766 72;
  • 49) 0.876 766 72 × 2 = 1 + 0.753 533 44;
  • 50) 0.753 533 44 × 2 = 1 + 0.507 066 88;
  • 51) 0.507 066 88 × 2 = 1 + 0.014 133 76;
  • 52) 0.014 133 76 × 2 = 0 + 0.028 267 52;
  • 53) 0.028 267 52 × 2 = 0 + 0.056 535 04;
  • 54) 0.056 535 04 × 2 = 0 + 0.113 070 08;
  • 55) 0.113 070 08 × 2 = 0 + 0.226 140 16;
  • 56) 0.226 140 16 × 2 = 0 + 0.452 280 32;
  • 57) 0.452 280 32 × 2 = 0 + 0.904 560 64;
  • 58) 0.904 560 64 × 2 = 1 + 0.809 121 28;
  • 59) 0.809 121 28 × 2 = 1 + 0.618 242 56;
  • 60) 0.618 242 56 × 2 = 1 + 0.236 485 12;
  • 61) 0.236 485 12 × 2 = 0 + 0.472 970 24;
  • 62) 0.472 970 24 × 2 = 0 + 0.945 940 48;
  • 63) 0.945 940 48 × 2 = 1 + 0.891 880 96;
  • 64) 0.891 880 96 × 2 = 1 + 0.783 761 92;

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:


0.000 282 495(10) =

0.0000 0000 0001 0010 1000 0011 0111 1010 1100 1001 0100 1001 1110 0000 0111 0011(2)

6. Positive number before normalization:

0.000 282 495(10) =

0.0000 0000 0001 0010 1000 0011 0111 1010 1100 1001 0100 1001 1110 0000 0111 0011(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:


0.000 282 495(10) =

0.0000 0000 0001 0010 1000 0011 0111 1010 1100 1001 0100 1001 1110 0000 0111 0011(2) =

0.0000 0000 0001 0010 1000 0011 0111 1010 1100 1001 0100 1001 1110 0000 0111 0011(2) × 20 =

1.0010 1000 0011 0111 1010 1100 1001 0100 1001 1110 0000 0111 0011(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 1000 0011 0111 1010 1100 1001 0100 1001 1110 0000 0111 0011


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 1000 0011 0111 1010 1100 1001 0100 1001 1110 0000 0111 0011 =

0010 1000 0011 0111 1010 1100 1001 0100 1001 1110 0000 0111 0011


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 1000 0011 0111 1010 1100 1001 0100 1001 1110 0000 0111 0011


Decimal number -0.000 282 495 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0011 - 0010 1000 0011 0111 1010 1100 1001 0100 1001 1110 0000 0111 0011

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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