-0.023 219 89 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.023 219 89(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
-0.023 219 89(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:

|-0.023 219 89| = 0.023 219 89


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: 0.023 219 89.

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) 0.023 219 89 × 2 = 0 + 0.046 439 78;
  • 2) 0.046 439 78 × 2 = 0 + 0.092 879 56;
  • 3) 0.092 879 56 × 2 = 0 + 0.185 759 12;
  • 4) 0.185 759 12 × 2 = 0 + 0.371 518 24;
  • 5) 0.371 518 24 × 2 = 0 + 0.743 036 48;
  • 6) 0.743 036 48 × 2 = 1 + 0.486 072 96;
  • 7) 0.486 072 96 × 2 = 0 + 0.972 145 92;
  • 8) 0.972 145 92 × 2 = 1 + 0.944 291 84;
  • 9) 0.944 291 84 × 2 = 1 + 0.888 583 68;
  • 10) 0.888 583 68 × 2 = 1 + 0.777 167 36;
  • 11) 0.777 167 36 × 2 = 1 + 0.554 334 72;
  • 12) 0.554 334 72 × 2 = 1 + 0.108 669 44;
  • 13) 0.108 669 44 × 2 = 0 + 0.217 338 88;
  • 14) 0.217 338 88 × 2 = 0 + 0.434 677 76;
  • 15) 0.434 677 76 × 2 = 0 + 0.869 355 52;
  • 16) 0.869 355 52 × 2 = 1 + 0.738 711 04;
  • 17) 0.738 711 04 × 2 = 1 + 0.477 422 08;
  • 18) 0.477 422 08 × 2 = 0 + 0.954 844 16;
  • 19) 0.954 844 16 × 2 = 1 + 0.909 688 32;
  • 20) 0.909 688 32 × 2 = 1 + 0.819 376 64;
  • 21) 0.819 376 64 × 2 = 1 + 0.638 753 28;
  • 22) 0.638 753 28 × 2 = 1 + 0.277 506 56;
  • 23) 0.277 506 56 × 2 = 0 + 0.555 013 12;
  • 24) 0.555 013 12 × 2 = 1 + 0.110 026 24;
  • 25) 0.110 026 24 × 2 = 0 + 0.220 052 48;
  • 26) 0.220 052 48 × 2 = 0 + 0.440 104 96;
  • 27) 0.440 104 96 × 2 = 0 + 0.880 209 92;
  • 28) 0.880 209 92 × 2 = 1 + 0.760 419 84;
  • 29) 0.760 419 84 × 2 = 1 + 0.520 839 68;
  • 30) 0.520 839 68 × 2 = 1 + 0.041 679 36;
  • 31) 0.041 679 36 × 2 = 0 + 0.083 358 72;
  • 32) 0.083 358 72 × 2 = 0 + 0.166 717 44;
  • 33) 0.166 717 44 × 2 = 0 + 0.333 434 88;
  • 34) 0.333 434 88 × 2 = 0 + 0.666 869 76;
  • 35) 0.666 869 76 × 2 = 1 + 0.333 739 52;
  • 36) 0.333 739 52 × 2 = 0 + 0.667 479 04;
  • 37) 0.667 479 04 × 2 = 1 + 0.334 958 08;
  • 38) 0.334 958 08 × 2 = 0 + 0.669 916 16;
  • 39) 0.669 916 16 × 2 = 1 + 0.339 832 32;
  • 40) 0.339 832 32 × 2 = 0 + 0.679 664 64;
  • 41) 0.679 664 64 × 2 = 1 + 0.359 329 28;
  • 42) 0.359 329 28 × 2 = 0 + 0.718 658 56;
  • 43) 0.718 658 56 × 2 = 1 + 0.437 317 12;
  • 44) 0.437 317 12 × 2 = 0 + 0.874 634 24;
  • 45) 0.874 634 24 × 2 = 1 + 0.749 268 48;
  • 46) 0.749 268 48 × 2 = 1 + 0.498 536 96;
  • 47) 0.498 536 96 × 2 = 0 + 0.997 073 92;
  • 48) 0.997 073 92 × 2 = 1 + 0.994 147 84;
  • 49) 0.994 147 84 × 2 = 1 + 0.988 295 68;
  • 50) 0.988 295 68 × 2 = 1 + 0.976 591 36;
  • 51) 0.976 591 36 × 2 = 1 + 0.953 182 72;
  • 52) 0.953 182 72 × 2 = 1 + 0.906 365 44;
  • 53) 0.906 365 44 × 2 = 1 + 0.812 730 88;
  • 54) 0.812 730 88 × 2 = 1 + 0.625 461 76;
  • 55) 0.625 461 76 × 2 = 1 + 0.250 923 52;
  • 56) 0.250 923 52 × 2 = 0 + 0.501 847 04;
  • 57) 0.501 847 04 × 2 = 1 + 0.003 694 08;
  • 58) 0.003 694 08 × 2 = 0 + 0.007 388 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:


0.023 219 89(10) =

0.0000 0101 1111 0001 1011 1101 0001 1100 0010 1010 1010 1101 1111 1110 10(2)

6. Positive number before normalization:

0.023 219 89(10) =

0.0000 0101 1111 0001 1011 1101 0001 1100 0010 1010 1010 1101 1111 1110 10(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 6 positions to the right, so that only one non zero digit remains to the left of it:


0.023 219 89(10) =

0.0000 0101 1111 0001 1011 1101 0001 1100 0010 1010 1010 1101 1111 1110 10(2) =

0.0000 0101 1111 0001 1011 1101 0001 1100 0010 1010 1010 1101 1111 1110 10(2) × 20 =

1.0111 1100 0110 1111 0100 0111 0000 1010 1010 1011 0111 1111 1010(2) × 2-6


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): -6

Mantissa (not normalized):
1.0111 1100 0110 1111 0100 0111 0000 1010 1010 1011 0111 1111 1010


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-6 + 2(11-1) - 1 =

(-6 + 1 023)(10) =

1 017(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;
  • 1 017 ÷ 2 = 508 + 1;
  • 508 ÷ 2 = 254 + 0;
  • 254 ÷ 2 = 127 + 0;
  • 127 ÷ 2 = 63 + 1;
  • 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) =

1017(10) =

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


Mantissa (normalized) =

1. 0111 1100 0110 1111 0100 0111 0000 1010 1010 1011 0111 1111 1010 =

0111 1100 0110 1111 0100 0111 0000 1010 1010 1011 0111 1111 1010


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 1001

Mantissa (52 bits) =
0111 1100 0110 1111 0100 0111 0000 1010 1010 1011 0111 1111 1010


Decimal number -0.023 219 89 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 1001 - 0111 1100 0110 1111 0100 0111 0000 1010 1010 1011 0111 1111 1010

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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