-0.057 036 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.057 036 9(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
-0.057 036 9(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:

|-0.057 036 9| = 0.057 036 9


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: 0.057 036 9.

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) 0.057 036 9 × 2 = 0 + 0.114 073 8;
  • 2) 0.114 073 8 × 2 = 0 + 0.228 147 6;
  • 3) 0.228 147 6 × 2 = 0 + 0.456 295 2;
  • 4) 0.456 295 2 × 2 = 0 + 0.912 590 4;
  • 5) 0.912 590 4 × 2 = 1 + 0.825 180 8;
  • 6) 0.825 180 8 × 2 = 1 + 0.650 361 6;
  • 7) 0.650 361 6 × 2 = 1 + 0.300 723 2;
  • 8) 0.300 723 2 × 2 = 0 + 0.601 446 4;
  • 9) 0.601 446 4 × 2 = 1 + 0.202 892 8;
  • 10) 0.202 892 8 × 2 = 0 + 0.405 785 6;
  • 11) 0.405 785 6 × 2 = 0 + 0.811 571 2;
  • 12) 0.811 571 2 × 2 = 1 + 0.623 142 4;
  • 13) 0.623 142 4 × 2 = 1 + 0.246 284 8;
  • 14) 0.246 284 8 × 2 = 0 + 0.492 569 6;
  • 15) 0.492 569 6 × 2 = 0 + 0.985 139 2;
  • 16) 0.985 139 2 × 2 = 1 + 0.970 278 4;
  • 17) 0.970 278 4 × 2 = 1 + 0.940 556 8;
  • 18) 0.940 556 8 × 2 = 1 + 0.881 113 6;
  • 19) 0.881 113 6 × 2 = 1 + 0.762 227 2;
  • 20) 0.762 227 2 × 2 = 1 + 0.524 454 4;
  • 21) 0.524 454 4 × 2 = 1 + 0.048 908 8;
  • 22) 0.048 908 8 × 2 = 0 + 0.097 817 6;
  • 23) 0.097 817 6 × 2 = 0 + 0.195 635 2;
  • 24) 0.195 635 2 × 2 = 0 + 0.391 270 4;
  • 25) 0.391 270 4 × 2 = 0 + 0.782 540 8;
  • 26) 0.782 540 8 × 2 = 1 + 0.565 081 6;
  • 27) 0.565 081 6 × 2 = 1 + 0.130 163 2;
  • 28) 0.130 163 2 × 2 = 0 + 0.260 326 4;
  • 29) 0.260 326 4 × 2 = 0 + 0.520 652 8;
  • 30) 0.520 652 8 × 2 = 1 + 0.041 305 6;
  • 31) 0.041 305 6 × 2 = 0 + 0.082 611 2;
  • 32) 0.082 611 2 × 2 = 0 + 0.165 222 4;
  • 33) 0.165 222 4 × 2 = 0 + 0.330 444 8;
  • 34) 0.330 444 8 × 2 = 0 + 0.660 889 6;
  • 35) 0.660 889 6 × 2 = 1 + 0.321 779 2;
  • 36) 0.321 779 2 × 2 = 0 + 0.643 558 4;
  • 37) 0.643 558 4 × 2 = 1 + 0.287 116 8;
  • 38) 0.287 116 8 × 2 = 0 + 0.574 233 6;
  • 39) 0.574 233 6 × 2 = 1 + 0.148 467 2;
  • 40) 0.148 467 2 × 2 = 0 + 0.296 934 4;
  • 41) 0.296 934 4 × 2 = 0 + 0.593 868 8;
  • 42) 0.593 868 8 × 2 = 1 + 0.187 737 6;
  • 43) 0.187 737 6 × 2 = 0 + 0.375 475 2;
  • 44) 0.375 475 2 × 2 = 0 + 0.750 950 4;
  • 45) 0.750 950 4 × 2 = 1 + 0.501 900 8;
  • 46) 0.501 900 8 × 2 = 1 + 0.003 801 6;
  • 47) 0.003 801 6 × 2 = 0 + 0.007 603 2;
  • 48) 0.007 603 2 × 2 = 0 + 0.015 206 4;
  • 49) 0.015 206 4 × 2 = 0 + 0.030 412 8;
  • 50) 0.030 412 8 × 2 = 0 + 0.060 825 6;
  • 51) 0.060 825 6 × 2 = 0 + 0.121 651 2;
  • 52) 0.121 651 2 × 2 = 0 + 0.243 302 4;
  • 53) 0.243 302 4 × 2 = 0 + 0.486 604 8;
  • 54) 0.486 604 8 × 2 = 0 + 0.973 209 6;
  • 55) 0.973 209 6 × 2 = 1 + 0.946 419 2;
  • 56) 0.946 419 2 × 2 = 1 + 0.892 838 4;
  • 57) 0.892 838 4 × 2 = 1 + 0.785 676 8;

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:


0.057 036 9(10) =

0.0000 1110 1001 1001 1111 1000 0110 0100 0010 1010 0100 1100 0000 0011 1(2)

6. Positive number before normalization:

0.057 036 9(10) =

0.0000 1110 1001 1001 1111 1000 0110 0100 0010 1010 0100 1100 0000 0011 1(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 5 positions to the right, so that only one non zero digit remains to the left of it:


0.057 036 9(10) =

0.0000 1110 1001 1001 1111 1000 0110 0100 0010 1010 0100 1100 0000 0011 1(2) =

0.0000 1110 1001 1001 1111 1000 0110 0100 0010 1010 0100 1100 0000 0011 1(2) × 20 =

1.1101 0011 0011 1111 0000 1100 1000 0101 0100 1001 1000 0000 0111(2) × 2-5


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): -5

Mantissa (not normalized):
1.1101 0011 0011 1111 0000 1100 1000 0101 0100 1001 1000 0000 0111


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-5 + 2(11-1) - 1 =

(-5 + 1 023)(10) =

1 018(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;
  • 1 018 ÷ 2 = 509 + 0;
  • 509 ÷ 2 = 254 + 1;
  • 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) =

1018(10) =

011 1111 1010(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. 1101 0011 0011 1111 0000 1100 1000 0101 0100 1001 1000 0000 0111 =

1101 0011 0011 1111 0000 1100 1000 0101 0100 1001 1000 0000 0111


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 1010

Mantissa (52 bits) =
1101 0011 0011 1111 0000 1100 1000 0101 0100 1001 1000 0000 0111


Decimal number -0.057 036 9 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 1010 - 1101 0011 0011 1111 0000 1100 1000 0101 0100 1001 1000 0000 0111

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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