64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: -1 271 304.176 73 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number -1 271 304.176 73(10) converted and written in 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:

|-1 271 304.176 73| = 1 271 304.176 73

2. First, convert to binary (in base 2) the integer part: 1 271 304.
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;
  • 1 271 304 ÷ 2 = 635 652 + 0;
  • 635 652 ÷ 2 = 317 826 + 0;
  • 317 826 ÷ 2 = 158 913 + 0;
  • 158 913 ÷ 2 = 79 456 + 1;
  • 79 456 ÷ 2 = 39 728 + 0;
  • 39 728 ÷ 2 = 19 864 + 0;
  • 19 864 ÷ 2 = 9 932 + 0;
  • 9 932 ÷ 2 = 4 966 + 0;
  • 4 966 ÷ 2 = 2 483 + 0;
  • 2 483 ÷ 2 = 1 241 + 1;
  • 1 241 ÷ 2 = 620 + 1;
  • 620 ÷ 2 = 310 + 0;
  • 310 ÷ 2 = 155 + 0;
  • 155 ÷ 2 = 77 + 1;
  • 77 ÷ 2 = 38 + 1;
  • 38 ÷ 2 = 19 + 0;
  • 19 ÷ 2 = 9 + 1;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

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.


1 271 304(10) =


1 0011 0110 0110 0000 1000(2)


4. Convert to binary (base 2) the fractional part: 0.176 73.

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) 0.176 73 × 2 = 0 + 0.353 46;
  • 2) 0.353 46 × 2 = 0 + 0.706 92;
  • 3) 0.706 92 × 2 = 1 + 0.413 84;
  • 4) 0.413 84 × 2 = 0 + 0.827 68;
  • 5) 0.827 68 × 2 = 1 + 0.655 36;
  • 6) 0.655 36 × 2 = 1 + 0.310 72;
  • 7) 0.310 72 × 2 = 0 + 0.621 44;
  • 8) 0.621 44 × 2 = 1 + 0.242 88;
  • 9) 0.242 88 × 2 = 0 + 0.485 76;
  • 10) 0.485 76 × 2 = 0 + 0.971 52;
  • 11) 0.971 52 × 2 = 1 + 0.943 04;
  • 12) 0.943 04 × 2 = 1 + 0.886 08;
  • 13) 0.886 08 × 2 = 1 + 0.772 16;
  • 14) 0.772 16 × 2 = 1 + 0.544 32;
  • 15) 0.544 32 × 2 = 1 + 0.088 64;
  • 16) 0.088 64 × 2 = 0 + 0.177 28;
  • 17) 0.177 28 × 2 = 0 + 0.354 56;
  • 18) 0.354 56 × 2 = 0 + 0.709 12;
  • 19) 0.709 12 × 2 = 1 + 0.418 24;
  • 20) 0.418 24 × 2 = 0 + 0.836 48;
  • 21) 0.836 48 × 2 = 1 + 0.672 96;
  • 22) 0.672 96 × 2 = 1 + 0.345 92;
  • 23) 0.345 92 × 2 = 0 + 0.691 84;
  • 24) 0.691 84 × 2 = 1 + 0.383 68;
  • 25) 0.383 68 × 2 = 0 + 0.767 36;
  • 26) 0.767 36 × 2 = 1 + 0.534 72;
  • 27) 0.534 72 × 2 = 1 + 0.069 44;
  • 28) 0.069 44 × 2 = 0 + 0.138 88;
  • 29) 0.138 88 × 2 = 0 + 0.277 76;
  • 30) 0.277 76 × 2 = 0 + 0.555 52;
  • 31) 0.555 52 × 2 = 1 + 0.111 04;
  • 32) 0.111 04 × 2 = 0 + 0.222 08;
  • 33) 0.222 08 × 2 = 0 + 0.444 16;
  • 34) 0.444 16 × 2 = 0 + 0.888 32;
  • 35) 0.888 32 × 2 = 1 + 0.776 64;
  • 36) 0.776 64 × 2 = 1 + 0.553 28;
  • 37) 0.553 28 × 2 = 1 + 0.106 56;
  • 38) 0.106 56 × 2 = 0 + 0.213 12;
  • 39) 0.213 12 × 2 = 0 + 0.426 24;
  • 40) 0.426 24 × 2 = 0 + 0.852 48;
  • 41) 0.852 48 × 2 = 1 + 0.704 96;
  • 42) 0.704 96 × 2 = 1 + 0.409 92;
  • 43) 0.409 92 × 2 = 0 + 0.819 84;
  • 44) 0.819 84 × 2 = 1 + 0.639 68;
  • 45) 0.639 68 × 2 = 1 + 0.279 36;
  • 46) 0.279 36 × 2 = 0 + 0.558 72;
  • 47) 0.558 72 × 2 = 1 + 0.117 44;
  • 48) 0.117 44 × 2 = 0 + 0.234 88;
  • 49) 0.234 88 × 2 = 0 + 0.469 76;
  • 50) 0.469 76 × 2 = 0 + 0.939 52;
  • 51) 0.939 52 × 2 = 1 + 0.879 04;
  • 52) 0.879 04 × 2 = 1 + 0.758 08;
  • 53) 0.758 08 × 2 = 1 + 0.516 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...)


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:


0.176 73(10) =


0.0010 1101 0011 1110 0010 1101 0110 0010 0011 1000 1101 1010 0011 1(2)


6. Positive number before normalization:

1 271 304.176 73(10) =


1 0011 0110 0110 0000 1000.0010 1101 0011 1110 0010 1101 0110 0010 0011 1000 1101 1010 0011 1(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 20 positions to the left, so that only one non zero digit remains to the left of it:


1 271 304.176 73(10) =


1 0011 0110 0110 0000 1000.0010 1101 0011 1110 0010 1101 0110 0010 0011 1000 1101 1010 0011 1(2) =


1 0011 0110 0110 0000 1000.0010 1101 0011 1110 0010 1101 0110 0010 0011 1000 1101 1010 0011 1(2) × 20 =


1.0011 0110 0110 0000 1000 0010 1101 0011 1110 0010 1101 0110 0010 0011 1000 1101 1010 0011 1(2) × 220


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): 20


Mantissa (not normalized):
1.0011 0110 0110 0000 1000 0010 1101 0011 1110 0010 1101 0110 0010 0011 1000 1101 1010 0011 1


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


20 + 2(11-1) - 1 =


(20 + 1 023)(10) =


1 043(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;
  • 1 043 ÷ 2 = 521 + 1;
  • 521 ÷ 2 = 260 + 1;
  • 260 ÷ 2 = 130 + 0;
  • 130 ÷ 2 = 65 + 0;
  • 65 ÷ 2 = 32 + 1;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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) =


1043(10) =


100 0001 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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =


1. 0011 0110 0110 0000 1000 0010 1101 0011 1110 0010 1101 0110 0010 0 0111 0001 1011 0100 0111 =


0011 0110 0110 0000 1000 0010 1101 0011 1110 0010 1101 0110 0010


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) =
100 0001 0011


Mantissa (52 bits) =
0011 0110 0110 0000 1000 0010 1101 0011 1110 0010 1101 0110 0010


The base ten decimal number -1 271 304.176 73 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
1 - 100 0001 0011 - 0011 0110 0110 0000 1000 0010 1101 0011 1110 0010 1101 0110 0010

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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