0.666 625 978 32 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.666 625 978 32(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
0.666 625 978 32(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. 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;

2. 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)


3. Convert to binary (base 2) the fractional part: 0.666 625 978 32.

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) 0.666 625 978 32 × 2 = 1 + 0.333 251 956 64;
  • 2) 0.333 251 956 64 × 2 = 0 + 0.666 503 913 28;
  • 3) 0.666 503 913 28 × 2 = 1 + 0.333 007 826 56;
  • 4) 0.333 007 826 56 × 2 = 0 + 0.666 015 653 12;
  • 5) 0.666 015 653 12 × 2 = 1 + 0.332 031 306 24;
  • 6) 0.332 031 306 24 × 2 = 0 + 0.664 062 612 48;
  • 7) 0.664 062 612 48 × 2 = 1 + 0.328 125 224 96;
  • 8) 0.328 125 224 96 × 2 = 0 + 0.656 250 449 92;
  • 9) 0.656 250 449 92 × 2 = 1 + 0.312 500 899 84;
  • 10) 0.312 500 899 84 × 2 = 0 + 0.625 001 799 68;
  • 11) 0.625 001 799 68 × 2 = 1 + 0.250 003 599 36;
  • 12) 0.250 003 599 36 × 2 = 0 + 0.500 007 198 72;
  • 13) 0.500 007 198 72 × 2 = 1 + 0.000 014 397 44;
  • 14) 0.000 014 397 44 × 2 = 0 + 0.000 028 794 88;
  • 15) 0.000 028 794 88 × 2 = 0 + 0.000 057 589 76;
  • 16) 0.000 057 589 76 × 2 = 0 + 0.000 115 179 52;
  • 17) 0.000 115 179 52 × 2 = 0 + 0.000 230 359 04;
  • 18) 0.000 230 359 04 × 2 = 0 + 0.000 460 718 08;
  • 19) 0.000 460 718 08 × 2 = 0 + 0.000 921 436 16;
  • 20) 0.000 921 436 16 × 2 = 0 + 0.001 842 872 32;
  • 21) 0.001 842 872 32 × 2 = 0 + 0.003 685 744 64;
  • 22) 0.003 685 744 64 × 2 = 0 + 0.007 371 489 28;
  • 23) 0.007 371 489 28 × 2 = 0 + 0.014 742 978 56;
  • 24) 0.014 742 978 56 × 2 = 0 + 0.029 485 957 12;
  • 25) 0.029 485 957 12 × 2 = 0 + 0.058 971 914 24;
  • 26) 0.058 971 914 24 × 2 = 0 + 0.117 943 828 48;
  • 27) 0.117 943 828 48 × 2 = 0 + 0.235 887 656 96;
  • 28) 0.235 887 656 96 × 2 = 0 + 0.471 775 313 92;
  • 29) 0.471 775 313 92 × 2 = 0 + 0.943 550 627 84;
  • 30) 0.943 550 627 84 × 2 = 1 + 0.887 101 255 68;
  • 31) 0.887 101 255 68 × 2 = 1 + 0.774 202 511 36;
  • 32) 0.774 202 511 36 × 2 = 1 + 0.548 405 022 72;
  • 33) 0.548 405 022 72 × 2 = 1 + 0.096 810 045 44;
  • 34) 0.096 810 045 44 × 2 = 0 + 0.193 620 090 88;
  • 35) 0.193 620 090 88 × 2 = 0 + 0.387 240 181 76;
  • 36) 0.387 240 181 76 × 2 = 0 + 0.774 480 363 52;
  • 37) 0.774 480 363 52 × 2 = 1 + 0.548 960 727 04;
  • 38) 0.548 960 727 04 × 2 = 1 + 0.097 921 454 08;
  • 39) 0.097 921 454 08 × 2 = 0 + 0.195 842 908 16;
  • 40) 0.195 842 908 16 × 2 = 0 + 0.391 685 816 32;
  • 41) 0.391 685 816 32 × 2 = 0 + 0.783 371 632 64;
  • 42) 0.783 371 632 64 × 2 = 1 + 0.566 743 265 28;
  • 43) 0.566 743 265 28 × 2 = 1 + 0.133 486 530 56;
  • 44) 0.133 486 530 56 × 2 = 0 + 0.266 973 061 12;
  • 45) 0.266 973 061 12 × 2 = 0 + 0.533 946 122 24;
  • 46) 0.533 946 122 24 × 2 = 1 + 0.067 892 244 48;
  • 47) 0.067 892 244 48 × 2 = 0 + 0.135 784 488 96;
  • 48) 0.135 784 488 96 × 2 = 0 + 0.271 568 977 92;
  • 49) 0.271 568 977 92 × 2 = 0 + 0.543 137 955 84;
  • 50) 0.543 137 955 84 × 2 = 1 + 0.086 275 911 68;
  • 51) 0.086 275 911 68 × 2 = 0 + 0.172 551 823 36;
  • 52) 0.172 551 823 36 × 2 = 0 + 0.345 103 646 72;
  • 53) 0.345 103 646 72 × 2 = 0 + 0.690 207 293 44;

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).


4. 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:


0.666 625 978 32(10) =

0.1010 1010 1010 1000 0000 0000 0000 0111 1000 1100 0110 0100 0100 0(2)

5. Positive number before normalization:

0.666 625 978 32(10) =

0.1010 1010 1010 1000 0000 0000 0000 0111 1000 1100 0110 0100 0100 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 1 positions to the right, so that only one non zero digit remains to the left of it:


0.666 625 978 32(10) =

0.1010 1010 1010 1000 0000 0000 0000 0111 1000 1100 0110 0100 0100 0(2) =

0.1010 1010 1010 1000 0000 0000 0000 0111 1000 1100 0110 0100 0100 0(2) × 20 =

1.0101 0101 0101 0000 0000 0000 0000 1111 0001 1000 1100 1000 1000(2) × 2-1


7. 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 0 (a positive number)

Exponent (unadjusted): -1

Mantissa (not normalized):
1.0101 0101 0101 0000 0000 0000 0000 1111 0001 1000 1100 1000 1000


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

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

(-1 + 1 023)(10) =

1 022(10)


9. 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;
  • 1 022 ÷ 2 = 511 + 0;
  • 511 ÷ 2 = 255 + 1;
  • 255 ÷ 2 = 127 + 1;
  • 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;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

1022(10) =

011 1111 1110(2)


11. 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. 0101 0101 0101 0000 0000 0000 0000 1111 0001 1000 1100 1000 1000 =

0101 0101 0101 0000 0000 0000 0000 1111 0001 1000 1100 1000 1000


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1111 1110

Mantissa (52 bits) =
0101 0101 0101 0000 0000 0000 0000 1111 0001 1000 1100 1000 1000


Decimal number 0.666 625 978 32 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1110 - 0101 0101 0101 0000 0000 0000 0000 1111 0001 1000 1100 1000 1000

Share

» Convert decimal 0.666 625 977 88 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 09, 2026 [September]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: September - by our visitors

» Improve your social skills: The Attention Economy - The Battlefield For Your Focus. Gemini Notebook YouTube Video of 7.50 minutes

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