-0.057 002 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.057 002 8(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
-0.057 002 8(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:

|-0.057 002 8| = 0.057 002 8


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: 0.057 002 8.

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) 0.057 002 8 × 2 = 0 + 0.114 005 6;
  • 2) 0.114 005 6 × 2 = 0 + 0.228 011 2;
  • 3) 0.228 011 2 × 2 = 0 + 0.456 022 4;
  • 4) 0.456 022 4 × 2 = 0 + 0.912 044 8;
  • 5) 0.912 044 8 × 2 = 1 + 0.824 089 6;
  • 6) 0.824 089 6 × 2 = 1 + 0.648 179 2;
  • 7) 0.648 179 2 × 2 = 1 + 0.296 358 4;
  • 8) 0.296 358 4 × 2 = 0 + 0.592 716 8;
  • 9) 0.592 716 8 × 2 = 1 + 0.185 433 6;
  • 10) 0.185 433 6 × 2 = 0 + 0.370 867 2;
  • 11) 0.370 867 2 × 2 = 0 + 0.741 734 4;
  • 12) 0.741 734 4 × 2 = 1 + 0.483 468 8;
  • 13) 0.483 468 8 × 2 = 0 + 0.966 937 6;
  • 14) 0.966 937 6 × 2 = 1 + 0.933 875 2;
  • 15) 0.933 875 2 × 2 = 1 + 0.867 750 4;
  • 16) 0.867 750 4 × 2 = 1 + 0.735 500 8;
  • 17) 0.735 500 8 × 2 = 1 + 0.471 001 6;
  • 18) 0.471 001 6 × 2 = 0 + 0.942 003 2;
  • 19) 0.942 003 2 × 2 = 1 + 0.884 006 4;
  • 20) 0.884 006 4 × 2 = 1 + 0.768 012 8;
  • 21) 0.768 012 8 × 2 = 1 + 0.536 025 6;
  • 22) 0.536 025 6 × 2 = 1 + 0.072 051 2;
  • 23) 0.072 051 2 × 2 = 0 + 0.144 102 4;
  • 24) 0.144 102 4 × 2 = 0 + 0.288 204 8;
  • 25) 0.288 204 8 × 2 = 0 + 0.576 409 6;
  • 26) 0.576 409 6 × 2 = 1 + 0.152 819 2;
  • 27) 0.152 819 2 × 2 = 0 + 0.305 638 4;
  • 28) 0.305 638 4 × 2 = 0 + 0.611 276 8;
  • 29) 0.611 276 8 × 2 = 1 + 0.222 553 6;
  • 30) 0.222 553 6 × 2 = 0 + 0.445 107 2;
  • 31) 0.445 107 2 × 2 = 0 + 0.890 214 4;
  • 32) 0.890 214 4 × 2 = 1 + 0.780 428 8;
  • 33) 0.780 428 8 × 2 = 1 + 0.560 857 6;
  • 34) 0.560 857 6 × 2 = 1 + 0.121 715 2;
  • 35) 0.121 715 2 × 2 = 0 + 0.243 430 4;
  • 36) 0.243 430 4 × 2 = 0 + 0.486 860 8;
  • 37) 0.486 860 8 × 2 = 0 + 0.973 721 6;
  • 38) 0.973 721 6 × 2 = 1 + 0.947 443 2;
  • 39) 0.947 443 2 × 2 = 1 + 0.894 886 4;
  • 40) 0.894 886 4 × 2 = 1 + 0.789 772 8;
  • 41) 0.789 772 8 × 2 = 1 + 0.579 545 6;
  • 42) 0.579 545 6 × 2 = 1 + 0.159 091 2;
  • 43) 0.159 091 2 × 2 = 0 + 0.318 182 4;
  • 44) 0.318 182 4 × 2 = 0 + 0.636 364 8;
  • 45) 0.636 364 8 × 2 = 1 + 0.272 729 6;
  • 46) 0.272 729 6 × 2 = 0 + 0.545 459 2;
  • 47) 0.545 459 2 × 2 = 1 + 0.090 918 4;
  • 48) 0.090 918 4 × 2 = 0 + 0.181 836 8;
  • 49) 0.181 836 8 × 2 = 0 + 0.363 673 6;
  • 50) 0.363 673 6 × 2 = 0 + 0.727 347 2;
  • 51) 0.727 347 2 × 2 = 1 + 0.454 694 4;
  • 52) 0.454 694 4 × 2 = 0 + 0.909 388 8;
  • 53) 0.909 388 8 × 2 = 1 + 0.818 777 6;
  • 54) 0.818 777 6 × 2 = 1 + 0.637 555 2;
  • 55) 0.637 555 2 × 2 = 1 + 0.275 110 4;
  • 56) 0.275 110 4 × 2 = 0 + 0.550 220 8;
  • 57) 0.550 220 8 × 2 = 1 + 0.100 441 6;

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:


0.057 002 8(10) =

0.0000 1110 1001 0111 1011 1100 0100 1001 1100 0111 1100 1010 0010 1110 1(2)

6. Positive number before normalization:

0.057 002 8(10) =

0.0000 1110 1001 0111 1011 1100 0100 1001 1100 0111 1100 1010 0010 1110 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:


0.057 002 8(10) =

0.0000 1110 1001 0111 1011 1100 0100 1001 1100 0111 1100 1010 0010 1110 1(2) =

0.0000 1110 1001 0111 1011 1100 0100 1001 1100 0111 1100 1010 0010 1110 1(2) × 20 =

1.1101 0010 1111 0111 1000 1001 0011 1000 1111 1001 0100 0101 1101(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 0010 1111 0111 1000 1001 0011 1000 1111 1001 0100 0101 1101


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 0010 1111 0111 1000 1001 0011 1000 1111 1001 0100 0101 1101 =

1101 0010 1111 0111 1000 1001 0011 1000 1111 1001 0100 0101 1101


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 0010 1111 0111 1000 1001 0011 1000 1111 1001 0100 0101 1101


Decimal number -0.057 002 8 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 1010 - 1101 0010 1111 0111 1000 1001 0011 1000 1111 1001 0100 0101 1101

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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