-38.123 412 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -38.123 412 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
-38.123 412 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:

|-38.123 412 9| = 38.123 412 9


2. First, convert to binary (in base 2) the integer part: 38.
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;
  • 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.

38(10) =

10 0110(2)


4. Convert to binary (base 2) the fractional part: 0.123 412 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) 0.123 412 9 × 2 = 0 + 0.246 825 8;
  • 2) 0.246 825 8 × 2 = 0 + 0.493 651 6;
  • 3) 0.493 651 6 × 2 = 0 + 0.987 303 2;
  • 4) 0.987 303 2 × 2 = 1 + 0.974 606 4;
  • 5) 0.974 606 4 × 2 = 1 + 0.949 212 8;
  • 6) 0.949 212 8 × 2 = 1 + 0.898 425 6;
  • 7) 0.898 425 6 × 2 = 1 + 0.796 851 2;
  • 8) 0.796 851 2 × 2 = 1 + 0.593 702 4;
  • 9) 0.593 702 4 × 2 = 1 + 0.187 404 8;
  • 10) 0.187 404 8 × 2 = 0 + 0.374 809 6;
  • 11) 0.374 809 6 × 2 = 0 + 0.749 619 2;
  • 12) 0.749 619 2 × 2 = 1 + 0.499 238 4;
  • 13) 0.499 238 4 × 2 = 0 + 0.998 476 8;
  • 14) 0.998 476 8 × 2 = 1 + 0.996 953 6;
  • 15) 0.996 953 6 × 2 = 1 + 0.993 907 2;
  • 16) 0.993 907 2 × 2 = 1 + 0.987 814 4;
  • 17) 0.987 814 4 × 2 = 1 + 0.975 628 8;
  • 18) 0.975 628 8 × 2 = 1 + 0.951 257 6;
  • 19) 0.951 257 6 × 2 = 1 + 0.902 515 2;
  • 20) 0.902 515 2 × 2 = 1 + 0.805 030 4;
  • 21) 0.805 030 4 × 2 = 1 + 0.610 060 8;
  • 22) 0.610 060 8 × 2 = 1 + 0.220 121 6;
  • 23) 0.220 121 6 × 2 = 0 + 0.440 243 2;
  • 24) 0.440 243 2 × 2 = 0 + 0.880 486 4;
  • 25) 0.880 486 4 × 2 = 1 + 0.760 972 8;
  • 26) 0.760 972 8 × 2 = 1 + 0.521 945 6;
  • 27) 0.521 945 6 × 2 = 1 + 0.043 891 2;
  • 28) 0.043 891 2 × 2 = 0 + 0.087 782 4;
  • 29) 0.087 782 4 × 2 = 0 + 0.175 564 8;
  • 30) 0.175 564 8 × 2 = 0 + 0.351 129 6;
  • 31) 0.351 129 6 × 2 = 0 + 0.702 259 2;
  • 32) 0.702 259 2 × 2 = 1 + 0.404 518 4;
  • 33) 0.404 518 4 × 2 = 0 + 0.809 036 8;
  • 34) 0.809 036 8 × 2 = 1 + 0.618 073 6;
  • 35) 0.618 073 6 × 2 = 1 + 0.236 147 2;
  • 36) 0.236 147 2 × 2 = 0 + 0.472 294 4;
  • 37) 0.472 294 4 × 2 = 0 + 0.944 588 8;
  • 38) 0.944 588 8 × 2 = 1 + 0.889 177 6;
  • 39) 0.889 177 6 × 2 = 1 + 0.778 355 2;
  • 40) 0.778 355 2 × 2 = 1 + 0.556 710 4;
  • 41) 0.556 710 4 × 2 = 1 + 0.113 420 8;
  • 42) 0.113 420 8 × 2 = 0 + 0.226 841 6;
  • 43) 0.226 841 6 × 2 = 0 + 0.453 683 2;
  • 44) 0.453 683 2 × 2 = 0 + 0.907 366 4;
  • 45) 0.907 366 4 × 2 = 1 + 0.814 732 8;
  • 46) 0.814 732 8 × 2 = 1 + 0.629 465 6;
  • 47) 0.629 465 6 × 2 = 1 + 0.258 931 2;
  • 48) 0.258 931 2 × 2 = 0 + 0.517 862 4;
  • 49) 0.517 862 4 × 2 = 1 + 0.035 724 8;
  • 50) 0.035 724 8 × 2 = 0 + 0.071 449 6;
  • 51) 0.071 449 6 × 2 = 0 + 0.142 899 2;
  • 52) 0.142 899 2 × 2 = 0 + 0.285 798 4;
  • 53) 0.285 798 4 × 2 = 0 + 0.571 596 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:


0.123 412 9(10) =

0.0001 1111 1001 0111 1111 1100 1110 0001 0110 0111 1000 1110 1000 0(2)

6. Positive number before normalization:

38.123 412 9(10) =

10 0110.0001 1111 1001 0111 1111 1100 1110 0001 0110 0111 1000 1110 1000 0(2)

7. Normalize the binary representation of the number.

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


38.123 412 9(10) =

10 0110.0001 1111 1001 0111 1111 1100 1110 0001 0110 0111 1000 1110 1000 0(2) =

10 0110.0001 1111 1001 0111 1111 1100 1110 0001 0110 0111 1000 1110 1000 0(2) × 20 =

1.0011 0000 1111 1100 1011 1111 1110 0111 0000 1011 0011 1100 0111 0100 00(2) × 25


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):
1.0011 0000 1111 1100 1011 1111 1110 0111 0000 1011 0011 1100 0111 0100 00


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) =

1 028(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;
  • 1 028 ÷ 2 = 514 + 0;
  • 514 ÷ 2 = 257 + 0;
  • 257 ÷ 2 = 128 + 1;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 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) =

1028(10) =

100 0000 0100(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 0000 1111 1100 1011 1111 1110 0111 0000 1011 0011 1100 0111 01 0000 =

0011 0000 1111 1100 1011 1111 1110 0111 0000 1011 0011 1100 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) =
100 0000 0100

Mantissa (52 bits) =
0011 0000 1111 1100 1011 1111 1110 0111 0000 1011 0011 1100 0111


Decimal number -38.123 412 9 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 100 0000 0100 - 0011 0000 1111 1100 1011 1111 1110 0111 0000 1011 0011 1100 0111

» Convert decimal -38.123 413 5 to 64 bit double precision IEEE 754 binary floating point representation standard

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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