0.022 137 89 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.022 137 89(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
0.022 137 89(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: 0.022 137 89.

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) 0.022 137 89 × 2 = 0 + 0.044 275 78;
  • 2) 0.044 275 78 × 2 = 0 + 0.088 551 56;
  • 3) 0.088 551 56 × 2 = 0 + 0.177 103 12;
  • 4) 0.177 103 12 × 2 = 0 + 0.354 206 24;
  • 5) 0.354 206 24 × 2 = 0 + 0.708 412 48;
  • 6) 0.708 412 48 × 2 = 1 + 0.416 824 96;
  • 7) 0.416 824 96 × 2 = 0 + 0.833 649 92;
  • 8) 0.833 649 92 × 2 = 1 + 0.667 299 84;
  • 9) 0.667 299 84 × 2 = 1 + 0.334 599 68;
  • 10) 0.334 599 68 × 2 = 0 + 0.669 199 36;
  • 11) 0.669 199 36 × 2 = 1 + 0.338 398 72;
  • 12) 0.338 398 72 × 2 = 0 + 0.676 797 44;
  • 13) 0.676 797 44 × 2 = 1 + 0.353 594 88;
  • 14) 0.353 594 88 × 2 = 0 + 0.707 189 76;
  • 15) 0.707 189 76 × 2 = 1 + 0.414 379 52;
  • 16) 0.414 379 52 × 2 = 0 + 0.828 759 04;
  • 17) 0.828 759 04 × 2 = 1 + 0.657 518 08;
  • 18) 0.657 518 08 × 2 = 1 + 0.315 036 16;
  • 19) 0.315 036 16 × 2 = 0 + 0.630 072 32;
  • 20) 0.630 072 32 × 2 = 1 + 0.260 144 64;
  • 21) 0.260 144 64 × 2 = 0 + 0.520 289 28;
  • 22) 0.520 289 28 × 2 = 1 + 0.040 578 56;
  • 23) 0.040 578 56 × 2 = 0 + 0.081 157 12;
  • 24) 0.081 157 12 × 2 = 0 + 0.162 314 24;
  • 25) 0.162 314 24 × 2 = 0 + 0.324 628 48;
  • 26) 0.324 628 48 × 2 = 0 + 0.649 256 96;
  • 27) 0.649 256 96 × 2 = 1 + 0.298 513 92;
  • 28) 0.298 513 92 × 2 = 0 + 0.597 027 84;
  • 29) 0.597 027 84 × 2 = 1 + 0.194 055 68;
  • 30) 0.194 055 68 × 2 = 0 + 0.388 111 36;
  • 31) 0.388 111 36 × 2 = 0 + 0.776 222 72;
  • 32) 0.776 222 72 × 2 = 1 + 0.552 445 44;
  • 33) 0.552 445 44 × 2 = 1 + 0.104 890 88;
  • 34) 0.104 890 88 × 2 = 0 + 0.209 781 76;
  • 35) 0.209 781 76 × 2 = 0 + 0.419 563 52;
  • 36) 0.419 563 52 × 2 = 0 + 0.839 127 04;
  • 37) 0.839 127 04 × 2 = 1 + 0.678 254 08;
  • 38) 0.678 254 08 × 2 = 1 + 0.356 508 16;
  • 39) 0.356 508 16 × 2 = 0 + 0.713 016 32;
  • 40) 0.713 016 32 × 2 = 1 + 0.426 032 64;
  • 41) 0.426 032 64 × 2 = 0 + 0.852 065 28;
  • 42) 0.852 065 28 × 2 = 1 + 0.704 130 56;
  • 43) 0.704 130 56 × 2 = 1 + 0.408 261 12;
  • 44) 0.408 261 12 × 2 = 0 + 0.816 522 24;
  • 45) 0.816 522 24 × 2 = 1 + 0.633 044 48;
  • 46) 0.633 044 48 × 2 = 1 + 0.266 088 96;
  • 47) 0.266 088 96 × 2 = 0 + 0.532 177 92;
  • 48) 0.532 177 92 × 2 = 1 + 0.064 355 84;
  • 49) 0.064 355 84 × 2 = 0 + 0.128 711 68;
  • 50) 0.128 711 68 × 2 = 0 + 0.257 423 36;
  • 51) 0.257 423 36 × 2 = 0 + 0.514 846 72;
  • 52) 0.514 846 72 × 2 = 1 + 0.029 693 44;
  • 53) 0.029 693 44 × 2 = 0 + 0.059 386 88;
  • 54) 0.059 386 88 × 2 = 0 + 0.118 773 76;
  • 55) 0.118 773 76 × 2 = 0 + 0.237 547 52;
  • 56) 0.237 547 52 × 2 = 0 + 0.475 095 04;
  • 57) 0.475 095 04 × 2 = 0 + 0.950 190 08;
  • 58) 0.950 190 08 × 2 = 1 + 0.900 380 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 - 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:


0.022 137 89(10) =

0.0000 0101 1010 1010 1101 0100 0010 1001 1000 1101 0110 1101 0001 0000 01(2)

5. Positive number before normalization:

0.022 137 89(10) =

0.0000 0101 1010 1010 1101 0100 0010 1001 1000 1101 0110 1101 0001 0000 01(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 6 positions to the right, so that only one non zero digit remains to the left of it:


0.022 137 89(10) =

0.0000 0101 1010 1010 1101 0100 0010 1001 1000 1101 0110 1101 0001 0000 01(2) =

0.0000 0101 1010 1010 1101 0100 0010 1001 1000 1101 0110 1101 0001 0000 01(2) × 20 =

1.0110 1010 1011 0101 0000 1010 0110 0011 0101 1011 0100 0100 0001(2) × 2-6


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): -6

Mantissa (not normalized):
1.0110 1010 1011 0101 0000 1010 0110 0011 0101 1011 0100 0100 0001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-6 + 2(11-1) - 1 =

(-6 + 1 023)(10) =

1 017(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;
  • 1 017 ÷ 2 = 508 + 1;
  • 508 ÷ 2 = 254 + 0;
  • 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;

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) =

1017(10) =

011 1111 1001(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. 0110 1010 1011 0101 0000 1010 0110 0011 0101 1011 0100 0100 0001 =

0110 1010 1011 0101 0000 1010 0110 0011 0101 1011 0100 0100 0001


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 1001

Mantissa (52 bits) =
0110 1010 1011 0101 0000 1010 0110 0011 0101 1011 0100 0100 0001


Decimal number 0.022 137 89 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1001 - 0110 1010 1011 0101 0000 1010 0110 0011 0101 1011 0100 0100 0001

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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