227.666 666 666 666 666 666 666 666 666 663 1 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 227.666 666 666 666 666 666 666 666 666 663 1(10) to 32 bit single precision IEEE 754 binary floating point representation standard (1 bit for sign, 8 bits for exponent, 23 bits for mantissa)

What are the steps to convert decimal number
227.666 666 666 666 666 666 666 666 666 663 1(10) to 32 bit single precision IEEE 754 binary floating point representation (1 bit for sign, 8 bits for exponent, 23 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 227.
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;
  • 227 ÷ 2 = 113 + 1;
  • 113 ÷ 2 = 56 + 1;
  • 56 ÷ 2 = 28 + 0;
  • 28 ÷ 2 = 14 + 0;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

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.

227(10) =

1110 0011(2)


3. Convert to binary (base 2) the fractional part: 0.666 666 666 666 666 666 666 666 666 663 1.

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) 0.666 666 666 666 666 666 666 666 666 663 1 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 326 2;
  • 2) 0.333 333 333 333 333 333 333 333 333 326 2 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 652 4;
  • 3) 0.666 666 666 666 666 666 666 666 666 652 4 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 304 8;
  • 4) 0.333 333 333 333 333 333 333 333 333 304 8 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 609 6;
  • 5) 0.666 666 666 666 666 666 666 666 666 609 6 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 219 2;
  • 6) 0.333 333 333 333 333 333 333 333 333 219 2 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 438 4;
  • 7) 0.666 666 666 666 666 666 666 666 666 438 4 × 2 = 1 + 0.333 333 333 333 333 333 333 333 332 876 8;
  • 8) 0.333 333 333 333 333 333 333 333 332 876 8 × 2 = 0 + 0.666 666 666 666 666 666 666 666 665 753 6;
  • 9) 0.666 666 666 666 666 666 666 666 665 753 6 × 2 = 1 + 0.333 333 333 333 333 333 333 333 331 507 2;
  • 10) 0.333 333 333 333 333 333 333 333 331 507 2 × 2 = 0 + 0.666 666 666 666 666 666 666 666 663 014 4;
  • 11) 0.666 666 666 666 666 666 666 666 663 014 4 × 2 = 1 + 0.333 333 333 333 333 333 333 333 326 028 8;
  • 12) 0.333 333 333 333 333 333 333 333 326 028 8 × 2 = 0 + 0.666 666 666 666 666 666 666 666 652 057 6;
  • 13) 0.666 666 666 666 666 666 666 666 652 057 6 × 2 = 1 + 0.333 333 333 333 333 333 333 333 304 115 2;
  • 14) 0.333 333 333 333 333 333 333 333 304 115 2 × 2 = 0 + 0.666 666 666 666 666 666 666 666 608 230 4;
  • 15) 0.666 666 666 666 666 666 666 666 608 230 4 × 2 = 1 + 0.333 333 333 333 333 333 333 333 216 460 8;
  • 16) 0.333 333 333 333 333 333 333 333 216 460 8 × 2 = 0 + 0.666 666 666 666 666 666 666 666 432 921 6;
  • 17) 0.666 666 666 666 666 666 666 666 432 921 6 × 2 = 1 + 0.333 333 333 333 333 333 333 332 865 843 2;
  • 18) 0.333 333 333 333 333 333 333 332 865 843 2 × 2 = 0 + 0.666 666 666 666 666 666 666 665 731 686 4;
  • 19) 0.666 666 666 666 666 666 666 665 731 686 4 × 2 = 1 + 0.333 333 333 333 333 333 333 331 463 372 8;
  • 20) 0.333 333 333 333 333 333 333 331 463 372 8 × 2 = 0 + 0.666 666 666 666 666 666 666 662 926 745 6;
  • 21) 0.666 666 666 666 666 666 666 662 926 745 6 × 2 = 1 + 0.333 333 333 333 333 333 333 325 853 491 2;
  • 22) 0.333 333 333 333 333 333 333 325 853 491 2 × 2 = 0 + 0.666 666 666 666 666 666 666 651 706 982 4;
  • 23) 0.666 666 666 666 666 666 666 651 706 982 4 × 2 = 1 + 0.333 333 333 333 333 333 333 303 413 964 8;
  • 24) 0.333 333 333 333 333 333 333 303 413 964 8 × 2 = 0 + 0.666 666 666 666 666 666 666 606 827 929 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).


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:


0.666 666 666 666 666 666 666 666 666 663 1(10) =

0.1010 1010 1010 1010 1010 1010(2)

5. Positive number before normalization:

227.666 666 666 666 666 666 666 666 666 663 1(10) =

1110 0011.1010 1010 1010 1010 1010 1010(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 7 positions to the left, so that only one non zero digit remains to the left of it:


227.666 666 666 666 666 666 666 666 666 663 1(10) =

1110 0011.1010 1010 1010 1010 1010 1010(2) =

1110 0011.1010 1010 1010 1010 1010 1010(2) × 20 =

1.1100 0111 0101 0101 0101 0101 0101 010(2) × 27


7. Up to this moment, there are the following elements that would feed into the 32 bit single precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): 7

Mantissa (not normalized):
1.1100 0111 0101 0101 0101 0101 0101 010


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(8-1) - 1 =

7 + 2(8-1) - 1 =

(7 + 127)(10) =

134(10)


9. Convert the adjusted exponent from the decimal (base 10) to 8 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 134 ÷ 2 = 67 + 0;
  • 67 ÷ 2 = 33 + 1;
  • 33 ÷ 2 = 16 + 1;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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) =

134(10) =

1000 0110(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 23 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. 110 0011 1010 1010 1010 1010 1010 1010 =

110 0011 1010 1010 1010 1010


12. The three elements that make up the number's 32 bit single precision IEEE 754 binary floating point representation:

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

Exponent (8 bits) =
1000 0110

Mantissa (23 bits) =
110 0011 1010 1010 1010 1010


Decimal number 227.666 666 666 666 666 666 666 666 666 663 1 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 1000 0110 - 110 0011 1010 1010 1010 1010

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How to convert decimal numbers from base ten to 32 bit single precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 32 bit single 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 base ten 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 of the previous dividing 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 previous multiplying operations, starting from the top of the constructed list 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, by shifting the decimal point (or if you prefer, the decimal mark) "n" positions either to the left or to the right, so that only one non zero digit remains to the left of the decimal point.
  • 7. Adjust the exponent in 8 bit excess/bias notation and then convert it from decimal (base 10) to 8 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(8-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign if the case) and adjust its length to 23 bits, either by removing the excess bits from the right (losing precision...) or by adding extra '0' bits 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 -25.347 from decimal system (base ten) to 32 bit single precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-25.347| = 25.347

  • 2. First convert the integer part, 25. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 25 ÷ 2 = 12 + 1;
    • 12 ÷ 2 = 6 + 0;
    • 6 ÷ 2 = 3 + 0;
    • 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:

    25(10) = 1 1001(2)

  • 4. Then convert the fractional part, 0.347. 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) 0.347 × 2 = 0 + 0.694;
    • 2) 0.694 × 2 = 1 + 0.388;
    • 3) 0.388 × 2 = 0 + 0.776;
    • 4) 0.776 × 2 = 1 + 0.552;
    • 5) 0.552 × 2 = 1 + 0.104;
    • 6) 0.104 × 2 = 0 + 0.208;
    • 7) 0.208 × 2 = 0 + 0.416;
    • 8) 0.416 × 2 = 0 + 0.832;
    • 9) 0.832 × 2 = 1 + 0.664;
    • 10) 0.664 × 2 = 1 + 0.328;
    • 11) 0.328 × 2 = 0 + 0.656;
    • 12) 0.656 × 2 = 1 + 0.312;
    • 13) 0.312 × 2 = 0 + 0.624;
    • 14) 0.624 × 2 = 1 + 0.248;
    • 15) 0.248 × 2 = 0 + 0.496;
    • 16) 0.496 × 2 = 0 + 0.992;
    • 17) 0.992 × 2 = 1 + 0.984;
    • 18) 0.984 × 2 = 1 + 0.968;
    • 19) 0.968 × 2 = 1 + 0.936;
    • 20) 0.936 × 2 = 1 + 0.872;
    • 21) 0.872 × 2 = 1 + 0.744;
    • 22) 0.744 × 2 = 1 + 0.488;
    • 23) 0.488 × 2 = 0 + 0.976;
    • 24) 0.976 × 2 = 1 + 0.952;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 23) 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:

    0.347(10) = 0.0101 1000 1101 0100 1111 1101(2)

  • 6. Summarizing - the positive number before normalization:

    25.347(10) = 1 1001.0101 1000 1101 0100 1111 1101(2)

  • 7. Normalize the binary representation of the number, shifting the decimal point 4 positions to the left so that only one non-zero digit stays to the left of the decimal point:

    25.347(10) =
    1 1001.0101 1000 1101 0100 1111 1101(2) =
    1 1001.0101 1000 1101 0100 1111 1101(2) × 20 =
    1.1001 0101 1000 1101 0100 1111 1101(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 32 bit single precision IEEE 754 binary floating point:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1001 0101 1000 1101 0100 1111 1101

  • 9. Adjust the exponent in 8 bit excess/bias notation and then convert it from decimal (base 10) to 8 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as already demonstrated above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(8-1) - 1 = (4 + 127)(10) = 131(10) =
    1000 0011(2)

  • 10. Normalize the mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal point) and adjust its length to 23 bits, by removing the excess bits from the right (losing precision...):

    Mantissa (not-normalized): 1.1001 0101 1000 1101 0100 1111 1101

    Mantissa (normalized): 100 1010 1100 0110 1010 0111

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 1000 0011

    Mantissa (23 bits) = 100 1010 1100 0110 1010 0111

  • Number -25.347, converted from the decimal system (base 10) to 32 bit single precision IEEE 754 binary floating point =
    1 - 1000 0011 - 100 1010 1100 0110 1010 0111