32bit IEEE 754: Decimal ↗ Single Precision Floating Point Binary: 12.342 Convert the Number to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard, From a Base 10 Decimal System Number

Number 12.342(10) converted and written in 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: 12.
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;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 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.


12(10) =


1100(2)


3. Convert to binary (base 2) the fractional part: 0.342.

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) 0.342 × 2 = 0 + 0.684;
  • 2) 0.684 × 2 = 1 + 0.368;
  • 3) 0.368 × 2 = 0 + 0.736;
  • 4) 0.736 × 2 = 1 + 0.472;
  • 5) 0.472 × 2 = 0 + 0.944;
  • 6) 0.944 × 2 = 1 + 0.888;
  • 7) 0.888 × 2 = 1 + 0.776;
  • 8) 0.776 × 2 = 1 + 0.552;
  • 9) 0.552 × 2 = 1 + 0.104;
  • 10) 0.104 × 2 = 0 + 0.208;
  • 11) 0.208 × 2 = 0 + 0.416;
  • 12) 0.416 × 2 = 0 + 0.832;
  • 13) 0.832 × 2 = 1 + 0.664;
  • 14) 0.664 × 2 = 1 + 0.328;
  • 15) 0.328 × 2 = 0 + 0.656;
  • 16) 0.656 × 2 = 1 + 0.312;
  • 17) 0.312 × 2 = 0 + 0.624;
  • 18) 0.624 × 2 = 1 + 0.248;
  • 19) 0.248 × 2 = 0 + 0.496;
  • 20) 0.496 × 2 = 0 + 0.992;
  • 21) 0.992 × 2 = 1 + 0.984;
  • 22) 0.984 × 2 = 1 + 0.968;
  • 23) 0.968 × 2 = 1 + 0.936;
  • 24) 0.936 × 2 = 1 + 0.872;

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...)


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:


0.342(10) =


0.0101 0111 1000 1101 0100 1111(2)


5. Positive number before normalization:

12.342(10) =


1100.0101 0111 1000 1101 0100 1111(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 3 positions to the left, so that only one non zero digit remains to the left of it:


12.342(10) =


1100.0101 0111 1000 1101 0100 1111(2) =


1100.0101 0111 1000 1101 0100 1111(2) × 20 =


1.1000 1010 1111 0001 1010 1001 111(2) × 23


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): 3


Mantissa (not normalized):
1.1000 1010 1111 0001 1010 1001 111


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =


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


3 + 2(8-1) - 1 =


(3 + 127)(10) =


130(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;
  • 130 ÷ 2 = 65 + 0;
  • 65 ÷ 2 = 32 + 1;
  • 32 ÷ 2 = 16 + 0;
  • 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) =


130(10) =


1000 0010(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. 100 0101 0111 1000 1101 0100 1111 =


100 0101 0111 1000 1101 0100


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 0010


Mantissa (23 bits) =
100 0101 0111 1000 1101 0100


The base ten decimal number 12.342 converted and written in 32 bit single precision IEEE 754 binary floating point representation:
0 - 1000 0010 - 100 0101 0111 1000 1101 0100

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