32bit IEEE 754: Decimal ↗ Single Precision Floating Point Binary: -128.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 11 Convert the Number to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard, From a Base 10 Decimal System Number

Number -128.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 11(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. Start with the positive version of the number:

|-128.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 11| = 128.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 11

2. First, convert to binary (in base 2) the integer part: 128.
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;
  • 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;

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.


128(10) =


1000 0000(2)


4. Convert to binary (base 2) the fractional part: 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 11.

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) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 11 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 22;
  • 2) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 22 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 44;
  • 3) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 44 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 88;
  • 4) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 88 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 76;
  • 5) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 76 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 52;
  • 6) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 52 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 04;
  • 7) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 04 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 08;
  • 8) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 08 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 16;
  • 9) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 16 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 32;
  • 10) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 32 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 776 64;
  • 11) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 776 64 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 553 28;
  • 12) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 553 28 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 106 56;
  • 13) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 106 56 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 213 12;
  • 14) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 213 12 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 426 24;
  • 15) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 426 24 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 852 48;
  • 16) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 852 48 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 704 96;
  • 17) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 704 96 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 409 92;
  • 18) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 409 92 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 110 819 84;
  • 19) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 110 819 84 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 221 639 68;
  • 20) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 221 639 68 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 443 279 36;
  • 21) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 443 279 36 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 886 558 72;
  • 22) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 886 558 72 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 773 117 44;
  • 23) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 773 117 44 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 546 234 88;
  • 24) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 546 234 88 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 092 469 76;

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...)


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:


0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 11(10) =


0.0001 1100 0111 0001 1100 0111(2)


6. Positive number before normalization:

128.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 11(10) =


1000 0000.0001 1100 0111 0001 1100 0111(2)

7. 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:


128.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 11(10) =


1000 0000.0001 1100 0111 0001 1100 0111(2) =


1000 0000.0001 1100 0111 0001 1100 0111(2) × 20 =


1.0000 0000 0011 1000 1110 0011 1000 111(2) × 27


8. 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 1 (a negative number)


Exponent (unadjusted): 7


Mantissa (not normalized):
1.0000 0000 0011 1000 1110 0011 1000 111


9. 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)


10. 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;

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) =


134(10) =


1000 0110(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 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. 000 0000 0001 1100 0111 0001 1100 0111 =


000 0000 0001 1100 0111 0001


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

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


Exponent (8 bits) =
1000 0110


Mantissa (23 bits) =
000 0000 0001 1100 0111 0001


The base ten decimal number -128.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 11 converted and written in 32 bit single precision IEEE 754 binary floating point representation:
1 - 1000 0110 - 000 0000 0001 1100 0111 0001

The latest decimal numbers converted from base ten to 32 bit single precision IEEE 754 floating point binary standard representation

Number -128.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 11 converted from decimal system (written in base ten) to 32 bit single precision IEEE 754 binary floating point representation standard Mar 29 06:05 UTC (GMT)
Number 1 101 100 110 084 converted from decimal system (written in base ten) to 32 bit single precision IEEE 754 binary floating point representation standard Mar 29 06:05 UTC (GMT)
Number 11 971 converted from decimal system (written in base ten) to 32 bit single precision IEEE 754 binary floating point representation standard Mar 29 06:05 UTC (GMT)
Number 6 152 221 753 932 768 217 403 converted from decimal system (written in base ten) to 32 bit single precision IEEE 754 binary floating point representation standard Mar 29 06:05 UTC (GMT)
Number 1 110 010 101 100 111 001 101 011 101 056 converted from decimal system (written in base ten) to 32 bit single precision IEEE 754 binary floating point representation standard Mar 29 06:05 UTC (GMT)
Number 1 000 011 000 110 111 010 000 000 000 084 converted from decimal system (written in base ten) to 32 bit single precision IEEE 754 binary floating point representation standard Mar 29 06:05 UTC (GMT)
Number 10 110 011 111 011 100 100 000 000 000 082 converted from decimal system (written in base ten) to 32 bit single precision IEEE 754 binary floating point representation standard Mar 29 06:05 UTC (GMT)
All base ten decimal numbers converted to 32 bit single precision IEEE 754 binary floating point