Convert the Number 45.24 to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number. Detailed Explanations

Number 45.24(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

The first steps we'll go through to make the conversion:

Convert to binary (to base 2) the integer part of the number.

Convert to binary the fractional part of the number.


1. First, convert to binary (in base 2) the integer part: 45.
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;
  • 45 ÷ 2 = 22 + 1;
  • 22 ÷ 2 = 11 + 0;
  • 11 ÷ 2 = 5 + 1;
  • 5 ÷ 2 = 2 + 1;
  • 2 ÷ 2 = 1 + 0;
  • 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.


45(10) =


10 1101(2)


3. Convert to binary (base 2) the fractional part: 0.24.

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) 0.24 × 2 = 0 + 0.48;
  • 2) 0.48 × 2 = 0 + 0.96;
  • 3) 0.96 × 2 = 1 + 0.92;
  • 4) 0.92 × 2 = 1 + 0.84;
  • 5) 0.84 × 2 = 1 + 0.68;
  • 6) 0.68 × 2 = 1 + 0.36;
  • 7) 0.36 × 2 = 0 + 0.72;
  • 8) 0.72 × 2 = 1 + 0.44;
  • 9) 0.44 × 2 = 0 + 0.88;
  • 10) 0.88 × 2 = 1 + 0.76;
  • 11) 0.76 × 2 = 1 + 0.52;
  • 12) 0.52 × 2 = 1 + 0.04;
  • 13) 0.04 × 2 = 0 + 0.08;
  • 14) 0.08 × 2 = 0 + 0.16;
  • 15) 0.16 × 2 = 0 + 0.32;
  • 16) 0.32 × 2 = 0 + 0.64;
  • 17) 0.64 × 2 = 1 + 0.28;
  • 18) 0.28 × 2 = 0 + 0.56;
  • 19) 0.56 × 2 = 1 + 0.12;
  • 20) 0.12 × 2 = 0 + 0.24;
  • 21) 0.24 × 2 = 0 + 0.48;
  • 22) 0.48 × 2 = 0 + 0.96;
  • 23) 0.96 × 2 = 1 + 0.92;
  • 24) 0.92 × 2 = 1 + 0.84;
  • 25) 0.84 × 2 = 1 + 0.68;
  • 26) 0.68 × 2 = 1 + 0.36;
  • 27) 0.36 × 2 = 0 + 0.72;
  • 28) 0.72 × 2 = 1 + 0.44;
  • 29) 0.44 × 2 = 0 + 0.88;
  • 30) 0.88 × 2 = 1 + 0.76;
  • 31) 0.76 × 2 = 1 + 0.52;
  • 32) 0.52 × 2 = 1 + 0.04;
  • 33) 0.04 × 2 = 0 + 0.08;
  • 34) 0.08 × 2 = 0 + 0.16;
  • 35) 0.16 × 2 = 0 + 0.32;
  • 36) 0.32 × 2 = 0 + 0.64;
  • 37) 0.64 × 2 = 1 + 0.28;
  • 38) 0.28 × 2 = 0 + 0.56;
  • 39) 0.56 × 2 = 1 + 0.12;
  • 40) 0.12 × 2 = 0 + 0.24;
  • 41) 0.24 × 2 = 0 + 0.48;
  • 42) 0.48 × 2 = 0 + 0.96;
  • 43) 0.96 × 2 = 1 + 0.92;
  • 44) 0.92 × 2 = 1 + 0.84;
  • 45) 0.84 × 2 = 1 + 0.68;
  • 46) 0.68 × 2 = 1 + 0.36;
  • 47) 0.36 × 2 = 0 + 0.72;
  • 48) 0.72 × 2 = 1 + 0.44;
  • 49) 0.44 × 2 = 0 + 0.88;
  • 50) 0.88 × 2 = 1 + 0.76;
  • 51) 0.76 × 2 = 1 + 0.52;
  • 52) 0.52 × 2 = 1 + 0.04;
  • 53) 0.04 × 2 = 0 + 0.08;

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:


0.24(10) =


0.0011 1101 0111 0000 1010 0011 1101 0111 0000 1010 0011 1101 0111 0(2)


5. Positive number before normalization:

45.24(10) =


10 1101.0011 1101 0111 0000 1010 0011 1101 0111 0000 1010 0011 1101 0111 0(2)


The last steps we'll go through to make the conversion:

Normalize the binary representation of the number.

Adjust the exponent.

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

Normalize the mantissa.


6. 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:


45.24(10) =


10 1101.0011 1101 0111 0000 1010 0011 1101 0111 0000 1010 0011 1101 0111 0(2) =


10 1101.0011 1101 0111 0000 1010 0011 1101 0111 0000 1010 0011 1101 0111 0(2) × 20 =


1.0110 1001 1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 10(2) × 25


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): 5


Mantissa (not normalized):
1.0110 1001 1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 10


8. 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)


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;
  • 1 028 ÷ 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;

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) =


1028(10) =


100 0000 0100(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, 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. 0110 1001 1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 10 1110 =


0110 1001 1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110


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) =
100 0000 0100


Mantissa (52 bits) =
0110 1001 1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110


The base ten decimal number 45.24 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 100 0000 0100 - 0110 1001 1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110

(64 bits IEEE 754)

Number 45.23 converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point representation = ?

Number 45.25 converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point representation = ?

Convert to 64 bit double precision IEEE 754 binary floating point representation standard

A number in 64 bit double precision IEEE 754 binary floating point standard representation requires three building elements: sign (it takes 1 bit and it's either 0 for positive or 1 for negative numbers), exponent (11 bits), mantissa (52 bits)

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Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

Available Base Conversions Between Decimal and Binary Systems

Conversions Between Decimal System Numbers (Written in Base Ten) and Binary System Numbers (Base Two and Computer Representation):


1. Integer -> Binary

2. Decimal -> Binary

3. Binary -> Integer

4. Binary -> Decimal