Convert the Number 1 440 025 to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard, From a Base 10 Decimal System Number. Detailed Explanations

Number 1 440 025(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)

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

Convert to binary (base 2) the integer number.


1. 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;
  • 1 440 025 ÷ 2 = 720 012 + 1;
  • 720 012 ÷ 2 = 360 006 + 0;
  • 360 006 ÷ 2 = 180 003 + 0;
  • 180 003 ÷ 2 = 90 001 + 1;
  • 90 001 ÷ 2 = 45 000 + 1;
  • 45 000 ÷ 2 = 22 500 + 0;
  • 22 500 ÷ 2 = 11 250 + 0;
  • 11 250 ÷ 2 = 5 625 + 0;
  • 5 625 ÷ 2 = 2 812 + 1;
  • 2 812 ÷ 2 = 1 406 + 0;
  • 1 406 ÷ 2 = 703 + 0;
  • 703 ÷ 2 = 351 + 1;
  • 351 ÷ 2 = 175 + 1;
  • 175 ÷ 2 = 87 + 1;
  • 87 ÷ 2 = 43 + 1;
  • 43 ÷ 2 = 21 + 1;
  • 21 ÷ 2 = 10 + 1;
  • 10 ÷ 2 = 5 + 0;
  • 5 ÷ 2 = 2 + 1;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the positive number.

Take all the remainders starting from the bottom of the list constructed above.


1 440 025(10) =


1 0101 1111 1001 0001 1001(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.


3. Normalize the binary representation of the number.

Shift the decimal mark 20 positions to the left, so that only one non zero digit remains to the left of it:


1 440 025(10) =


1 0101 1111 1001 0001 1001(2) =


1 0101 1111 1001 0001 1001(2) × 20 =


1.0101 1111 1001 0001 1001(2) × 220


4. 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): 20


Mantissa (not normalized):
1.0101 1111 1001 0001 1001


5. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =


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


20 + 2(8-1) - 1 =


(20 + 127)(10) =


147(10)


6. 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;
  • 147 ÷ 2 = 73 + 1;
  • 73 ÷ 2 = 36 + 1;
  • 36 ÷ 2 = 18 + 0;
  • 18 ÷ 2 = 9 + 0;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

7. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =


147(10) =


1001 0011(2)


8. 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 adding the necessary number of zeros to the right.


Mantissa (normalized) =


1. 0101 1111 1001 0001 1001 000 =


010 1111 1100 1000 1100 1000


9. 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) =
1001 0011


Mantissa (23 bits) =
010 1111 1100 1000 1100 1000


The base ten decimal number 1 440 025 converted and written in 32 bit single precision IEEE 754 binary floating point representation:
0 - 1001 0011 - 010 1111 1100 1000 1100 1000

(32 bits IEEE 754)

Number 1 440 024 converted from decimal system (base 10) to 32 bit single precision IEEE 754 binary floating point representation = ?

Number 1 440 026 converted from decimal system (base 10) to 32 bit single precision IEEE 754 binary floating point representation = ?

Convert to 32 bit single 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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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:

Example: convert the negative number -25.347 from decimal system (base ten) to 32 bit single 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