The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 1000 0010 - 100 0101 1000 0101 0010 1011 Converted and Written as a Base Ten Decimal System Number (Float)

0 - 1000 0010 - 100 0101 1000 0101 0010 1011: 32 bit single precision IEEE 754 binary floating point standard representation number converted to decimal system (base ten)

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

Convert the exponent from binary (from base 2) to decimal (in base 10).

Adjust the exponent.

Convert the mantissa from binary (from base 2) to decimal (in base 10).

1. Identify the elements that make up the binary representation of the number:

The first bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.
0


The next 8 bits contain the exponent:
1000 0010


The last 23 bits contain the mantissa:
100 0101 1000 0101 0010 1011


2. Convert the exponent from binary (from base 2) to decimal (in base 10).

The exponent is allways a positive integer.

1000 0010(2) =


1 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 0 × 20 =


128 + 0 + 0 + 0 + 0 + 0 + 2 + 0 =


128 + 2 =


130(10)

3. Adjust the exponent.

Subtract the excess bits: 2(8 - 1) - 1 = 127,

that is due to the 8 bit excess/bias notation.


The exponent, adjusted = 130 - 127 = 3


4. Convert the mantissa from binary (from base 2) to decimal (in base 10).

The mantissa represents the fractional part of the number (what comes after the whole part of the number, separated from it by a comma).


100 0101 1000 0101 0010 1011(2) =

1 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 1 × 2-5 + 0 × 2-6 + 1 × 2-7 + 1 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 0 × 2-12 + 1 × 2-13 + 0 × 2-14 + 1 × 2-15 + 0 × 2-16 + 0 × 2-17 + 1 × 2-18 + 0 × 2-19 + 1 × 2-20 + 0 × 2-21 + 1 × 2-22 + 1 × 2-23 =


0.5 + 0 + 0 + 0 + 0.031 25 + 0 + 0.007 812 5 + 0.003 906 25 + 0 + 0 + 0 + 0 + 0.000 122 070 312 5 + 0 + 0.000 030 517 578 125 + 0 + 0 + 0.000 003 814 697 265 625 + 0 + 0.000 000 953 674 316 406 25 + 0 + 0.000 000 238 418 579 101 562 5 + 0.000 000 119 209 289 550 781 25 =


0.5 + 0.031 25 + 0.007 812 5 + 0.003 906 25 + 0.000 122 070 312 5 + 0.000 030 517 578 125 + 0.000 003 814 697 265 625 + 0.000 000 953 674 316 406 25 + 0.000 000 238 418 579 101 562 5 + 0.000 000 119 209 289 550 781 25 =


0.543 126 463 890 075 683 593 75(10)

5. Put all the numbers into expression to calculate the single precision floating point decimal value:

(-1)Sign × (1 + Mantissa) × 2(Adjusted exponent) =


(-1)0 × (1 + 0.543 126 463 890 075 683 593 75) × 23 =


1.543 126 463 890 075 683 593 75 × 23 =


12.345 011 711 120 605 468 75

0 - 1000 0010 - 100 0101 1000 0101 0010 1011 converted from a 32 bit single precision IEEE 754 binary floating point standard representation number to a decimal system number, written in base ten (float) = 12.345 011 711 120 605 468 75(10)

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

Number 0 - 1000 0010 - 100 0101 1000 0101 0010 1010 converted from 32 bit single precision IEEE 754 binary floating point standard representation to decimal system written in base ten (float) = ?

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Convert 32 bit single precision IEEE 754 binary floating point standard numbers to base ten decimal system (float)



A number in 32 bit single precision IEEE 754 binary floating point standard representation...

... requires three building elements: the sign (it takes 1 bit and it's either 0 for positive or 1 for negative numbers), the exponent (8 bits) and the mantissa (23 bits)

The latest 32 bit single precision IEEE 754 floating point binary standard numbers converted and written as decimal system numbers (in base ten, float)

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All 32 bit single precision IEEE 754 binary floating point representation numbers converted to base ten decimal numbers (float)

How to convert numbers from 32 bit single precision IEEE 754 binary floating point standard to decimal system in base 10

Follow the steps below to convert a number from 32 bit single precision IEEE 754 binary floating point representation to base 10 decimal system:

Example: convert the number 1 - 1000 0001 - 100 0001 0000 0010 0000 0000 from 32 bit single precision IEEE 754 binary floating point system to base 10 decimal system (float):

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