Mathwords logoMathwords

Fractional Part — Definition, Formula & Examples

The fractional part of a number is the portion that remains after you remove the whole-number part. For example, the fractional part of 7.83 is 0.83.

For any real number xx, the fractional part, denoted {x}\{x\}, is defined as x−⌊x⌋x - \lfloor x \rfloor, where ⌊x⌋\lfloor x \rfloor is the greatest integer less than or equal to xx (the floor function). The fractional part always satisfies 0≤{x}<10 \leq \{x\} < 1.

Key Formula

{x}=x−⌊x⌋\{x\} = x - \lfloor x \rfloor
Where:
  • {x}\{x\} = The fractional part of x
  • xx = Any real number
  • ⌊x⌋\lfloor x \rfloor = The floor of x (greatest integer ≤ x)

How It Works

To find the fractional part, subtract the integer part from the original number. For positive numbers, this is straightforward: just drop everything before the decimal point. For negative numbers, be careful — the floor of −3.25-3.25 is −4-4, not −3-3, so the fractional part is −3.25−(−4)=0.75-3.25 - (-4) = 0.75. The result is always a value from 0 (inclusive) up to but not including 1.

Worked Example

Problem: Find the fractional part of 5.72.
Step 1: Identify the floor (integer part) of the number.
⌊5.72⌋=5\lfloor 5.72 \rfloor = 5
Step 2: Subtract the floor from the original number.
{5.72}=5.72−5=0.72\{5.72\} = 5.72 - 5 = 0.72
Answer: The fractional part of 5.72 is 0.72.

Why It Matters

The fractional part appears when you convert improper fractions to mixed numbers — the remainder over the denominator gives you the fractional part. It also shows up in programming and computer science when you need to separate a number into its whole and decimal components.

Common Mistakes

Mistake: Assuming the fractional part of a negative number like −3.25-3.25 is −0.25-0.25 or 0.250.25 by simply reading the digits after the decimal point.
Correction: Use the floor function: ⌊−3.25⌋=−4\lfloor -3.25 \rfloor = -4, so {−3.25}=−3.25−(−4)=0.75\{-3.25\} = -3.25 - (-4) = 0.75. The fractional part is always between 0 and 1.