Data Precision & Representation

Explore why 0.1 + 0.2 ≠ 0.3 in computers, and understand the fundamental limits of number representation.

The Famous 0.1 + 0.2 Problem

Try this in any programming language's console:

0.1 + 0.2 =

This is NOT a bug. It's a fundamental limitation of how computers represent decimal numbers in binary.

Precision Explorer

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Actual Stored Value
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Error

Why Does This Happen?

The Binary Problem

Computers use binary (base-2). Just like 1/3 = 0.333... in decimal (never ends), many simple decimal fractions are infinite in binary.

Example
0.1 in decimal = 0.0001100110011... in binary (repeating forever)

Finite Storage

Computers have limited memory. They must truncate these infinite sequences, causing small errors.

A 64-bit floating point number (standard) has about 15-17 significant decimal digits of precision.

Which Numbers Are Exact?

Numbers that can be expressed as fractions with powers of 2 in the denominator:

  • 0.5 = 1/2 ✓
  • 0.25 = 1/4 ✓
  • 0.125 = 1/8 ✓
  • 0.1 = 1/10 ✗ (10 = 2×5, has factor of 5)

Precision vs Accuracy

Precision

How many digits can be stored. More precision = more decimal places.

Example: 3.14159265358979 has high precision (many digits)

Accuracy

How close to the true value. High precision doesn't guarantee accuracy.

Example: 3.14000000000000 is precise but less accurate than 3.14159

Practical Implications

Never compare floating points for equality!

// BAD: May fail unexpectedly
if (0.1 + 0.2 === 0.3) { ... }

// GOOD: Use epsilon comparison
if (Math.abs((0.1 + 0.2) - 0.3) < 0.0001) { ... }

For money, use integers!

Store cents (integers) instead of dollars (decimals). $19.99 → 1999 cents.

Scientific calculations accumulate error

Each operation can introduce tiny errors. Over millions of calculations, this adds up.