Decimal to Fraction

Convert decimal numbers to their exact fraction representation

Decimal → Fraction Calculator

What is calculated?

This tool converts a decimal number to a fraction. The decimal is analyzed and converted to an exact fraction; the result is automatically simplified.

Enter decimal number
Example: 2.45 = 2 9/20 = 49/20
Fraction representation
Decimal will be converted to a simplified fraction

Decimal → Fraction Info

Properties

Decimal → Fraction conversion:

Place-value system Automatic simplification Exact representation Mixed numbers

Note: Finite decimals can always be represented exactly as fractions.

Examples
Simple: 0.5 = ¹⁄₂
Decimal: 0.25 = ¹⁄₄
Mixed: 2.75 = 2³⁄₄
Complex: 0.125 = ¹⁄₈

Formulas & Rules for Decimal → Fraction Conversion

Principle
\[0.abc = \frac{abc}{1000}\] Use place-value system
Mixed numbers
\[n.abc = n + \frac{abc}{1000}\] Integer and fractional parts
Simplify
\[\frac{abc}{1000} = \frac{abc \div \text{gcd}}{1000 \div \text{gcd}}\] Fully reduce
Improper fraction
\[n\frac{a}{b} = \frac{nb + a}{b}\] Conversion possible
One decimal place
\[0.a = \frac{a}{10}\] Tenths
Two decimal places
\[0.ab = \frac{ab}{100}\] Hundredths
Three decimal places
\[0.abc = \frac{abc}{1000}\] Thousandths
Negative numbers
\[-n.abc = -\frac{\text{numerator}}{\text{denominator}}\] Preserve sign

Step-by-step Example

Example: Convert 2.45 to a fraction
Step 1: Write fractional part as fraction
\[2.45 = 2 + 0.45 = 2 + \frac{45}{100}\] \[= 2\frac{45}{100}\]

0.45 = 45 hundredths = 45/100

Step 2: Simplify fraction
\[\frac{45}{100} = \frac{45 \div 5}{100 \div 5} = \frac{9}{20}\] \[2\frac{45}{100} = 2\frac{9}{20}\]

gcd(45, 100) = 5, therefore divide by 5.

Step 3: Optional - as improper fraction
\[2\frac{9}{20} = \frac{2 \times 20 + 9}{20} = \frac{40 + 9}{20} = \frac{49}{20}\]
Step 4: Result
\[2.45 = 2\frac{9}{20} = \color{blue}{\frac{49}{20}}\]
More examples
Simple:
\[0.5 = \frac{5}{10} = \frac{1}{2}\]
Quarter:
\[0.25 = \frac{25}{100} = \frac{1}{4}\]
Eighth:
\[0.125 = \frac{125}{1000} = \frac{1}{8}\]
Mixed:
\[3.75 = 3\frac{3}{4} = \frac{15}{4}\]
Place value table
Decimal places:
• 1 place: Tenths (÷ 10)
• 2 places: Hundredths (÷ 100)
• 3 places: Thousandths (÷ 1000)
• n places: ÷ 10ⁿ
Repeating decimals
Repeating decimals like 0.333... = ¹⁄₃ require special techniques and are not covered here.
Steps of decimal → fraction conversion
1. Fractional part as fraction
2. Find gcd
3. Fully simplify
4. Optional transform

The place-value system is the key to converting decimals to fractions.

Applications of decimal → fraction conversion

Decimal → fraction conversion is useful in many areas:

Precise measurements
  • Exact dimensions
  • Engineering tolerances
  • Scientific measurements
  • Construction and craftsmanship
Cooking & Recipes
  • Recipe conversions
  • Adjust portion sizes
  • Ingredient ratios
  • Baking and cooking quantities
Mathematical education
  • Develop number sense
  • Understand fraction arithmetic
  • Learn place-value system
  • Exact vs. approximate calculations
Engineering & Programming
  • Gear ratios
  • Algorithm development
  • Digital signal processing
  • CAD applications

Mathematical context

Description

Converting decimals to fractions uses the place-value system and reveals the fundamental relationship between different number representations. Finite decimals correspond to rational numbers with denominators that are powers of ten. Reduction by the greatest common divisor gives the simplest fractional form and exposes the number's mathematical structure.

Summary

Decimal → fraction conversion connects practical decimal notation with the exact fractional representation of rational numbers. It's essential for precise calculations, mathematical understanding and bridging everyday use with theoretical mathematics.

More Fraction Calculators

Add Fractions  •  Compare Fractions  •  Decimal to Fraction  •  Divide Fractions  •  Fraction to Decimal  •  Fraction to Percent  •  Egyptian Fraction  •  Mixed number to improper Fraction  •  Multiply Fractions  •  Percent to Fraction  •  Simplify Fraction  •  Subtract Fractions  •