Decimal to Fraction Calculator
Convert terminating and repeating decimals into exact simplified fractions, mixed numbers, and fractional inch measurements with step-by-step GCD factorization.
| Conversion Metric | Calculated Value | Mathematical Context & Formulation |
|---|---|---|
| Input Decimal Value | 0.625 | Original floating-point representation |
| Unreduced Fraction | 625/1000 | Decimal digits written over power of 10 |
| Greatest Common Divisor (GCD) | 125 | Highest integer dividing both numerator and denominator |
| Simplified Fraction (Lowest Terms) | 5/8 | Numerator and denominator reduced to coprime integers |
| Mixed Number Representation | 5/8 | Proper fraction |
| Percentage Equivalent | 62.5% | Decimal multiplied by 100% |
| Scientific Notation | 6.25 × 10⁻¹ | Normalized exponential notation |
| Nearest 1/16 Inch Mark | 5/8" | Exact match |
| Nearest 1/32 Inch Mark | 5/8" | Exact match |
| Nearest 1/64 Inch Mark | 5/8" | Exact match |
Fraction Form: 5/8 (Mixed: 5/8) | Percentage: 62.5%.
Illustrative Walkthrough: Converting 0.625 to a Simplified Fraction
0.625, there are three digits to the right of the decimal point (tenths, hundredths, thousandths).625 / 1000.1000 mod 625 = 375, 625 mod 375 = 250, 375 mod 250 = 125, 250 mod 125 = 0. The GCD is 125.625 ÷ 125 = 5 and 1000 ÷ 125 = 8. The simplified proper fraction is 5/8.The Mathematical Logic Behind Decimal-to-Fraction Conversion
Every terminating decimal represents a rational number—a quantity that can be expressed as the ratio of two integers a / b. Converting a terminating decimal to a fraction follows a 3-step algebraic proof:
- Step 1 (Base-10 Expansion): Count the number of decimal digits d following the decimal point. Multiply the decimal by 10ᵈ to clear the fractional portion, and set 10ᵈ as the denominator. For example, 2.75 has 2 decimal digits, yielding
275 / 100. - Step 2 (Coprime Factorization): Calculate the Greatest Common Divisor (GCD) of the numerator and denominator using the Euclidean algorithm. For 275 and 100, GCD(275, 100) = 25.
- Step 3 (Reduction to Lowest Terms): Divide both parts by the GCD:
(275 ÷ 25) / (100 ÷ 25) = 11/4. When written as a mixed number,11 ÷ 4 = 2 with a remainder of 3, giving 2 3/4.
Converting Repeating (Periodic) Decimals Using Algebra
Repeating decimals (such as 0.333... or 0.1666...) have infinite decimal digits and cannot be placed over a simple power of 10. Instead, mathematicians use the algebraic cancellation method:
- Case A: Pure Repeating Decimals (e.g., x = 0.333...): Let x = 0.333... Since 1 digit repeats, multiply by 10¹:
10x = 3.333...Subtract the original equation:10x − x = 3.333... − 0.333... ⇒ 9x = 3. Dividing both sides givesx = 3/9 = 1/3. - Case B: Mixed Repeating Decimals (e.g., x = 0.1666...): The non-repeating part has 1 digit (d = 1) and the repeating part has 1 digit (r = 1). Multiply by 10^(d+r) = 10² = 100:
100x = 16.666...Multiply by 10^d = 10¹ = 10:10x = 1.666...Subtract:100x − 10x = 16.666... − 1.666... ⇒ 90x = 15. Dividing yieldsx = 15/90 = 1/6.
Decimal to Fractional Inches in Machining & Construction
In American architecture, carpentry, and fabrication, blueprint dimensions frequently mix decimal engineering callouts (e.g., 0.4375") with physical ruler graduations based on powers of 2 (1/2", 1/4", 1/8", 1/16", 1/32", 1/64"):
- To find the nearest fractional mark: Multiply the decimal fraction by the target denominator (e.g., 0.4375 × 16 = 7.000). Round to the nearest whole integer (7), giving 7/16" with 0.000" error.
- Handling non-exact dimensions: For a decimal like 0.390", multiplying by 16 gives
0.390 × 16 = 6.24 ⇒ 6/16" = 3/8"with an error of +0.015". Checking 32nds:0.390 × 32 = 12.48 ⇒ 13/32"with an error of −0.016". Checking 64ths:0.390 × 64 = 24.96 ⇒ 25/64"with a negligible error of −0.0006".
The power of 10 is determined directly by the number of decimal places (digits to the right of the decimal point). If there is 1 decimal place (e.g., 0.4), use 10¹ = 10 (4/10). If there are 2 decimal places (e.g., 0.25), use 10² = 100 (25/100). If there are 3 decimal places (e.g., 0.625), use 10³ = 1000 (625/1000). Each additional decimal digit multiplies the denominator by another factor of 10.
A proper fraction has a numerator smaller than its denominator (e.g., 5/8), representing a value strictly between 0 and 1. An improper fraction has a numerator greater than or equal to its denominator (e.g., 11/4), representing a value equal to or greater than 1. A mixed number expresses an improper fraction as a whole integer combined with a proper fraction (e.g., 2 3/4).
Our calculator applies the Euclidean algorithm to compute the Greatest Common Divisor (GCD/GCF) between the raw numerator and denominator. It then divides both the top and bottom by this common factor. Because the GCD represents the largest integer that divides both numbers without leaving a remainder, dividing by it guarantees the resulting fraction is completely irreducible (coprime).
Denominators consisting of 9s occur because subtracting equations in base-10 algebra cancels the infinite repeating tail. Multiplying by 10 shifts the decimal 1 place; subtracting the original equation yields 10x − x = 9x. Similarly, for a 2-digit repeating cycle, multiplying by 100 and subtracting leaves 100x − x = 99x. For mixed repeating decimals, multiplying by powers of 10 to align the repeating parts produces differences like 100x − 10x = 90x or 1000x − 10x = 990x.
No. Only rational numbers (terminating decimals like 0.125 and periodic repeating decimals like 0.666...) can be expressed as an exact fraction of two integers. Irrational numbers—such as π (3.14159...), √2 (1.41421...), and the golden ratio φ (1.61803...)—possess non-terminating, non-repeating decimal expansions that can never be converted into exact rational fractions.