Simplest Form
Fractions & DecimalsA fraction is in simplest form when the numerator and denominator share no common factor other than 1.
Formula
\frac{a}{b} \text{ where } \gcd(a,b) = 1
Definition
A fraction is in simplest form (also called lowest terms) when you cannot divide both the top and bottom by any number other than $1$. It is the smallest, cleanest version of a fraction.
Example
$6/8$ is NOT in simplest form because you can divide both by $2$ to get $3/4$. But $3/4$ IS in simplest form because $3$ and $4$ share no common factor other than $1$.
Key Insight
Simplest form makes fractions easier to understand and compare. It's like giving someone your address as "123 Main St" instead of "246/2 Main St" - same place, much clearer.
Definition
A fraction $a/b$ is in simplest form (lowest terms) when $\gcd(a, b) = 1$, meaning the numerator and denominator are coprime. To reduce a fraction, divide both numerator and denominator by their GCD: $(a/\gcd)/(b/\gcd)$.
Example
Reduce $36/48$. $\gcd(36, 48)$: $36 = 2^2 \times 3^2$, $48 = 2^4 \times 3$. $\gcd = 2^2 \times 3 = 12$. So $36/48 = (36/12)/(48/12) = 3/4$.
Key Insight
The Euclidean algorithm efficiently computes GCD for large numbers: $\gcd(48, 36) = \gcd(36, 12) = \gcd(12, 0) = 12$. This beats listing all factors and is the basis for modern cryptographic key generation.
Definition
The simplest form (or reduced form) of a rational number $a/b$ is the unique representative $(a', b')$ of its equivalence class with $b' > 0$ and $\gcd(a', b') = 1$. Existence and uniqueness follow from the fundamental theorem of arithmetic and properties of the GCD.
Example
For any Gaussian integer fraction $(a+bi)/(c+di)$ in $\mathbb{Z}[i]$, the analogue of simplest form requires dividing by the GCD in $\mathbb{Z}[i]$ (a Euclidean domain). The norm function $N(a+bi) = a^2+b^2$ serves as the "size" in the Euclidean algorithm.
Key Insight
Coprimality ($\gcd = 1$) is the condition for $a/b$ to be in simplest form, and it directly controls the period of the decimal expansion: $1/b$ terminates iff $b$ has only factors of $2$ and $5$; otherwise the period length equals the multiplicative order of $10$ modulo ($b$ after removing $2$s and $5$s).