A repeating decimal is a rational number. This classification stems from the formal definition of rational numbers and the algebraic process that allows any repeating decimal to be converted into a simple fraction. This conversion demonstrates that the number is a ratio of two integers, which is the defining characteristic of a rational number.
Defining Rational Numbers and Repeating Decimals
A rational number is formally defined as any number that can be expressed as the quotient, or fraction, $p/q$, where $p$ and $q$ are integers and the denominator $q$ is not zero. This definition includes all integers, terminating decimals, and repeating decimals.
A repeating decimal, also known as a recurring decimal, is a decimal representation of a number where a sequence of one or more digits repeats infinitely. For instance, the fraction $1/3$ results in the repeating decimal $0.333…$, often written as $0.\overline{3}$. Another example is $1/11$, which produces $0.090909…$, or $0.\overline{09}$. The existence of this predictable, repeating pattern links these decimals back to the structure of a fraction.
The Proof: Converting a Repeating Decimal to a Fraction
The algebraic method for converting a repeating decimal into a fraction proves these numbers are rational. This process uses the repeating nature of the decimal to eliminate the infinite tail of digits through subtraction. Consider $x = 0.\overline{3}$, or $x = 0.333…$.
The next step involves multiplying the equation by a power of 10 corresponding to the number of repeating digits; since only one digit repeats, we multiply by 10. This yields $10x = 3.333…$. Subtracting the original equation ($x = 0.333…$) from the new one effectively cancels out the infinite repeating part.
The subtraction results in $10x – x = 3.333… – 0.333…$, which simplifies to $9x = 3$. Solving for $x$ gives the fraction $x = 3/9$, which reduces to $1/3$.
The same method applies when multiple digits repeat, such as with $0.\overline{12}$. We set $x = 0.121212…$. Because two digits repeat, we multiply the equation by 100, resulting in $100x = 12.121212…$.
Subtracting the original equation ($x$) from the multiplied equation ($100x$) eliminates the repeating decimal tail: $100x – x = 12.121212… – 0.121212…$. This simplifies to $99x = 12$. Solving for $x$ yields the fraction $x = 12/99$, which reduces to $4/33$. The ability to consistently transform any repeating decimal into the $p/q$ form confirms their classification as rational numbers.
The Difference: When Decimals Become Irrational
Irrational numbers cannot be expressed as a ratio of two integers. In decimal form, they are non-terminating and non-repeating. Their digits continue infinitely without settling into a predictable sequence.
Classic examples include the mathematical constant pi ($\pi$), which begins $3.14159…$, and the square root of two ($\sqrt{2}$), which starts $1.414213…$. The lack of a repeating pattern means the algebraic subtraction method used for rational numbers is impossible to perform, as there is no power of 10 that can align the decimal points and eliminate the infinite, non-repeating tail, confirming they cannot be written as a simple fraction.