Rational and irrational numbers Knowledge-Based Study

Study notes

1. What is a Rational Number?

   A rational number is any number that can be written in the form $\frac{a}{b}$, where $a$ and $b$ are integers, and $b \neq 0$.

   The word rational comes from the word ratio, referring to a comparison or division between two integers.

   Forms of Rational Numbers

   • Integers: Every integer is rational because it can be placed over 1. For example, $7 = \frac{7}{1}$ and $-4 = \frac{-4}{1}$.
   • Fractions: Proper and improper fractions like $\frac{3}{5}$ and $-\frac{11}{4}$.
   • Terminating Decimals: Decimals that come to an end, such as $0.25 = \frac{1}{4}$ or $-1.8 = -\frac{9}{5}$.
   • Repeating Decimals: Decimals that repeat indefinitely in a regular pattern, such as $0.\overline{3} = \frac{1}{3}$ or $0.\overline{142857} = \frac{1}{7}$.

   What is an Irrational Number?

   An irrational number is a real number that cannot be expressed as a ratio of two integers $\frac{a}{b}$. When written as a decimal, an irrational number continues indefinitely without terminating and without repeating any pattern.

   Common Examples of Irrational Numbers

   • Roots of Non-Perfect Squares: Values such as $\sqrt{2} \approx 1.414213...$, $\sqrt{3}$, and $\sqrt{10}$. Because no integer squared equals 2, $\sqrt{2}$ cannot be simplified into an integer or fraction.
   • Mathematical Constants: The number $\pi \approx 3.141592...$ is irrational because its decimal expansion never terminates and never repeats in a fixed cycle.
   • Constructed Non-Repeating Decimals: Patterns such as $0.101001000100001...$, where the number of zeros increases each time, never repeat a predictable block of digits.

   How to Distinguish Rational from Irrational

   To classify any real number, examine its decimal form and its exact simplified value:

   • Check for perfect squares in radicals: $\sqrt{49} = 7$, which is rational. In contrast, $\sqrt{50} = 5\sqrt{2}$, which contains an irrational component and is therefore irrational.
   • Approximations vs. Exact Values: The fraction $\frac{22}{7}$ and the decimal $3.14$ are rational numbers often used to approximate $\pi$, but $\pi$ itself remains irrational.
   • Decimal Patterns: If the digits stop or establish a fixed repeating loop, the number is rational. If the digits continue forever with no repeating cycle, the number is irrational.