Identify rational and irrational numbers

Learning Objectives Define rational numbers and provide examples. Define irrational numbers and provide examples. Distinguish between terminating, repeating, and non-terminating/non-repeating decimals. Classify any given real number as rational or irrational. Explain the reasoning behind their classification of numbers. Identify square roots of non-perfect squares as irrational numbers. Have you ever wondered why some numbers seem to go on forever without a pattern, while others stop neatly or repeat? 🤔 Let's explore the fascinating world of numbers! In this lesson, you'll learn to categorize numbers into two main groups: rational and irrational. Understanding these classifications is fundamental to advanced mathematics, helping you make sense of equations, measu...

TermDefinitionExample Rational NumberA number that can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q$ is not zero. Rational numbers include all integers, fractions, terminating decimals, and repeating decimals.$5$ (which is $\frac{5}{1}$), $\frac{3}{4}$, $0.75$, $0.333...$ (which is $\frac{1}{3}$) Irrational NumberA number that cannot be expressed as a simple fraction $\frac{p}{q}$. When written as a decimal, an irrational number goes on forever without repeating any pattern (non-terminating and non-repeating).$\sqrt{2}$, $\pi$ (pi), $0.123456789101112...$ Terminating DecimalA decimal that has a finite number of digits after the decimal point. It stops.$0.5$, $3.14$, $0.25$ Repeating DecimalA decimal that has a digit or a block of digits that repeats infi...

Rational Number Criterion A number $x$ is rational if and only if it can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. This rule applies to all integers (e.g., $7 = \frac{7}{1}$), fractions, terminating decimals (e.g., $0.25 = \frac{1}{4}$), and repeating decimals (e.g., $0.\overline{6} = \frac{2}{3}$). If you can express it as a simple fraction, it's rational. Irrational Number Criterion A number $x$ is irrational if and only if it cannot be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. This means its decimal representation is non-terminating and non-repeating. This rule helps identify numbers like $\pi$ or the square roots of non-perfect squares. If a decimal goes on forever without a repeating...