๐ What are Rational Numbers?
Rational numbers are numbers that can be expressed as a fraction, where both the numerator (the top number) and the denominator (the bottom number) are integers (whole numbers), and the denominator is not zero. Think of it as any number you can write as a simple fraction.
- โ Definition: A number that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
- ๐ข Examples: 2 (which can be written as $\frac{2}{1}$), 0.5 (which can be written as $\frac{1}{2}$), -3 (which can be written as $\frac{-3}{1}$), 0.333... (which can be written as $\frac{1}{3}$).
- โ Decimal Representation: Rational numbers have decimal representations that either terminate (end) or repeat.
โพ๏ธ What are Irrational Numbers?
Irrational numbers are numbers that cannot be expressed as a fraction of two integers. Their decimal representations are non-terminating and non-repeating. They go on forever without a repeating pattern.
- โ Definition: A number that cannot be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers.
- ๐งฎ Examples: $\pi$ (approximately 3.14159...), $\sqrt{2}$ (approximately 1.41421...), $e$ (approximately 2.71828...).
- ๐ Decimal Representation: Irrational numbers have decimal representations that are non-terminating and non-repeating.
๐ Rational vs. Irrational Numbers: A Detailed Comparison
| Feature |
Rational Numbers |
Irrational Numbers |
| Definition |
Can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers ($q \neq 0$). |
Cannot be expressed as a fraction of two integers. |
| Decimal Representation |
Terminating or repeating decimals. |
Non-terminating and non-repeating decimals. |
| Examples |
$\frac{1}{2}$, 0.75, -5, 0.333... |
$\sqrt{2}$, $\pi$, $e$ |
| Set of Numbers |
Belongs to the set of rational numbers, denoted by $\mathbb{Q}$. |
Belongs to the set of irrational numbers, often denoted by $\mathbb{I}$. |
| Operations |
Arithmetic operations (+, -, *, /) between two rational numbers result in a rational number (except division by zero). |
Arithmetic operations between two irrational numbers may or may not result in an irrational number. |
๐ Key Takeaways
- โ
Fractions: If you can write a number as a fraction of two integers, it's rational.
- โพ๏ธ Decimal Patterns: Terminating or repeating decimals are rational; non-terminating, non-repeating decimals are irrational.
- ๐ก Common Irrationals: Remember $\pi$ and square roots of non-perfect squares (like $\sqrt{2}$) are common irrational numbers.
- ๐ง Think Critically: Not all numbers are easy to categorize at first glance, but understanding the definitions will help you classify them correctly.