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📅 Updated 2026📖 17 Problems⏱️ ~45 min read🏷️ Number Systems
📚 Understanding Rational & Irrational Numbers
1. What is a Rational Number?
A rational number is any number that can be expressed in the form
\( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
In other words, a rational number is a ratio of two integers.
The set of all rational numbers is denoted by \( \mathbb{Q} \).
2. What is an Irrational Number?
An irrational number is a number that cannot be expressed as a ratio of two integers.
Their decimal expansions are non-terminating and non-repeating.
Examples: \( \pi, e, \sqrt{2}, \sqrt{3}, \sqrt{5} \)
The set of irrational numbers is not closed under addition or multiplication
(e.g., \( \sqrt{2} + (-\sqrt{2}) = 0 \), which is rational).
🔑 Key Insight: Every rational number has a decimal expansion that either
terminates (like \( \frac{1}{4} = 0.25 \)) or repeats
(like \( \frac{1}{3} = 0.\overline{3} \)). Irrational numbers never terminate and never repeat.
3. Closure Property
A set is said to be closed under an operation if applying that operation
to any two elements of the set always produces an element that also belongs to the same set.
Formally: A set \( S \) is closed under operation \( * \) if for all
\( a, b \in S \), we have \( a * b \in S \).
➕ Addition
\( \mathbb{N}, \mathbb{W}, \mathbb{Z}, \mathbb{Q}, \mathbb{R} \) are all closed under addition.
✖️ Multiplication
\( \mathbb{N}, \mathbb{W}, \mathbb{Z}, \mathbb{Q}, \mathbb{R} \) are all closed under multiplication.
➖ Subtraction
\( \mathbb{Z}, \mathbb{Q}, \mathbb{R} \) are closed. \( \mathbb{N} \) and \( \mathbb{W} \) are not closed.
➗ Division
Only \( \mathbb{Q} \) and \( \mathbb{R} \) are closed (excluding division by zero). \( \mathbb{N}, \mathbb{W}, \mathbb{Z} \) are not closed.
4. Identity Elements
Additive Identity: \( 0 \) — because \( a + 0 = 0 + a = a \) for any \( a \).
Multiplicative Identity: \( 1 \) — because \( a \times 1 = 1 \times a = a \) for any \( a \).
5. Inverse Elements
Additive Inverse: For any \( a \), the number \( -a \) such that \( a + (-a) = 0 \).
Multiplicative Inverse: For any \( a \neq 0 \), the number \( \frac{1}{a} \) such that
\( a \times \frac{1}{a} = 1 \). Zero has no multiplicative inverse.
6. Commutative & Associative Properties
Commutative: \( a * b = b * a \) (order doesn't matter).
Associative: \( (a * b) * c = a * (b * c) \) (grouping doesn't matter).
Addition and multiplication are both commutative and associative on \( \mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R} \).
Subtraction and division are neither commutative nor associative.
