Real Numbers

Rational numbers are numbers that can

Rational numbers are numbers that can be expressed in the form p/q, where p and q are integers and q ≠ 0. Examples include 2/3, -5/7, 4, and 0. The de

Irrational numbers are real numbers

Irrational numbers are real numbers that cannot be expressed as p/q (where p and q are integers with q ≠ 0). Examples include √2, √3, π, and (2 + √5).

We prove √2 is irrational using

We prove √2 is irrational using contradiction: Assume √2 = p/q (in reduced form). Then 2q² = p², meaning p² is even, so p is even. Let p = 2t, then 2q

A surd is an irrational root

A surd is an irrational root of a positive rational number. For example, √7, ∛5, and ⁴√8 are surds because their values are irrational. However, √4 =

A surd is in simplest form

A surd is in simplest form when the radicand has no perfect square factors. For example: √48 = √(16 × 3) = √16 × √3 = 4√3. Similarly, √98 = √(49 × 2)