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Rational numbers are numbers that can be expressed as a quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. These numbers are denoted as fractions like \( \frac{a}{b} \), where \(a\) and \(b\) are integers and \(b eq 0\). Common examples include \( \frac{1}{2} \), \(-3\), and \(\frac{7}{4}\).
One of the distinguishing features of rational numbers is that they have either a terminating or repeating decimal form. For instance, \( \frac{1}{4} = 0.25 \) and \( \frac{1}{3} = 0.333... \) both demonstrate decimal patterns unique to rational numbers.

• Always expressed as a fraction of two integers
• Denominator cannot be zero
• Have terminating or repeating decimals

When rationalizing denominators, the goal is to eliminate radicals or irrational numbers from the denominator to make it a rational number, which simplifies calculations and expressions.

Irrational numbers are numbers that cannot be written as a simple fraction of two integers. These numbers have non-terminating, non-repeating decimal expansions, meaning that they go on forever without repeating any pattern. Classic examples of irrational numbers include \( \pi \), \( e \), and square roots of non-perfect squares such as \( \sqrt{2} \) and \( \sqrt{5} \).
These numbers are intriguing because they fill the gap between rational numbers on the number line, providing a dense and complete set of real numbers. While they can't be expressed as exact fractions, they play a crucial role in mathematics, especially in geometry and calculus.

• Cannot be expressed as a fraction of two integers
• Decimal form is non-terminating and non-repeating
• Include numbers like \( \sqrt{5} \), \( \pi \), and \( e \)

To rationalize a denominator containing an irrational number, multiply both the numerator and denominator by the irrational component, transforming it into a rational number.

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, \( \sqrt{16} = 4 \) because \( 4 \times 4 = 16 \). Square roots can be both rational and irrational, depending on whether the original number is a perfect square.
Perfect squares, like 4, 9, and 16, have whole numbers as their square roots, while non-perfect squares, such as 2, 3, or 5, have irrational square roots like \( \sqrt{2} \), \( \sqrt{3} \), and \( \sqrt{5} \). These irrational square roots can be essential in various mathematical operations, including geometry and algebra.

• Square root is a value that produces the original number when squared
• Perfect squares have rational square roots, while non-perfect squares have irrational square roots
• Irrational square roots are often encountered in roots of non-perfect squares

In the case of rationalizing denominators, square roots are used because multiplying a square root by itself converts an irrational number into a rational one, as shown when \( \sqrt{5} \times \sqrt{5} = 5 \).