Irrational Number

Definition

An irrational number is a type of real number that cannot be expressed as a simple fraction, meaning it cannot be written in the form of \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\). These numbers have non-repeating, non-terminating decimal expansions, making them distinct from rational numbers. Common examples of irrational numbers include the square root of any prime number and the mathematical constant \(\pi\).

5 Must Know Facts For Your Next Test

1. Irrational numbers cannot be exactly represented as fractions, which distinguishes them from rational numbers.
2. Famous examples of irrational numbers include \(\sqrt{2}\) and \(\pi\), both of which have decimal expansions that go on forever without repeating.
3. When simplifying square roots, if the radicand (the number under the square root) is not a perfect square, the result will be an irrational number.
4. The set of irrational numbers is dense in the real numbers, meaning between any two real numbers, there exists an irrational number.
5. Irrational numbers are crucial in geometry and calculus, often appearing in calculations involving circles and certain functions.

Review Questions

• Compare and contrast rational and irrational numbers. What are some key differences?
  • Rational numbers can be expressed as a fraction of two integers, while irrational numbers cannot be written in this way. This means that rational numbers have decimal expansions that either terminate or repeat, whereas irrational numbers have non-terminating and non-repeating decimal expansions. For example, while \(\frac{1}{2}\) is rational with a decimal expansion of 0.5, \(\sqrt{2}\) is irrational because its decimal goes on infinitely without repeating.
• What role do irrational numbers play in simplifying square roots? Give an example.
  • When simplifying square roots, if the radicand is not a perfect square, the result will be an irrational number. For instance, simplifying \(\sqrt{8}\) involves factoring it into \(\sqrt{4} imes \sqrt{2}\), which simplifies to \(2\sqrt{2}\). Here, \(\sqrt{2}\) is an irrational number since its decimal form does not terminate or repeat.
• Evaluate how the existence of irrational numbers impacts mathematical concepts such as limits and continuity in calculus.
  • The presence of irrational numbers enriches mathematical analysis by allowing for more complex functions and limits. For example, when calculating limits involving trigonometric functions or exponential growth, irrational numbers frequently emerge. This complexity helps define continuity in functions where irrational values can approach certain points but never actually reach them. Thus, understanding irrational numbers is fundamental to grasping advanced concepts like limits and continuity in calculus.

Related terms

Rational Number: A rational number is a number that can be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\).

Square Root: The square root of a number \(x\) is a value that, when multiplied by itself, gives the original number \(x\).

Real Numbers: Real numbers include all the rational and irrational numbers, representing all possible values along the number line.