Irrational numbers

Definition

Irrational numbers are real numbers that cannot be expressed as a simple fraction or ratio of two integers. They have non-repeating, non-terminating decimal expansions, which means their decimal representation goes on forever without repeating any pattern. This characteristic sets them apart from rational numbers and connects them to various concepts such as decimal representations, the ordering of real numbers, and methods of proof.

5 Must Know Facts For Your Next Test

1. Irrational numbers include famous constants such as π (pi) and √2, which are widely recognized in mathematics.
2. The decimal expansion of an irrational number never repeats or terminates, making it impossible to express exactly as a fraction.
3. Irrational numbers play a crucial role in various mathematical fields, including geometry and calculus, due to their properties.
4. Between any two rational numbers, there are infinitely many irrational numbers, highlighting the density of irrationals in the real number line.
5. Proof by contradiction is often used to establish the irrationality of certain numbers, demonstrating that assuming they are rational leads to a logical inconsistency.

Review Questions

• How can you differentiate between rational and irrational numbers based on their decimal representations?
  • Rational numbers have decimal representations that either terminate or repeat a pattern, which allows them to be expressed as a fraction of two integers. In contrast, irrational numbers have non-terminating and non-repeating decimal expansions. For example, while 0.75 (a rational number) can be written as 3/4, √2 is irrational because its decimal form goes on indefinitely without repeating.
• Discuss the significance of irrational numbers within the real number system in relation to their density.
  • Irrational numbers are significant because they fill in the gaps between rational numbers on the real number line. No matter how close two rational numbers are, there are infinitely many irrational numbers between them. This characteristic illustrates the concept of density in real numbers; it emphasizes that both rational and irrational numbers coexist within the same continuum, enriching our understanding of numerical relationships.
• Evaluate how proof by contradiction can be applied to demonstrate that certain numbers are irrational and its implications in mathematics.
  • Proof by contradiction is an essential method in mathematics for establishing the irrationality of certain numbers. For instance, to show that √2 is irrational, one assumes it can be expressed as a fraction a/b in simplest form. Following this assumption leads to a contradiction regarding the evenness of integers involved. This method not only reinforces the understanding of irrationality but also emphasizes the logical structure and consistency required in mathematical proofs, influencing various areas including number theory and algebra.

Related terms

Rational Numbers:

Numbers that can be expressed as a ratio of two integers, where the denominator is not zero, such as 1/2 or -3.

Real Numbers: The set of all numbers that include both rational and irrational numbers, encompassing all the numbers on the number line.

Transcendental Numbers:

A type of irrational number that is not the root of any non-zero polynomial equation with rational coefficients, like π and e.