Irrational number
Definition and Properties
An irrational number is a real number that cannot be expressed as a ratio of two integers. In other words, it cannot be written in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \). The decimal expansion of an irrational number is non-terminating and non-repeating. This distinguishes them from rational numbers, whose decimal expansions either terminate or repeat periodically.
Historical Background
The concept of irrational numbers dates back to ancient Greece. The discovery is often attributed to the Pythagorean philosopher Hippasus, who demonstrated that the square root of 2 cannot be expressed as a ratio of two integers. This finding was revolutionary and controversial, as it challenged the Pythagorean belief that all numbers are rational.
Types of Irrational Numbers
Irrational numbers can be broadly classified into two types: algebraic and transcendental.
Algebraic Irrational Numbers
An algebraic irrational number is a root of a non-zero polynomial equation with rational coefficients that cannot be expressed as a ratio of integers. For example, the square roots of non-perfect squares, such as \( \sqrt{2} \) and \( \sqrt{3} \), are algebraic irrational numbers. These numbers satisfy polynomial equations like \( x^2 - 2 = 0 \) and \( x^2 - 3 = 0 \), respectively.
Transcendental Numbers
Transcendental numbers are irrational numbers that are not roots of any non-zero polynomial equation with rational coefficients. The most famous examples are π and e. The number π is the ratio of the circumference of a circle to its diameter, and e is the base of the natural logarithm. Both numbers have been proven to be transcendental, meaning they cannot be solutions to any algebraic equation with rational coefficients.
Mathematical Properties
Density
Irrational numbers are dense in the real number line. This means that between any two real numbers, there exists an irrational number. This property is shared with rational numbers, making both sets dense in the real numbers.
Cardinality
The set of irrational numbers is uncountably infinite, which means its cardinality is greater than that of the set of rational numbers, which is countably infinite. This was first proven by Georg Cantor using his famous diagonal argument.
Closure Properties
The set of irrational numbers is not closed under addition, subtraction, multiplication, or division. For example, the sum of \( \sqrt{2} \) and \( -\sqrt{2} \) is 0, a rational number. However, certain operations involving irrational numbers can still result in irrational numbers. For instance, the product of \( \sqrt{2} \) and \( \sqrt{3} \) is \( \sqrt{6} \), which is irrational.
Examples of Irrational Numbers
Square Roots
The most straightforward examples of irrational numbers are the square roots of non-perfect squares. For instance, \( \sqrt{2} \), \( \sqrt{3} \), and \( \sqrt{5} \) are all irrational. These numbers cannot be expressed as a ratio of two integers and have non-terminating, non-repeating decimal expansions.
Famous Constants
Two of the most well-known irrational numbers are π and e. The number π (approximately 3.14159) is the ratio of a circle's circumference to its diameter and is ubiquitous in mathematics and physics. The number e (approximately 2.71828) is the base of the natural logarithm and appears in various contexts, such as compound interest and the study of exponential growth and decay.
Golden Ratio
The golden ratio, often denoted by the Greek letter φ (phi), is another famous irrational number. It is defined as the positive solution to the quadratic equation \( x^2 - x - 1 = 0 \), which is \( \frac{1 + \sqrt{5}}{2} \). The golden ratio appears in various areas of mathematics, art, architecture, and nature.
Proofs of Irrationality
Square Root of 2
One of the most famous proofs of irrationality is that of \( \sqrt{2} \). Assume, for contradiction, that \( \sqrt{2} \) is rational, meaning it can be written as \( \frac{a}{b} \) in its lowest terms. Then \( \sqrt{2} = \frac{a}{b} \) implies \( 2 = \frac{a^2}{b^2} \), or \( 2b^2 = a^2 \). This means \( a^2 \) is even, so \( a \) must be even. Let \( a = 2k \). Substituting back, we get \( 2b^2 = (2k)^2 \), or \( 2b^2 = 4k^2 \), which simplifies to \( b^2 = 2k^2 \). Thus, \( b^2 \) is even, so \( b \) must be even. But this contradicts the assumption that \( \frac{a}{b} \) is in lowest terms. Hence, \( \sqrt{2} \) is irrational.
Transcendence of e and π
The proofs of the transcendence of e and π are more complex and involve advanced techniques from number theory and complex analysis. The transcendence of e was first proven by Charles Hermite in 1873, and the transcendence of π was proven by Ferdinand von Lindemann in 1882. These proofs showed that these numbers are not only irrational but also cannot be roots of any non-zero polynomial equation with rational coefficients.
Applications of Irrational Numbers
Irrational numbers play a crucial role in various fields of mathematics and science.
Geometry
In geometry, irrational numbers often appear as lengths of line segments that cannot be expressed as rational numbers. For example, the diagonal of a unit square is \( \sqrt{2} \), an irrational number. The golden ratio also appears in the proportions of various geometric shapes and figures.
Calculus
In calculus, the number e is fundamental in the study of exponential functions and logarithms. The function \( e^x \) is unique in that it is its own derivative, making it essential in solving differential equations and modeling growth and decay processes.
Physics
In physics, irrational numbers frequently appear in measurements and constants. For example, the period of a simple pendulum involves the square root of the length of the pendulum divided by the acceleration due to gravity, both of which can result in irrational numbers.