irrational number

Examples of irrational number in the following topics:

• Zeroes of Polynomial Functions With Rational Coefficients

  • In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.
  • A real number that is not rational is called irrational.
  • Irrational numbers include √2, π, and e.
  • The decimal expansion of an irrational number continues forever without repeating.
  • Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.

• Real Numbers, Functions, and Graphs

  • Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced.
  • The real numbers include all the rational numbers, such as the integer -5 and the fraction $\displaystyle \frac{4}{3}$, and all the irrational numbers such as $\sqrt{2}$ (1.41421356… the square root of two, an irrational algebraic number) and $\pi$ (3.14159265…, a transcendental number).
  • The real line can be thought of as a part of the complex plane, and correspondingly, complex numbers include real numbers as a special case.
  • An example is the function that relates each real number $x$ to its square: $f(x)= x^{2}$.
  • Here, the domain is the entire set of real numbers and the function maps each real number to its square.

• Zeroes of Polynomial Functions with Real Coefficients

  • A real number is any rational or irrational number, such as -5, 4/3, or even √2.
  • An example of a non-real number would be √-1.
  • Even though all polynomials have roots, not all roots are real numbers.

• Research Examples

  • Berk (1974) uses game theory (http://en.wikipedia.org/wiki/game_Theory) to suggest that even a panic in a burning theater can reflect rational calculation: If members of the audience decide that it is more rational to run to the exits than to walk, the result may look like an animal-like stampede without in fact being irrational.
  • In a series of empirical studies of assemblies of people, McPhail (1991) argues that crowds vary along a number of dimensions, and that traditional stereotypes of emotionality and unanimity often do not describe what happens in crowds.

• Radical Functions

  • If the square root of a number is taken, the result is a number which when squared gives the first number.
  • The cube root is the number which, when cubed, or multiplied by itself and then multiplied by itself again, gives back the original number.
  • If a root of a whole number is squared root, which is not itself the square of a rational number, the answer will have an infinite number of decimal places.
  • Such a number is described as irrational and is defined as a number which cannot be written as a rational number: $\displaystyle \frac {a}{b}$, where $a$ and $b$ are integers.
  • The result of taking the square root is written with the approximately equal sign because the result is an irrational value which cannot be written in decimal notation exactly.

• Natural Logarithms

  • The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828.
  • The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.

• Generalized Anxiety Disorder

  • Generalized anxiety disorder is characterized by chronic anxiety that is excessive, uncontrollable, and often irrational.
  • Generalized anxiety disorder (GAD) is characterized by chronic anxiety that is excessive, uncontrollable, often irrational, and disproportionate to the actual object of concern.
  • In order for GAD to be diagnosed, a person must experience excessive anxiety and worry—more days than not—for at least 6 months and about a number of events or activities (such as work or school performance).
  • One of the main characteristics of GAD is excessive, constant, often irrational worry that impedes the individual's normal daily functioning.

• Introduction to Rational Functions

  • Neither the coefficients of the polynomials, nor the values taken by the function, are necessarily rational numbers.
  • However, the adjective "irrational" is not generally used for functions.
  • Note that the function itself is rational, even though the value of $f(x)$ is irrational for all $x$.
  • Since this condition cannot be satisfied by a real number, the domain of the function is all real numbers.

• The Number e

  • The number $e$ is an important mathematical constant, approximately equal to $2.71828$.
  • The number $e$, sometimes called the natural number or Euler's number, is an important mathematical constant, approximately equal to 2.71828.
  • There are a number of different definitions of the number $e$.
  • The number $e$ is very important in mathematics, alongside $0, 1, i, \, \text{and} \, \pi.$ All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity, which (amazingly) states that $e^{i\pi}+1=0.$ Like the constant $\pi$, $e$ is irrational (it cannot be written as a  ratio of integers), and it is transcendental (it is not a root of any non-zero polynomial with rational coefficients).
  • One of the many places the number $e$ plays a role in mathematics is in the formula for compound interest.

• Obsessive-Compulsive Disorder

  • Usually, the person knows that such thoughts and urges are irrational and thus tries to suppress or ignore them, but has an extremely difficult time doing so.
  • They may also include such mental acts as counting, praying, or reciting something to oneself, as well as nervous rituals like rouching a doorknob or opening and closing a door a certain number of times before leaving a room.
  • The acts of those who have OCD may appear paranoid and potentially psychotic, or disconnected from reality; however, OCD sufferers generally recognize their obsessions and compulsions as irrational.
  • A person will tend to recognize the obsessions as idiosyncratic or irrational, but still must perform them.