rational number

Examples of rational number in the following topics:

• Zeroes of Polynomial Functions With Rational Coefficients

  • Since q may be equal to 1, every integer is a rational number.
  • Moreover, any repeating or terminating decimal represents a rational number.
  • A real number that is not rational is called irrational.
  • Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.
  • Therefore zero is a rational number, but division by zero is undefined.

• The Intermediate Value Theorem

  • The theorem depends on (and is actually equivalent to) the completeness of the real numbers.
  • It is false for the rational numbers Q.
  • However there is no rational number x such that f(x) = 0, because √2 is irrational.
  • The function is defined for all real numbers x ≠ −2 and is continuous at every such point.
  • A graph of a rational function, .

• Introduction to Rational Functions

  • A rational function is one such that $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$; the domain of a rational function can be calculated.
  • Neither the coefficients of the polynomials, nor the values taken by the function, are necessarily rational numbers.
  • Note that every polynomial function is a rational function with $Q(x) = 1$.
  • Factorizing the numerator and denominator of rational function helps to identify singularities of algebraic rational functions.
  • Since this condition cannot be satisfied by a real number, the domain of the function is all real numbers.

• Rational Exponents

  • A rational exponent is a rational number that provides another method for writing roots.
  • For example, an $n$th root of a number $b$ is a number $x$ such that $x^n = b$.
  • where $b$ is a real number and the rational exponent $\frac{m}{n}$ is a fraction in lowest terms.
  • The following are rules for operations on numbers with rational exponents.
  • This expression can be rewritten using the rule for dividing numbers with rational exponents:

• Domains of Rational and Radical Functions

  • Rational and radical expressions have restrictions on their domains which can be found algebraically or graphically.
  • A rational expression is one which can be written as the ratio of two polynomial functions.
  • Despite being called a rational expression, neither the coefficients of the polynomials nor the values taken by the function are necessarily rational numbers.
  • The domain of a rational expression of is the set of all points for which the denominator is not zero.
  • To find the domain of a rational function, set the denominator equal to zero and solve.  

• Integer Coefficients and the Rational Zeros Theorem

  • In algebra, the Rational Zero Theorem, or Rational Root Theorem, or Rational Root Test, states a constraint on rational solutions (also known as zeros, or roots) of the polynomial equation
  • Since any integer has only a finite number of divisors, the rational root theorem provides us with a finite number of candidates for rational roots.
  • Once we have found all the rational zeros (and counted their multiplicity, for example, by dividing using long division), we know the number of irrational and complex roots.
  • We can use the Rational Root Test to see whether this root is rational.
  • Use the Rational Zeros Theorem to find all possible rational roots of a polynomial

• Simplifying, Multiplying, and Dividing Rational Expressions

  • Just like a fraction involving numbers, a rational expression can be simplified, multiplied, and divided.
  • As a first example, consider the rational expression $\frac { 3x^3 }{ x }$.
  • For a simple example, consider the following, where a rational expression is multiplied by a fraction of whole numbers:
  • We follow the same rules to multiply two rational expressions together.
  • Dividing rational expressions follows the same rules as dividing fractions.

• Rational Equations

  • A rational equation sets two rational expressions equal to each other and involves unknown values that make the equation true.
  • For an equation that involves two fractions or rational expressions, cross-multiplying is a helpful strategy for simplifying the equation or determining the value of a variable.
  • Notice that the rational expressions on both sides of the equal sign have the same denominator.
  • If you have a rational equation where the denominators on either side of the equation are the same, then their respective numerators must also be the same value, even though they might be expressed in different terms.

• Rational Algebraic Expressions

  • Here, we will show you a systematic method for finding least common denominators—a method that works with rational expressions just as well as it does with numbers.
  • For each of the denominators, we find all the prime factors—i.e., the prime numbers that multiply to give that number.
  • Similarly, any number whose prime factors include a 2, a 3, and a 5 will be a multiple of 30.
  • When applying this strategy to rational expressions, first look at the denominators of the two rational expressions and see if they are the same.
  • The rational expressions therefore become:

• Polynomial and Rational Functions as Models

  • Polynomial and rational functions are both relatively accurate and easy to use.
  • They can take on only a limited number of shapes and are particularly ill-suited to modeling asymptotes.
  • To deal with the asymptotic problems of polynomials, we also use rational functions:
  • A rational function is the ratio of two polynomial functions and has the following form:
  • Polynomials and rational functions are used for approximation in many everyday devices.