real number

(noun)

An element of the set of real numbers. The set of real numbers include the rational numbers and the irrational numbers, but not all complex numbers.

Related Terms

  • inequality
  • imaginary number
  • complex
  • root

Examples of real number in the following topics:

  • Interval Notation

    • Interval notation uses parentheses and brackets to describe sets of real numbers and their endpoints.
    • A "real interval" is a set of real numbers such that any number that lies between two numbers in the set is also included in the set.
    • Other examples of intervals include the set of all real numbers and the set of all negative real numbers.
    • An interval is said to be bounded if both of its endpoints are real numbers.
    • Representations of open and closed intervals on the real number line.
  • Introduction to Complex Numbers

    • A complex number has the form $a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.
    • A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number.
    • It is beneficial to think of the set of complex numbers as an extension of the set of real numbers.
    • Complex numbers allow for solutions to certain equations that have no real number solutions.
    • has no solution if we restrict ourselves to the real numbers, since the square of a real number is never negative.
  • Absolute Value

    • Absolute value can be thought of as the distance of a real number from zero.
    • In mathematics, the absolute value (sometimes called the modulus) of a real number $a$ is denoted $\left | a \right |$.
    • Therefore, $\left | a \right |>0$ for all numbers.
    • When applied to the difference between real numbers, the absolute value represents the distance between the numbers on a number line.
    • The absolute values of 5 and -5 shown on a number line.
  • Addition and Subtraction of Complex Numbers

    • Complex numbers can be added and subtracted by adding the real parts and imaginary parts separately.
    • Note that this is always possible since the real and imaginary parts are real numbers, and real number addition is defined and understood.
    • Note that it is possible for two non-real complex numbers to add to a real number.
    • However, two real numbers can never add to be a non-real complex number.
    • Calculate the sums and differences of complex numbers by adding the real parts and the imaginary parts separately
  • Multiplication of Complex Numbers

    • Any time an $i^2$ appears in a calculation, it can be replaced by the real number $-1.$
    • Two complex numbers can be multiplied to become another complex number.
    • Note that this last multiplication yields a real number, since:
    • Note that if a number has a real part of $0$, then the FOIL method is not necessary.
    • Note that it is possible for two nonreal complex numbers to multiply together to be a real number.
  • Zeros of Polynomial Functions with Real Coefficients

    • This section specifically deals with polynomials that have real coefficients.
    • A real number is any rational or irrational number, such as $-5$, $\frac {4}{3}$, or even $\sqrt 2$.
    • (An example of a non-real number would be $\sqrt -1$.)
    • Even though all polynomials have roots, not all roots are real numbers.
    • Some roots can be complex, but no matter how many of the roots are real or complex, there are always as many roots as there are powers in the function.
  • The Fundamental Theorem of Algebra

    • Some polynomials with real coefficients, like $x^2 + 1$, have no real zeros.
    • Every polynomial of odd degree with real coefficients has a real zero.
    • In particular, since every real number is also a complex number, every polynomial with real coefficients does admit a complex root.
    • The complex conjugate root theorem says that if a complex number $a+bi$ is a zero of a polynomial with real coefficients, then its complex conjugate $a-bi$ is also a zero of this polynomial.
    • Therefore, a polynomial of even degree admits an even number of real roots, and a polynomial of odd degree admits an odd number of real roots (counted with multiplicity).
  • Complex Conjugates and Division

    • Geometrically, z* is the "reflection" of z about the real axis (as shown in the figure below).
    • The real and imaginary parts of a complex number can be extracted using the conjugate, respectively:
    • Moreover, a complex number is real if and only if it equals its conjugate.
    • The division of two complex numbers is defined in terms of complex multiplication (described above) and real division.
    • Neither the real part c nor the imaginary part d of the denominator can be equal to zero for division to be defined.
  • Addition, Subtraction, and Multiplication

    • Complex numbers are added by adding the real and imaginary parts; multiplication follows the rule $i^2=-1$.
    • Complex numbers are added by adding the real and imaginary parts of the summands.
    • The multiplication of two complex numbers is defined by the following formula:
    • Addition of two complex numbers can be done geometrically by constructing a parallelogram.
    • Discover the similarities between arithmetic operations on complex numbers and binomials
  • Imaginary Numbers

    • There is no real value such that when multiplied by itself it results in a negative value.
    • This means that there is no real value of $x$ that would make $x^2 =-1$ a true statement.
    • That is where imaginary numbers come in.
    • Specifically, the imaginary number, $i$, is defined as the square root of -1: thus, $i=\sqrt{-1}$.
    • We can write the square root of any negative number in terms of $i$.
Subjects
  • Accounting
  • Algebra
  • Art History
  • Biology
  • Business
  • Calculus
  • Chemistry
  • Communications
  • Economics
  • Finance
  • Management
  • Marketing
  • Microbiology
  • Physics
  • Physiology
  • Political Science
  • Psychology
  • Sociology
  • Statistics
  • U.S. History
  • World History
  • Writing

Except where noted, content and user contributions on this site are licensed under CC BY-SA 4.0 with attribution required.