real number

(noun)

a value that represents a quantity along a continuous line

Related Terms

Examples of real number in the following topics:

  • 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.

  • Expressing Functions as Power Functions

    • A power function is a function of the form $f(x) = cx^r$ where $c$ and $r$ are constant real numbers.
    • A power function is a function of the form $f(x) = cx^r$ where $c$ and $r$ are constant real numbers.
    • The domain of a power function can sometimes be all real numbers, but generally a non-negative value is used to avoid problems with simplifying.
    • The Taylor series of a real or complex-valued function $f(x)$ that is infinitely differentiable in a neighborhood of a real or complex number $a$ is the power series:
    • Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial.

  • Convergence of Series with Positive Terms

    • For a sequence $\{a_n\}$, where $a_n$ is a non-negative real number for every $n$, the sum $\sum_{n=0}^{\infty}a_n$ can either converge or diverge to $\infty$.
    • For a sequence $\{a_n\}$, where $a_n$ is a non-negative real number for every $n$, the sequence of partial sums
    • It is possible to "visualize" its convergence on the real number line?

  • Intermediate Value Theorem

    • For a real-valued continuous function $f$ on the interval $[a,b]$ and a number $u$ between $f(a)$ and $f(b)$, there is a $c \in [a,b]$ such that $f(c)=u$.
    • Version I: If $f$ is a real-valued continuous function on the interval $[a, b]$, and $u$ is a number between $f(a)$ and $f(b)$, then there is a $c \in [a, b]$ such that $f(c) = u$.
    • Version 2: Suppose that $f : [a, b] \to R$ is continuous and that u is a real number satisfying $f(a) < u < f(b)$ or $f(a) > u > f(b)$.
    • Version 3: Suppose that $I$ is an interval $[a, b]$ in the real numbers $\mathbb{R}$ and that $f : I \to R$ is a continuous function.
    • The theorem depends on (and is actually equivalent to) the completeness of the real numbers.

  • Improper Integrals

    • An Improper integral is the limit of a definite integral as an endpoint of the integral interval approaches either a real number or $\infty$ or $-\infty$.
    • An improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or $\infty$ or $-\infty$ or, in some cases, as both endpoints approach limits.

  • Summing an Infinite Series

    • For any infinite sequence of real or complex numbers, the associated series is defined as the ordered formal sum
    • An infinite sequence of real numbers shown in blue dots.

  • Logarithmic Functions

    • The logarithm of a number is the exponent by which another fixed value must be raised to produce that number.
    • The logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number.
    • This definition assumes that we know exactly what we mean by 'raising a real positive number to a real power'.
    • Making this proviso, if the base b is any positive number except 1, and the number x is greater than zero, there is always a real number y that solves the equation: $b^{y} = x$ so the logarithm is well defined.
    • To define the logarithm, the base b must be a positive real number not equal to 1 and x must be a positive number.

  • Power Series

    • These power series arise primarily in real and complex analysis, but also occur in combinatorics (under the name of generating functions) and in electrical engineering (under the name of the $Z$-transform).
    • The familiar decimal notation for real numbers can also be viewed as an example of a power series, with integer coefficients, but with the argument $x$ fixed at $\frac{1}{10}$.
    • In number theory, the concept of $p$-adic numbers is also closely related to that of a power series.
    • If $c$ is not the only convergent point, then there is always a number $r$ with 0 < r ≤ ∞ such that the series converges whenever $\left| x-c \right| r$.
    • The number $r$ is called the radius of convergence of the power series.

  • Taylor Polynomials

    • The Taylor series of a real or complex-valued function $f(x)$ that is infinitely differentiable in a neighborhood of a real or complex number $a$ is the power series
    • Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial.

  • Precise Definition of a Limit

    • Suppose $f:R \rightarrow R$ is defined on the real line and $p,L \in R$.
    • For every real $\varepsilon > 0$, there exists a real $\delta > 0$ such that for all real $x$, $\varepsilon > 0$ $0 < \left | x-p \right | < \delta$ implies $\left | f(x) - L \right | < \varepsilon$.
    • For an arbitrarily small $\varepsilon$, there always exists a large enough number $N$ such that when $x$ approaches $N$, $\left | f(x)-L \right | < \varepsilon$.
    • For an arbitrarily small $\epsilon$, there always exists a large enough number $N$ such that when $x$ approaches $N$, $\left | f(x)-L \right | < \varepsilon$.