real numbers

Examples of real numbers 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.

• 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.

• 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

• 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.

• 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).

• 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 $i$.