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.

• 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 $i$.