complex

(adjective)

a number, of the form $a+bi$, where $a$ and $b$ are real numbers and $i$ is the square root of $-1$.

Related Terms

Examples of complex in the following topics:

  • Roots of Complex Numbers

  • Exponentials With Complex Arguments: Euler's Formula

  • Trigonometry and Complex Numbers: De Moivre's Theorem

  • Complex Logarithms

  • Complex Conjugates and Division

    • The complex conjugate of x + yi is x - yi, and the division of two complex numbers can be defined using the complex conjugate.
    • The complex conjugate of the complex number z = x + yi is defined as x - yi.
    • The reciprocal of a nonzero complex number $z = x + yi$ is given by
    • The division of two complex numbers is defined in terms of complex multiplication (described above) and real division.
    • Practice dividing complex numbers by multiplying both the numerator and denominator by the complex conjugate of the denominator

  • Introduction to Complex Numbers

    • Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
    • The complex number $a+bi$ can be identified with the point $(a,b)$.
    • Thus, for example, complex number $-2+3i$ would be associated with the point $(-2,3)$ and would be plotted in the complex plane as shown below.
    • 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.
    • The complex number $-2+3i$ is plotted in the complex plane, $2$ to the left on the real axis, and $3$ up on the imaginary axis.

  • What Happens When a Function Has a Complex Argument?

  • Complex Conjugates

    • The complex conjugate of the number $a+bi$ is $a-bi$.
    • The complex conjugate (sometimes just called the conjugate) of a complex number $a+bi$ is the complex number $a-bi$.
    • The symbol for the complex conjugate of $z$ is $\overline{z}$.
    • One important fact about conjugates is that whenever a complex number is a root of polynomial, its complex conjugate is a root as well.
    • Simplifying gives the two complex numbers $-1/2+(\sqrt{3}/2)i$ and $-1/2-(\sqrt{3}/2)i$ , which are complex conjugates of each other.

  • Addition, Subtraction, and Multiplication

    • Complex numbers are added by adding the real and imaginary parts of the summands.
    • Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtained by building a parallelogram, three of whose vertices are O, A, and B (as shown in ).
    • 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

  • Addition and Subtraction of Complex Numbers

    • Complex numbers can be added and subtracted by adding the real parts and imaginary parts separately.
    • Complex numbers can be added and subtracted to produce other complex numbers.
    • In a similar fashion, complex numbers can be subtracted.
    • 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.