complex numbers

Examples of complex numbers in the following topics:

• Roots of Complex Numbers

• Trigonometry and Complex Numbers: De Moivre's Theorem

• 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.
  • The complex number $a+bi$ can be identified with the point $(a,b)$.
  • 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.

• 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.

• 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

• 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:
  • We then combine these to write our complex number in standard form.
  • Note that it is possible for two nonreal complex numbers to multiply together to be a real number.

• 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.
  • Specifically, conjugating twice gives the original complex number: z** = z .
  • Moreover, a complex number is real if and only if it equals its conjugate.
  • The reciprocal of a nonzero complex number $z = x + yi$ is given by

• Division of Complex Numbers

  • Division of complex numbers is accomplished by multiplying by the multiplicative inverse of the denominator.
  • We have seen how to add, subtract, and multiply complex numbers, but it remains to learn how to divide them.
  • For complex numbers, the multiplicative inverse can be deduced using the complex conjugate.
  • We have already seen that multiplying a complex number $z=a+bi$ with its complex conjugate $\overline{z}=a-bi$ gives the real number $a^2+b^2$.
  • Suppose you wanted to divide the complex number $z=2+3i$ by the number $1+2i$.

• Complex Numbers in Polar Coordinates

  • Complex numbers can be represented in polar coordinates using the formula $a+bi=re^{i\theta}$.
  • This leads to a way to visualize multiplying and dividing complex numbers geometrically.
  • Sometimes it is helpful to think of complex numbers in a different geometric way.
  • When written this way, it now becomes easier to multiply and divide complex numbers.
  • Explain how to represent complex numbers in polar coordinates and why it is useful to do so

• 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 $a-bi$.
  • Since the conjugate of a conjugate is the original complex number, we say that the two numbers are conjugates of each other.
  • The number $\sqrt{a^2+b^2}$ is called the length or the modulus of the complex number $z=a+bi$.
  • Explain how to find a complex number's conjugate and what it is used for