An introduction to adding, subtracting and multiplying complex numbers. [Algebra: Lesson 34 of 86]

A demonstration of calculating i, raised to arbitrarily high exponents. [Algebra: Lesson 32 of 86]

A demonstration of dividing complex numbers and complex conjugates. [Algebra: Lesson 35 of 86]

An introduction to i and imaginary numbers. [Algebra: Lesson 31 of 8]

An explanation of i as the principal square root of -l. [Algebra: Lesson 33 of 86]

This textbook covers topics such as Trigonometry and Right Angles, Circular Functions, Trigonometric Identities, Inverse Functions, Trigonometric Equations, Triangles and Vectors, as well as Polar Equations and Complex Numbers. It can also be used in conjunction with other directed courses in Mathematical Analysis or Linear Algebra as a full course in Precalculus. This digital textbook was reviewed for its alignment with California content standards.

This is a textbook for an introductory course in complex analysis. It has been used for the undergraduate complex analysis course at Georgia Tech and at a few other places.

This course is an introduction to complex analysis, or the theory of the analytic functions of a complex variable. Put differently, complex analysis is the theory of the differentiation and integration of functions that depend on one complex variable. Because of the algebraic properties of the complex numbers and the inherently geometric flavor of complex analysis, this course will feel quite different from Real Analysis, although many of the same concepts, such as open sets, metrics, and limits will reappear. Upon successful completion of this course, the student will be able to: manipulate complex numbers in various representations, define fundamental topological concepts in the context of the complex plane, and define and calculate limits and derivatives of functions of a complex variable; represent analytic functions as power series on their domains and verify that they are well-defined; define a branch of the complex logarithm; classify singularities and find Laurent series for meromorphic functions; state and prove fundamental results, including Cauchy’s Theorem and Cauchy’s Integral Formula, the Fundamental Theorem of Algebra, Morera’s Theorem and Liouville’s Theorem; use them to prove related results; calculate contour integrals; calculate definite integrals on the real line using the Residue Theorem; define linear fractional transformations and prove their essential characteristics; find the image of a region under a conformal mapping; state, prove, and use the Open Mapping Theorem. This free course may be completed online at any time. (Mathematics 243)

Revise and reinforce concepts and language associated with complex numbers.

This module sets out to instruct about complex numbers: what they are, what they mean, how to manipulate them, and the different ways to describe them (i.e. polar form). The second half of this module proposes to introduce the characteristics of complex vectors and matrices and how they compare to the laws governing standard vectors and matrices.

You may have met complex numbers before, but not had experience in manipulating them. This unit gives an accessible introduction to complex numbers, which are very important in science and technology, as well as mathematics. The unit includes definitions, concepts and techniques which will be very helpful and interesting to a wide variety of people with a reasonable background in algebra and trigonometry.

Study of ordinary differential equations, including modeling of physical problems and interpretation of their solutions. Standard solution methods for single first-order equations, including graphical and numerical methods. Higher-order forced linear equations with constant coefficients. Complex numbers and exponentials. Matrix methods for first-order linear systems with constant coefficients. Non-linear autonomous systems; phase plane analysis. Fourier series; Laplace transforms.

These are exercises to practice topics covered in modules regarding complex numbers, vectors, matrices, complex functions, and complex differentiation.

A teacher's guide to complex numbers.

This module provides practice problems related to development of concepts related to complex numbers.

This module provides practice problems related to complex numbers.

This module provides practice problems related to i and imaginary numbers.

This module provides practice problems related to solving quadratics containing imaginary numbers.

This module provides sample problems related to developing concepts related to the square root of -1.