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Imaginary unit

From AoPSWiki

The imaginary unit, i=\sqrt{-1}, is the fundamental component of all complex numbers. In fact, it is a complex number itself. It has a magnitude of 1, and can be written as 1 \text{cis } \left(\frac{\pi}{2}\right). Any complex number can be expressed as a+bi for some real numbers a and b.

Contents

Trigonometric function cis

Main article: cis

The trigonometric function \text{cis } x is also defined as e^{ix} or \cos x + i\sin x.

Series

When i is used in an exponential series, it repeats at every four terms:

  1. i^1=\sqrt{-1}
  2. i^2=\sqrt{-1}\cdot\sqrt{-1}=-1
  3. i^3=-1\cdot i=-i
  4. i^4=-i\cdot i=-i^2=-(-1)=1
  5. i^5=1\cdot i=i

This has many useful properties.

Use in factorization

i is often very helpful in factorization. For example, consider the difference of squares: (a+b)(a-b)=a^2-b^2. With i, it is possible to factor the otherwise-unfactorisable a^2+b^2 into (a+bi)(a-bi).

Problems

Introductory

Intermediate

  • The equation z^6+z^3+1 has complex roots with argument \theta between 90^\circ and 180^\circ in the complex plane. Determine the degree measure of \theta. (Source)

Olympiad

  • Let A\in\mathcal M_2(R) and P\in R[X] with no real roots. If \det(P(A)) = 0 , show that P(A) = O_2. (Source)

See also

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