Introduction to Complex Numbers (DP IB Applications & Interpretation (AI): HL): Revision Note

Cartesian form

What is an imaginary number?

• Up until now, equations like $x^2=-1$ have “no real solutions”

  • The solutions are $x=\pm \sqrt{-1}$

    • but you cannot square root a negative number

• To extend this idea, mathematicians have defined one of the square roots of negative one as $\mathrm{i}$

  • This is called an imaginary number

    • $\sqrt{-1}=\mathrm{i}$

    • ${\mathrm{i}}^2=-1$

• The square roots of other negative numbers can be found by rewriting them as a multiple of  $\sqrt{-1}$

  • using $\sqrt{ab}=\sqrt{a}\times \sqrt{b}$

    • e.g. $\sqrt{-4}=\sqrt{4\times \left(-1\right)}=\sqrt{4}\sqrt{-1}=2\mathrm{i}$

What is a complex number?

• Complex numbers have both a real part and an imaginary part

  • For example: $3+4\mathrm{i}$

    • The real part is 3

    • The imaginary part is 4

    • The imaginary part does not include the '$\mathrm{i}$'

• Complex numbers are often denoted by $z$

  • The real part of $z$ is $\mathrm{Re}\left(z\right)$

  • The imaginary part of $z$ is $\mathrm{Im}\left(z\right)$

• The set of all complex numbers is given the symbol $\mathrm{ℂ}$

• Complex numbers can have one part

  • e.g. $5$ is purely real

    • and real numbers are a subset of complex numbers

  • e.g. $5\mathrm{i}$ is purely imaginary

    • and imaginary numbers are a subset of complex numbers

• Two complex numbers are equal if, and only if, both the real and imaginary parts are identical.

  • For example, $3+2\mathrm{i}$ and $3-2\mathrm{i}$ are not equal

What is Cartesian form?

• The form $z=a+b\mathrm{i}$ is known as Cartesian form

  • $a,b\in \mathrm{ℝ}$

• In general, for $z=a+b\mathrm{i}$

  • $\mathrm{Re}\left(z\right)=a$

  • $\mathrm{Im}\left(z\right)=b$