Algebra 2 - Simplifying Complex Numbers

Complex Numbers

• We know that $\sqrt{25}=\pm 5$, but what if we want to find the value of $\sqrt{-25}$. Is it possible to compute its value?
• The answer is YES. But, the result will be a complex number.
• $\sqrt{-25}=\sqrt{25\times \left(-1\right)}=\sqrt{{\left(5\right)}^2\times \left(-1\right)}=5\sqrt{-1}=5i$ where $i=\sqrt{-1}$ is also called an imaginary unit.

• In general, a complex number is represented as $z=a+ib$, where $a=Re\left\{z\right\}=$real part and $b=Im\left\{z\right\}=$imaginary part.

• An example of a complex number written in standard form is $z=2-3i$.

Powers of Imaginary Unit [i]

• We know that $i=\sqrt{-1}$.

• • $i^2=-1$
  • $i^3=-i$
  • $i^4=1$
  • $i^{4k}=1$, where $k$ is any integer.
  • $i^{4k+1}=i$
  • $i^{4k+2}=-1$
  • $i^{4k+3}=-i$

Solved Examples

Example 1: Simplify $i^{33}$.

Solution: $i^{33}=$$i^{4\times 8+1}=$$i^{4\times 8}·i=$$\left(1\right)·i$$=i$

 

Example 2: Simplify $i^{2034}$.

Solution: $i^{2034}=$$i^{4\times 508+2}=$$i^{4\times 508}·i^2=$$\left(1\right)·\left(-1\right)=$$-1$

 

Example 3: Simplify $i^{-87}$.

Solution: $i^{-87}=$$\frac{1}{i^{87}}=$$\frac{1}{i^{4\times 21+3}}=$$\frac{1}{i^{4\times 21}·i^3}=$$\frac{i^4}{\left(1\right)·i^3}=$$i$

 

Example 4: Simplify $\frac{5}{i}$.

Solution: $\frac{5}{i}=$$5\times \frac{1}{i}=$$5\times \frac{i^4}{i}=$$5\times i^3=$$5\times \left(-i\right)=$$-5i$

Cheat Sheet

• When 'i'  is raised to the power of any integer, the result can be 1, i, –1, or – i.
• Simplifying complex numbers means transforming the given complex number into the standard form $a+ib$.

Blunder Areas

• $i^0\ne i$; $i^0\ne 0$. Rather $i^0=1$.