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# Algebra 2 - Simplifying Complex Numbers

## Complex Numbers

• We know that $\sqrt{25}=±5$, 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.
• $\sqrt{-25}=\sqrt{25×\left(-1\right)}=\sqrt{{\left(5\right)}^{2}×\left(-1\right)}=5\sqrt{-1}=5i$ 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}=$${i}^{4×8+1}=$${i}^{4×8}·i=$$\left(1\right)·i$$=i$

Example 2: Simplify ${i}^{2034}$.

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

Example 3: Simplify ${i}^{-87}$.

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

Example 4: Simplify $\frac{5}{i}$.

Solution: $\frac{5}{i}=$$5×\frac{1}{i}=$$5×\frac{{i}^{4}}{i}=$$5×{i}^{3}=$$5×\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 ${i}^{0}=1$.