Powers of i

Let's take a look at powers of i.

LaTeX: i^1=i $i^1=i$

LaTeX: i^2=-1 $i^2=-1$

LaTeX: i^3=-i $i^3=-i$

LaTeX: i^4=1 $i^4=1$

LaTeX: i^5=i $i^5=i$

LaTeX: i^6=-1 $i^6=-1$

LaTeX: i^7=-i $i^7=-i$

LaTeX: i^8=1 $i^8=1$

Powers of i cycle through the same four outcomes, $LaTeX: i,\:-1,\:-i\:,\:1$ $i,\:-1,\:-i\:,\:1$ . Thus $LaTeX: i^4=i^8=i^{12}=...=i^{4n}=1$ $i^4=i^8=i^{12}=...=i^{4n}=1$ for any whole number n. Any power of i with an exponent that is a multiple of 4 has a value of 1. As with real numbers, LaTeX: i^0=1. $i^0=1.$

This pattern allows us to simplify any power of i.

Examples:

Simplify $LaTeX: i^{103}$ $i^{103}$

Since 100 is a multiple of 4, $LaTeX: i^{100}=1$ $i^{100}=1$ .

Thus, $LaTeX: i^{103}=i^{100}\cdot i^3=1\cdot\left(-i\right)=-i$ $i^{103}=i^{100}\cdot i^3=1\cdot\left(-i\right)=-i$

Simplify $LaTeX: i^{-11}$ $i^{-11}$

Since 12 is a multiple of four, I can multiply by 1 in the form of $LaTeX: i^{12}$ $i^{12}$ .

$LaTeX: i^{-11}=i^{-11}\left(i^{12}\right)=i^{-11+12}=i^1=i$ $i^{-11}=i^{-11}\left(i^{12}\right)=i^{-11+12}=i^1=i$

Complex numbers can be confusing if you have never dealt with them before. You may want to read the ebook section on them, which is on pages 123-129 in your ebook before working on Homework 1.3 on the next page.