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Powers of i

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Powers of i

Study Guide

Key Definition

The imaginary unit $i$ is defined as $\sqrt{-1}$. The powers of $i$ repeat every four: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$.

Important Notes

The powers of $i$ repeat every four cycles.
$i^2 = -1$ is a fundamental property.
$i^3 = -i$ shows how powers build on each other.
$i^4 = 1$ resets the cycle.
Use the cycle to simplify powers of $i$.

Mathematical Notation

$i$ is the imaginary unit

$\sqrt{-1}$ is the definition of square root of $i$

$i^2 = -1$ represents the square of $i$

$i^3 = -i$ is $i$ cubed

$i^4 = 1$ resets to one

Remember to use proper notation when solving problems

Why It Works

The pattern of powers of $i$ simplifies calculations with complex numbers by repeating every four powers, which helps in reducing higher powers of $i$ to basic values.

Remember

The powers cycle: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$. Use this cycle to simplify expressions.

Quick Reference

Cycle Property:$i^n = i^{(n \mod 4)}$

Understanding Powers of i

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Simple explanation with the basic cycle: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is $i^3$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are coding a game that needs to calculate $i^6$ for special effects. How would you simplify this?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Can you find a pattern in the powers of $i$ to predict $i^{100}$?

Challenge Quiz

Single Choice Quiz

Advanced

What is $i^{23}$?

Recap

Watch & Learn

Review key concepts and takeaways