1. Equations & Inequalities

Powers of i

1. Equations & Inequalities

Powers of i: Videos & Practice Problems

Topic summary

Understanding the powers of the imaginary unit $I^{1} is essential, as they cycle through four values: I^{1}, I^{2} = -1, I^{3} = -I, and I^{4} = 1 . To evaluate higher powers, determine if the$ exponent is divisible by 4. If so, the result is 1; otherwise, find the remainder to simplify calculations, such as $I^{22} = -1$ and $I^{67} = -I$ .

concept

Powers of i

Video duration:

Powers of i Video Summary

The imaginary unit $ i $ is defined as the square root of negative one, $ i = \sqrt{-1} $. When raising $ i $ to various powers, a pattern emerges that simplifies calculations significantly. Understanding the first few powers of $ i $ allows us to easily determine higher powers.

Starting with the basic powers:

$ i^1 = i $
$ i^2 = i \cdot i = \sqrt{-1} \cdot \sqrt{-1} = -1 $
$ i^3 = i^2 \cdot i = -1 \cdot i = -i $
$ i^4 = i^2 \cdot i^2 = (-1) \cdot (-1) = 1 $

Continuing this pattern, we can express higher powers of $ i $ in terms of these four results. For example:

$ i^5 = i^4 \cdot i = 1 \cdot i = i $
$ i^6 = i^4 \cdot i^2 = 1 \cdot (-1) = -1 $
$ i^7 = i^4 \cdot i^3 = 1 \cdot (-i) = -i $
$ i^8 = i^4 \cdot i^4 = 1 \cdot 1 = 1 $

This reveals a repeating cycle every four powers: $ i, -1, -i, 1 $. Therefore, any power of $ i $ can be simplified by finding the remainder when the exponent is divided by 4:

For any integer $ n $:

If $ n \mod 4 = 0 $, then $ i^n = 1 $
If $ n \mod 4 = 1 $, then $ i^n = i $
If $ n \mod 4 = 2 $, then $ i^n = -1 $
If $ n \mod 4 = 3 $, then $ i^n = -i $

This cyclical nature allows for quick calculations of any power of $ i $, making it a powerful tool in complex number arithmetic.

concept

Higher Powers of i

Video duration:

Higher Powers of i Video Summary

Understanding the powers of the imaginary unit $ i $ is essential in complex number calculations. The powers of $ i $ cycle through four distinct values: $ i $, $-1$, $-i$, and $1$. This cyclical nature allows for efficient computation of higher powers without the need to manually count through each power.

To simplify calculations, we can express any power of $ i $ in terms of $ i^4 $, which equals $ 1 $. For instance, to evaluate $ i^{20} $, we can rewrite it as $ (i^4)^5 $. Since $ i^4 = 1 $, it follows that $ i^{20} = 1^5 = 1 $.

For powers that are not multiples of four, such as $ i^{22} $, we can still utilize the cyclical pattern. We start by determining how many complete sets of $ i^4 $ fit into the exponent. In this case, $ i^{22} $ can be expressed as $ (i^4)^5 \cdot i^2 $. Since $ i^2 = -1 $, we find that $ i^{22} = 1 \cdot (-1) = -1 $.

To further streamline the process, we can determine the remainder when the exponent is divided by $ 4 $. If the exponent is evenly divisible by $ 4 $, the result is $ 1 $. For example, $ i^{100} $ can be evaluated by checking if $ 100 \div 4 $ has a remainder. Since $ 100 $ is divisible by $ 4 $, $ i^{100} = 1 $.

For exponents that yield a remainder, we can find the equivalent power of $ i $ based on the remainder. For instance, when calculating $ i^{67} $, dividing $ 67 $ by $ 4 $ gives a quotient of $ 16 $ and a remainder of $ 3 $. Thus, $ i^{67} = i^3 $. Knowing that $ i^3 = -i $, we conclude that $ i^{67} = -i $.

In summary, to evaluate powers of $ i $, determine if the exponent is divisible by $ 4 $. If it is, the result is $ 1 $. If not, calculate the remainder and use it to find the corresponding power of $ i $ from the cycle: $ i^1 = i $, $ i^2 = -1 $, $ i^3 = -i $, and $ i^0 = 1 $. This method provides a quick and efficient way to handle complex exponentiation.

Problem

Simplify the power of $i$ .
$i^{1003}$

$i$

$-1$

$-i$

Problem

Simplify the power of $i$ .
$i^{85}$

$i$

-1

$-i$

Do you want more practice?

We have more practice problems on Powers of i

Here’s what students ask on this topic:

What is the imaginary unit 'i' and how do its powers cycle?

The imaginary unit 'i' is defined as the square root of -1. Its powers cycle through four values: i, -1, -i, and 1. This cyclical pattern occurs every four powers, making it easier to calculate higher powers of i. Specifically, i¹ = i, i² = -1, i³ = -i, and i⁴ = 1. Understanding this cycle allows us to simplify calculations involving powers of i, especially when dealing with large exponents. By recognizing this repeating pattern, we can express any power of i in terms of these four values, facilitating quick and efficient computation.

How can I calculate higher powers of 'i' using divisibility by 4?

To calculate higher powers of 'i', check if the exponent is divisible by 4. If it is, the result is 1, as i⁴ = 1. If not, divide the exponent by 4 to find the remainder. The remainder determines the equivalent lower power of i: a remainder of 1 corresponds to i, 2 to -1, and 3 to -i. For example, i²² has a remainder of 2 when divided by 4, so it equals -1. Similarly, i⁶⁷ has a remainder of 3, equating to -i. This method simplifies calculations by reducing the exponent to one of the four known values.

What is the pattern for powers of 'i' and how does it help in calculations?

The pattern for powers of 'i' is a cycle of four values: i, -1, -i, and 1. This pattern repeats every four powers, which helps simplify calculations for higher powers of 'i'. By recognizing this cycle, we can express any power of 'i' in terms of these four values, making it easier to compute large exponents. For instance, i²⁰ can be expressed as (i⁴)⁵ = 1. This cyclical nature allows us to quickly determine the value of 'i' raised to any power by checking divisibility by 4 and using the remainder to find the corresponding value.

How do you simplify i¹⁰⁰ using the powers of 'i' cycle?

To simplify i¹⁰⁰, use the cycle of powers of 'i'. First, check if 100 is divisible by 4. Since 100 divided by 4 equals 25 with no remainder, i¹⁰⁰ simplifies to 1, as i⁴ = 1. This method leverages the cyclical pattern of powers of 'i', which repeats every four powers: i, -1, -i, and 1. By recognizing this cycle, we can efficiently compute large powers of 'i' without manually cycling through each power, saving time and effort in calculations.

What is the remainder method for calculating powers of 'i'?

The remainder method for calculating powers of 'i' involves dividing the exponent by 4 and using the remainder to determine the equivalent lower power of 'i'. If the exponent is divisible by 4, the result is 1. If not, the remainder will be 1, 2, or 3, corresponding to i, -1, or -i, respectively. For example, dividing 22 by 4 gives a remainder of 2, so i²² equals -1. Similarly, dividing 67 by 4 gives a remainder of 3, so i⁶⁷ equals -i. This method simplifies calculations by reducing the exponent to one of the four known values.