1. Equations and Inequalities

Powers of i

1. Equations and Inequalities

Powers of i: Videos & Practice Problems

Topic summary

Understanding the powers of the imaginary unit i reveals a repeating cycle of four values: i, -1, -i, and 1. To simplify calculations for higher powers, determine if the exponent is divisible by 4. If yes, the result is 1. If not, find the remainder when divided by 4 to identify the corresponding value in the cycle. For example, i²² yields -1, while i⁶⁷ results in -i. This method streamlines evaluating complex powers of i.

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Powers of i

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Powers of i Video Summary

The imaginary unit $ i $ is defined as the square root of negative one, which leads to interesting properties when raised to various powers. Understanding these powers is essential for simplifying complex expressions in mathematics.

Starting with the basic powers of $ i $, we find:

1. $ i^1 = i $

2. $ i^2 = \sqrt{-1}^2 = -1 $

3. $ i^3 = i^2 \cdot i = -1 \cdot i = -i $

4. $ i^4 = (i^2)^2 = (-1)^2 = 1 $

From these calculations, a pattern emerges. The powers of $ i $ cycle every four terms:

- $ i^1 = i $

- $ i^2 = -1 $

- $ i^3 = -i $

- $ i^4 = 1 $

Continuing this pattern, we can express higher powers of $ i $ in terms of these four values. 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)^2 = 1^2 = 1 $

This cyclical nature means that any power of $ i $ can be simplified by finding the remainder when the exponent is divided by 4. The results will always correspond to one of the four values: $ i $, $ -1 $, $ -i $, or $ 1 $. For instance, to find $ i^{100} $, we calculate $ 100 \mod 4 $, which gives a remainder of 0, indicating that $ i^{100} = 1 $.

In summary, the powers of the imaginary unit $ i $ repeat every four terms, allowing for straightforward simplification of even very high powers. This understanding is crucial for working with complex numbers and their applications in various mathematical contexts.

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Higher Powers of i

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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 multiply through the cycle.

To simplify calculations, we can express any power of $ i $ in terms of $ i^4 $, which equals $ 1 $. For instance, when calculating $ i^{20} $, we recognize that $ 20 $ is a multiple of $ 4 $ (specifically, $ 4 \times 5 $), leading to the conclusion that $ i^{20} = (i^4)^5 = 1^5 = 1 $.

For powers that are not multiples of $ 4 $, such as $ i^{22} $, we can determine the result by finding the remainder when the exponent is divided by $ 4 $. Dividing $ 22 $ by $ 4 $ gives a quotient of $ 5 $ and a remainder of $ 2 $. Thus, $ i^{22} $ can be expressed as $ i^2 $, which equals $-1$. Therefore, $ i^{22} = -1 $.

To generalize this process, when faced with a power of $ i $, first check if the exponent is evenly divisible by $ 4 $. If it is, the result is $ 1 $. If not, divide the exponent by $ 4 $ and note the remainder. The possible remainders are $ 0 $, $ 1 $, $ 2 $, or $ 3 $, corresponding to the values $ 1 $, $ i $, $-1$, and $-i$ respectively. For example, for $ i^{67} $, dividing $ 67 $ by $ 4 $ yields a remainder of $ 3 $, leading to $ i^{67} = i^3 = -i $.

This method of using the remainder when dividing by $ 4 $ streamlines the process of calculating higher powers of $ i $, making it a valuable tool in complex number arithmetic.

Problem

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

$i$

$-1$

$-i$

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

$i$

-1

$-i$

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Here’s what students ask on this topic:

What are the powers of i and how do they repeat?

The powers of the imaginary unit i follow a repeating cycle of four values: i, -1, -i, and 1. Specifically, i¹ = i, i² = -1, i³ = -i, and i⁴ = 1. This pattern repeats every four powers. For example, i⁵ = i, i⁶ = -1, i⁷ = -i, and i⁸ = 1. Understanding this cycle helps simplify calculations for higher powers of i.

How do you calculate i to the power of 100?

To calculate i¹⁰⁰, you can use the repeating cycle of the powers of i. First, determine if 100 is divisible by 4. Since 100 ÷ 4 = 25 with no remainder, i¹⁰⁰ corresponds to i^4k where k is an integer. Since i⁴ = 1, i¹⁰⁰ = 1.

What is the value of i to the power of 67?

To find i⁶⁷, divide 67 by 4 and find the remainder. 67 ÷ 4 = 16 with a remainder of 3. Therefore, i⁶⁷ is equivalent to i³. Since i³ = -i, i⁶⁷ = -i.

How do you simplify higher powers of i?

To simplify higher powers of i, determine if the exponent is divisible by 4. If it is, the result is 1. If not, divide the exponent by 4 and use the remainder to find the corresponding value in the cycle. For example, i²² has a remainder of 2 when divided by 4, so i²² = i² = -1.