Properties of Imaginary Unit

Oct 13, 2025 by windailys.com 28 views

Understanding the Problem

The problem asks us to find an equivalent expression for $i^{233}$ , where $i$ is the imaginary unit, defined as the square root of -1. This means that $i^2 = -1$ . We need to simplify the expression $i^{233}$ to one of the given options: A. 1, B. -1, C. $i$ , or D. $-i$ .

Properties of Imaginary Unit

Before we proceed, let's recall some properties of the imaginary unit $i$ :

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

Simplifying $i^{233}$

We can simplify $i^{233}$ by using the properties of the imaginary unit. Since $i^4 = 1$ , we can write $i^{233}$ as:

i^{233} = i^{4 \cdot 58 + 1} = (i^4)^{58} \cdot i^1 = 1^{58} \cdot i = i

Conclusion

Therefore, the expression equivalent to $i^{233}$ is $\boxed{C. i}$ .

Additional Examples

To further illustrate the concept, let's consider a few more examples:

$i^{234} = i^{4 \cdot 58 + 2} = (i^4)^{58} \cdot i^2 = 1^{58} \cdot (-1) = -1$
$i^{235} = i^{4 \cdot 58 + 3} = (i^4)^{58} \cdot i^3 = 1^{58} \cdot (-i) = -i$
$i^{236} = i^{4 \cdot 59} = (i^4)^{59} = 1^{59} = 1$

Final Thoughts

In conclusion, the expression equivalent to $i^{233}$ is $\boxed{C. i}$ . We simplified the expression using the properties of the imaginary unit, specifically the fact that $i^4 = 1$ . This technique can be applied to simplify expressions involving powers of $i$ .

Frequently Asked Questions

Q: What is the imaginary unit $i$ ? A: The imaginary unit $i$ is defined as the square root of -1, denoted by $i = \sqrt{-1}$ .
Q: What are the properties of the imaginary unit $i$ $i$ ? A: The properties of the imaginary unit $i$ $i$ are:
- $i^2 = -1$
- $i^3 = i^2 \cdot i = -i$
- $i^4 = (i^2)^2 = (-1)^2 = 1$
Q: How do you simplify expressions involving powers of $i$ ? A: To simplify expressions involving powers of $i$ , you can use the properties of the imaginary unit, specifically the fact that $i^4 = 1$ .

Understanding the Imaginary Unit

The imaginary unit $i$ is a fundamental concept in mathematics, particularly in algebra and calculus. It is defined as the square root of -1, denoted by $i = \sqrt{-1}$ . In this article, we will explore the properties of the imaginary unit and how to simplify expressions involving powers of $i$ .

Q&A: Imaginary Unit and Powers of i

Q: What is the imaginary unit $i$ ?

A: The imaginary unit $i$ is defined as the square root of -1, denoted by $i = \sqrt{-1}$ .

Q: What are the properties of the imaginary unit $i$ ?

A: The properties of the imaginary unit $i$ are: + $i^2 = -1$ + $i^3 = i^2 \cdot i = -i$ + $i^4 = (i^2)^2 = (-1)^2 = 1$ + $i^5 = i^4 \cdot i = 1 \cdot i = i$ + $i^6 = i^4 \cdot i^2 = 1 \cdot (-1) = -1$ + $i^7 = i^4 \cdot i^3 = 1 \cdot (-i) = -i$ + $i^8 = (i^4)^2 = 1^2 = 1$

Q: How do you simplify expressions involving powers of $i$ ?

A: To simplify expressions involving powers of $i$ , you can use the properties of the imaginary unit, specifically the fact that $i^4 = 1$ . For example, to simplify $i^{233}$ , you can write it as:

i^{233} = i^{4 \cdot 58 + 1} = (i^4)^{58} \cdot i^1 = 1^{58} \cdot i = i

Q: What is the pattern for powers of $i$ ?

A: The pattern for powers of $i$ is: + $i^1 = i$ + $i^2 = -1$ + $i^3 = -i$ + $i^4 = 1$ + $i^5 = i$ + $i^6 = -1$ + $i^7 = -i$ + $i^8 = 1$

Q: How do you handle negative powers of $i$ ?

A: To handle negative powers of $i$ , you can use the property that $i^{-n} = \frac{1}{i^n}$ . For example, to simplify $i^{-1}$ , you can write it as:

i^{-1} = \frac{1}{i^1} = \frac{1}{i}

Q: What is the relationship between $i$ and complex numbers?

A: The imaginary unit $i$ is a fundamental component of complex numbers. A complex number is a number of the form $a + bi$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: How do you add and subtract complex numbers?

A: To add and subtract complex numbers, you can use the following rules: + $(a + bi) + (c + di) = (a + c) + (b + d)i$ + $(a + bi) - (c + di) = (a - c) + (b - d)i$

Q: How do you multiply complex numbers?

A: To multiply complex numbers, you can use the following rule: + $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

Q: How do you divide complex numbers?

A: To divide complex numbers, you can use the following rule: + $\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$

Conclusion

In conclusion, the imaginary unit $i$ is a fundamental concept in mathematics, particularly in algebra and calculus. Understanding the properties of the imaginary unit and how to simplify expressions involving powers of $i$ is essential for working with complex numbers. We hope this Q&A article has provided you with a better understanding of the imaginary unit and powers of $i$ .