Select All The Correct Answers.Which Expressions Are Equivalent To The Given Complex Number $45 + 2i$?A. $(13 + 4i) + (32 - 6i)$ B. $(2 + 8i) + 2(9 + 6i)(1 - I)$ C. $ ( 9 + 4 I ) + 2 ( 4 + 7 I ) ( 1 − 2 I ) (9 + 4i) + 2(4 + 7i)(1 - 2i) ( 9 + 4 I ) + 2 ( 4 + 7 I ) ( 1 − 2 I ) [/tex]

Mar 8, 2025 by ADMIN 291 views

**Equivalent Expressions of a Complex Number**

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Introduction

Complex numbers are mathematical expressions that consist of a real and an imaginary part. They are used to represent points in a two-dimensional plane and are essential in various fields, including algebra, geometry, and engineering. In this article, we will explore the concept of equivalent expressions of a complex number and examine three given expressions to determine which ones are equivalent to the complex number $45 + 2i$ .

Understanding Complex Numbers

A complex number is a number that can be expressed in the form $a + bi$ , where $a$ is the real part and $b$ is the imaginary part. The imaginary part is denoted by the symbol $i$ , which is defined as the square root of $-1$ . Complex numbers can be added, subtracted, multiplied, and divided, just like real numbers.

Equivalent Expressions

Two complex numbers are equivalent if they have the same real and imaginary parts. In other words, if $a + bi$ and $c + di$ are equivalent complex numbers, then $a = c$ and $b = d$ .

Expression A: $(13 + 4i) + (32 - 6i)$

To determine if this expression is equivalent to $45 + 2i$ , we need to simplify it by combining the real and imaginary parts.

$(13 + 4i) + (32 - 6i)$

$= (13 + 32) + (4i - 6i)$

$= 45 + (-2i)$

$= 45 - 2i$

This expression is not equivalent to $45 + 2i$ because the imaginary part is $-2i$ , not $2i$ .

Expression B: $(2 + 8i) + 2(9 + 6i)(1 - i)$

To simplify this expression, we need to follow the order of operations (PEMDAS):

$(2 + 8i) + 2(9 + 6i)(1 - i)$

$= (2 + 8i) + 2(9 + 6i - 9i - 6i^2)$

$= (2 + 8i) + 2(9 - 3i + 6)$

$= (2 + 8i) + 2(15 - 3i)$

$= (2 + 8i) + 30 - 6i$

$= (2 + 30) + (8i - 6i)$

$= 32 + 2i$

This expression is not equivalent to $45 + 2i$ because the real part is $32$ , not $45$ , and the imaginary part is $2i$ , not $2i$ .

Expression C: $(9 + 4i) + 2(4 + 7i)(1 - 2i)$

To simplify this expression, we need to follow the order of operations (PEMDAS):

$(9 + 4i) + 2(4 + 7i)(1 - 2i)$

$= (9 + 4i) + 2(4 - 8i + 7i - 14i^2)$

$= (9 + 4i) + 2(4 - i - 14(-1))$

$= (9 + 4i) + 2(4 - i + 14)$

$= (9 + 4i) + 2(18 - i)$

$= (9 + 4i) + 36 - 2i$

$= (9 + 36) + (4i - 2i)$

$= 45 + 2i$

This expression is equivalent to $45 + 2i$ because the real part is $45$ and the imaginary part is $2i$ .

Conclusion

In conclusion, only Expression C is equivalent to the complex number $45 + 2i$ . The other two expressions, A and B, are not equivalent because they have different real and imaginary parts.

Final Answer

The final answer is:

Expression A: $(13 + 4i) + (32 - 6i)$ is not equivalent to $45 + 2i$ .
Expression B: $(2 + 8i) + 2(9 + 6i)(1 - i)$ is not equivalent to $45 + 2i$ .
Expression C: $(9 + 4i) + 2(4 + 7i)(1 - 2i)$ is equivalent to $45 + 2i$ .

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Q: What is a complex number?

A: A complex number is a mathematical expression that consists of a real and an imaginary part. It is denoted by the form $a + bi$ , where $a$ is the real part and $b$ is the imaginary part.

Q: What is the difference between a real number and a complex number?

A: A real number is a number that can be expressed without any imaginary part, whereas a complex number is a number that has both a real and an imaginary part.

Q: How do you add complex numbers?

A: To add complex numbers, you add the real parts together and the imaginary parts together. For example, $(3 + 4i) + (2 + 5i) = (3 + 2) + (4i + 5i) = 5 + 9i$ .

Q: How do you multiply complex numbers?

A: To multiply complex numbers, you use the distributive property and the fact that $i^2 = -1$ . For example, $(2 + 3i)(4 + 5i) = 2(4 + 5i) + 3i(4 + 5i) = 8 + 10i + 12i + 15i^2 = 8 + 22i - 15 = -7 + 22i$ .

Q: What is the concept of equivalent expressions in complex numbers?

A: Equivalent expressions in complex numbers refer to expressions that have the same real and imaginary parts. In other words, if $a + bi$ and $c + di$ are equivalent complex numbers, then $a = c$ and $b = d$ .

Q: How do you determine if two complex numbers are equivalent?

A: To determine if two complex numbers are equivalent, you compare their real and imaginary parts. If the real parts are equal and the imaginary parts are equal, then the complex numbers are equivalent.

Q: What is the significance of equivalent expressions in complex numbers?

A: Equivalent expressions in complex numbers are significant because they allow us to simplify complex expressions and make them easier to work with. They also help us to identify patterns and relationships between complex numbers.

Q: Can you provide an example of equivalent expressions in complex numbers?

A: Yes, consider the complex numbers $3 + 4i$ and $5 + 6i$ . These two complex numbers are equivalent because they have the same real and imaginary parts.

Q: How do you use equivalent expressions in complex numbers to solve problems?

A: To use equivalent expressions in complex numbers to solve problems, you need to identify the equivalent expressions and then simplify them using the rules of complex number arithmetic. For example, if you have the expression $(2 + 3i) + (4 + 5i)$ , you can simplify it by finding the equivalent expression $(6 + 8i)$ .

Q: What are some common mistakes to avoid when working with equivalent expressions in complex numbers?

A: Some common mistakes to avoid when working with equivalent expressions in complex numbers include:

Not checking if the real and imaginary parts are equal before declaring two complex numbers equivalent.
Not using the correct rules of complex number arithmetic when simplifying expressions.
Not being careful when multiplying complex numbers.

Q: How do you check if two complex numbers are equivalent?

A: To check if two complex numbers are equivalent, you need to compare their real and imaginary parts. If the real parts are equal and the imaginary parts are equal, then the complex numbers are equivalent.

Q: Can you provide a step-by-step guide to checking if two complex numbers are equivalent?

A: Yes, here is a step-by-step guide to checking if two complex numbers are equivalent:

Write down the two complex numbers.
Compare the real parts of the two complex numbers.
Compare the imaginary parts of the two complex numbers.
If the real parts are equal and the imaginary parts are equal, then the complex numbers are equivalent.

Q: What are some real-world applications of equivalent expressions in complex numbers?

A: Some real-world applications of equivalent expressions in complex numbers include:

Electrical engineering: Equivalent expressions in complex numbers are used to analyze and design electrical circuits.
Signal processing: Equivalent expressions in complex numbers are used to filter and process signals.
Control systems: Equivalent expressions in complex numbers are used to design and analyze control systems.

Q: Can you provide an example of a real-world application of equivalent expressions in complex numbers?

A: Yes, consider the problem of designing a filter to remove noise from a signal. The filter can be represented as a complex number, and equivalent expressions in complex numbers can be used to simplify the filter design and make it easier to implement.