Calculate { (8+i)(-5+3i)$}$.3. Calculate { (3+2i)^2$}$.

Introduction

Complex numbers are an extension of the real number system, which includes both real and imaginary numbers. They are used to represent points in a two-dimensional plane and have numerous applications in mathematics, physics, and engineering. In this article, we will focus on calculating complex number expressions, specifically the products and squares of complex numbers.

Calculating Complex Number Products

2. Calculate {(8+i)(-5+3i)$}$

To calculate the product of two complex numbers, we can use the distributive property, which states that for any complex numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. We can also use the fact that $i^2 = -1$.

Let's start by calculating the product of $(8+i)$ and $(-5+3i)$.

{(8+i)(-5+3i) = 8(-5+3i) + i(-5+3i)$

Using the distributive property, we can expand the expression as follows:

[$= -40 + 24i - 5i + 3i^2$

Since $i^2 = -1$, we can substitute this value into the expression:

[$= -40 + 24i - 5i - 3$

Combining like terms, we get:

[$= -43 + 19i$

Therefore, the product of $(8+i)$ and $(-5+3i)$ is $-43+19i$.

3. Calculate [(3+2i)^2\$}

To calculate the square of a complex number, we can use the formula $(a+bi)^2 = a^2 + 2abi + (bi)^2$. We can also use the fact that $i^2 = -1$.

Let's start by calculating the square of $(3+2i)$.

[$(3+2i)^2 = 3^2 + 2(3)(2i) + (2i)^2$

Using the formula, we can expand the expression as follows:

[$= 9 + 12i + 4i^2$

Since $i^2 = -1$, we can substitute this value into the expression:

[$= 9 + 12i - 4$

Combining like terms, we get:

[$= 5 + 12i$

Therefore, the square of $(3+2i)$ is $5+12i$.

Conclusion

In this article, we have calculated two complex number expressions: the product of $(8+i)$ and $(-5+3i)$, and the square of $(3+2i)$. We have used the distributive property and the fact that $i^2 = -1$ to simplify the expressions. The results are $-43+19i$ and $5+12i$, respectively.

Applications of Complex Numbers

Complex numbers have numerous applications in mathematics, physics, and engineering. Some of the key applications include:

• Electrical Engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
• Signal Processing: Complex numbers are used to represent signals and analyze their frequency content.
• Control Systems: Complex numbers are used to analyze the stability and behavior of control systems.
• Navigation: Complex numbers are used to represent GPS coordinates and calculate distances and directions.

Real-World Examples

1. AC Circuits: Complex numbers are used to represent AC circuits and analyze their behavior. For example, a circuit with a resistance of 10 ohms and an inductance of 5 henries can be represented as a complex number $10 + 5i$.
2. Signal Processing: Complex numbers are used to represent signals and analyze their frequency content. For example, a signal with a frequency of 100 Hz and an amplitude of 10 can be represented as a complex number $10e^{i2\pi 100t}$.
3. Control Systems: Complex numbers are used to analyze the stability and behavior of control systems. For example, a control system with a gain of 2 and a time constant of 5 seconds can be represented as a complex number $2e^{-5t}$.

Conclusion

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Q&A: Complex Numbers

Q: What is a complex number?

A: A complex number is a number that can be expressed in the form $a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, which satisfies $i^2 = -1$.

Q: What is the imaginary unit $i$?

A: The imaginary unit $i$ is a mathematical concept that is used to extend the real number system to the complex number system. It is defined as the square root of $-1$, denoted by $i = \sqrt{-1}$.

Q: How do I add complex numbers?

A: To add complex numbers, you can simply add the real parts and the imaginary parts separately. For example, if you have two complex numbers $a+bi$ and $c+di$, their sum is $(a+c)+(b+d)i$.

Q: How do I subtract complex numbers?

A: To subtract complex numbers, you can simply subtract the real parts and the imaginary parts separately. For example, if you have two complex numbers $a+bi$ and $c+di$, their difference is $(a-c)+(b-d)i$.

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you can use the distributive property and the fact that $i^2 = -1$. For example, if you have two complex numbers $a+bi$ and $c+di$, their product is $(ac-bd)+(ad+bc)i$.

Q: How do I divide complex numbers?

A: To divide complex numbers, you can multiply the numerator and denominator by the conjugate of the denominator. For example, if you have two complex numbers $a+bi$ and $c+di$, their quotient is $\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}$.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number $a+bi$ is $a-bi$. It is denoted by $\overline{a+bi}$.

Q: How do I find the magnitude of a complex number?

A: The magnitude of a complex number $a+bi$ is given by $\sqrt{a^2+b^2}$. It is denoted by $|a+bi|$.

Q: How do I find the argument of a complex number?

A: The argument of a complex number $a+bi$ is given by $\tan^{-1}\left(\frac{b}{a}\right)$. It is denoted by $\arg(a+bi)$.

Q: What is the polar form of a complex number?

A: The polar form of a complex number $a+bi$ is given by $re^{i\theta}$, where $r = \sqrt{a^2+b^2}$ and $\theta = \tan^{-1}\left(\frac{b}{a}\right)$.

Q: How do I convert a complex number from rectangular form to polar form?

A: To convert a complex number from rectangular form to polar form, you can use the formulas $r = \sqrt{a^2+b^2}$ and $\theta = \tan^{-1}\left(\frac{b}{a}\right)$.

Q: How do I convert a complex number from polar form to rectangular form?

A: To convert a complex number from polar form to rectangular form, you can use the formulas $a = r\cos\theta$ and $b = r\sin\theta$.

Conclusion

In this article, we have answered some of the most frequently asked questions about complex numbers. We have covered topics such as adding, subtracting, multiplying, and dividing complex numbers, as well as finding the magnitude and argument of a complex number. We have also discussed the polar form of a complex number and how to convert between rectangular and polar forms. We hope that this article has been helpful in understanding complex numbers and how to work with them.