Subtract { (5-2i)-(1+8i)$}$.Which Property Allows You To Write The Expression As ${ 5-2i-1-8i\$} ?A. Associative B. Commutative C. Distributive D. Identity What Is The Real Part Of The Difference? What Is The Imaginary Part Of

Introduction

In mathematics, complex numbers are a fundamental concept that extends the real number system to include numbers with both real and imaginary parts. When working with complex numbers, it's essential to understand the properties that govern their operations, such as addition, subtraction, multiplication, and division. In this article, we'll focus on subtracting complex numbers and explore the properties that allow us to simplify expressions.

Subtracting Complex Numbers

To subtract complex numbers, we need to understand the concept of complex number subtraction. The subtraction of two complex numbers is defined as the difference between their real parts and the difference between their imaginary parts. For example, given two complex numbers $a + bi$ and $c + di$, their difference is defined as:

$$(a + bi) - (c + di) = (a - c) + (b - d)i$$

Property of Subtraction

Now, let's consider the expression $(5-2i)-(1+8i)$. To simplify this expression, we can use the property of subtraction that allows us to write the expression as $5-2i-1-8i$. This property is known as the Commutative Property of Addition.

The commutative property of addition states that the order of the numbers being added does not change the result. In other words, for any two numbers $a$ and $b$, we have:

$$a + b = b + a$$

In the context of complex numbers, this property can be applied to the subtraction of complex numbers as follows:

$$(5-2i)-(1+8i) = 5-2i-1-8i$$

Real and Imaginary Parts

To find the real part of the difference, we need to subtract the real parts of the two complex numbers:

$$5 - 1 = 4$$

To find the imaginary part of the difference, we need to subtract the imaginary parts of the two complex numbers:

$$-2i - 8i = -10i$$

Therefore, the real part of the difference is $4$, and the imaginary part of the difference is $-10i$.

Conclusion

In conclusion, when subtracting complex numbers, we can use the commutative property of addition to simplify the expression. By applying this property, we can rewrite the expression as the difference between the real parts and the difference between the imaginary parts. This allows us to find the real and imaginary parts of the difference, which is essential in many mathematical applications.

Answer to the Question

The correct answer to the question is B. Commutative. The commutative property of addition allows us to write the expression $(5-2i)-(1+8i)$ as $5-2i-1-8i$.

Properties of Complex Numbers

In addition to the commutative property of addition, there are several other properties of complex numbers that are essential to understand. These properties include:

• Associative Property of Addition: The associative property of addition states that the order in which we add three or more numbers does not change the result. For example, for any three numbers $a$, $b$, and $c$, we have:

  $$(a + b) + c = a + (b + c)$$

• Distributive Property: The distributive property states that the multiplication of a complex number by a sum of two or more complex numbers can be distributed to each of the complex numbers. For example, for any complex number $a + bi$ and any two complex numbers $c + di$ and $e + fi$, we have:

  $$(a + bi)(c + di) = ac + adi + bci - bd$$

• Identity Property: The identity property states that there exists a complex number that does not change the result when added to or multiplied by another complex number. For example, for any complex number $a + bi$, we have:

  $$(a + bi) + 0 = a + bi$$

  $$(a + bi) \cdot 1 = a + bi$$

Real-World Applications

Complex numbers have numerous real-world applications in various fields, including:

• 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.
• Navigation: Complex numbers are used to represent positions and velocities in navigation systems.
• Computer Graphics: Complex numbers are used to represent 3D transformations and analyze their behavior.

Conclusion

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?

A: The imaginary unit, denoted by $i$, is a mathematical concept that is used to extend the real number system to include numbers with both real and imaginary parts. It is defined as the square root of $-1$, i.e., $i^2 = -1$.

Q: How do I add complex numbers?

A: To add complex numbers, we simply add the real parts and the imaginary parts separately. For example, given two complex numbers $a + bi$ and $c + di$, their sum is defined as:

$$(a + bi) + (c + di) = (a + c) + (b + d)i$$

Q: How do I subtract complex numbers?

A: To subtract complex numbers, we can use the commutative property of addition to simplify the expression. For example, given two complex numbers $a + bi$ and $c + di$, their difference is defined as:

$$(a + bi) - (c + di) = (a - c) + (b - d)i$$

Q: How do I multiply complex numbers?

A: To multiply complex numbers, we can use the distributive property to multiply the real and imaginary parts separately. For example, given two complex numbers $a + bi$ and $c + di$, their product is defined as:

$$(a + bi)(c + di) = ac + adi + bci - bd$$

Q: How do I divide complex numbers?

A: To divide complex numbers, we can use the concept of complex conjugates. The complex conjugate of a complex number $a + bi$ is defined as $a - bi$. To divide two complex numbers, we can multiply the numerator and denominator by the complex conjugate of the denominator. For example, given two complex numbers $a + bi$ and $c + di$, their quotient is defined as:

$$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}$$

Q: What is the modulus of a complex number?

A: The modulus of a complex number $a + bi$ is defined as the distance from the origin to the point $(a, b)$ in the complex plane. It is denoted by $|a + bi|$ and is calculated as:

$$|a + bi| = \sqrt{a^2 + b^2}$$

Q: What is the argument of a complex number?

A: The argument of a complex number $a + bi$ is defined as the angle between the positive real axis and the line segment joining the origin to the point $(a, b)$ in the complex plane. It is denoted by $\arg(a + bi)$ and is calculated as:

$$\arg(a + bi) = \tan^{-1}\left(\frac{b}{a}\right)$$

Q: What are the properties of complex numbers?

A: Complex numbers have several properties, including:

• Commutative Property of Addition: The order of the numbers being added does not change the result.
• Associative Property of Addition: The order in which we add three or more numbers does not change the result.
• Distributive Property: The multiplication of a complex number by a sum of two or more complex numbers can be distributed to each of the complex numbers.
• Identity Property: There exists a complex number that does not change the result when added to or multiplied by another complex number.

Q: What are the real-world applications of complex numbers?

A: Complex numbers have numerous real-world applications in various fields, including:

• 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.
• Navigation: Complex numbers are used to represent positions and velocities in navigation systems.
• Computer Graphics: Complex numbers are used to represent 3D transformations and analyze their behavior.

Conclusion

In conclusion, complex numbers are a fundamental concept in mathematics that extends the real number system to include numbers with both real and imaginary parts. They have several properties, including the commutative, associative, distributive, and identity properties. Complex numbers have numerous real-world applications in various fields, including electrical engineering, signal processing, navigation, and computer graphics.