Which Values Of $x$ And $y$ Would Make The Following Expression Represent A Real Number?$(4+5i)(x+yi)$A. $ X = 4 , Y = 5 X=4, Y=5 X = 4 , Y = 5 [/tex]B. $x=-4, Y=0$C. $x=4, Y=-5$D. $ X = 0 , Y = 5 X=0, Y=5 X = 0 , Y = 5 [/tex]

Introduction

In mathematics, complex numbers are a fundamental concept that extends the real number system to include numbers with both real and imaginary parts. The expression $(4+5i)(x+yi)$ represents the product of two complex numbers, where $i$ is the imaginary unit. In this article, we will explore the conditions under which this expression represents a real number.

Understanding Complex Numbers

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 the equation $i^2 = -1$. The real part of a complex number is $a$, and the imaginary part is $b$. Complex numbers can be added, subtracted, multiplied, and divided, just like real numbers.

The Expression $(4+5i)(x+yi)$

The expression $(4+5i)(x+yi)$ represents the product of two complex numbers. To simplify this expression, we can use the distributive property of multiplication over addition:

$$(4+5i)(x+yi) = 4x + 4yi + 5xi + 5yi^2$$

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

$$4x + 4yi + 5xi - 5y$$

Combining like terms, we get:

$$(4x - 5y) + (4y + 5x)i$$

Real Number Representation

For the expression $(4x - 5y) + (4y + 5x)i$ to represent a real number, the imaginary part must be equal to zero. This means that:

$$4y + 5x = 0$$

Solving for $x$ and $y$

We can solve for $x$ and $y$ by rearranging the equation:

$$5x = -4y$$

$$x = -\frac{4}{5}y$$

This equation represents a line in the $xy$-plane. To find the values of $x$ and $y$ that satisfy this equation, we can substitute different values of $y$ into the equation.

Checking the Options

Let's check the options to see if they satisfy the equation:

A. $x = 4, y = 5$

Substituting these values into the equation, we get:

$$5(4) = -4(5)$$

$$20 = -20$$

This is not true, so option A is not correct.

B. $x = -4, y = 0$

Substituting these values into the equation, we get:

$$5(-4) = -4(0)$$

$$-20 = 0$$

This is not true, so option B is not correct.

C. $x = 4, y = -5$

Substituting these values into the equation, we get:

$$5(4) = -4(-5)$$

$$20 = 20$$

This is true, so option C is correct.

D. $x = 0, y = 5$

Substituting these values into the equation, we get:

$$5(0) = -4(5)$$

$$0 = -20$$

This is not true, so option D is not correct.

Conclusion

$$$$$$

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 real and imaginary parts.

Q: What is the imaginary unit $i$?

A: The imaginary unit $i$ is a mathematical concept that satisfies the equation $i^2 = -1$. It is used to extend the real number system to include complex numbers.

Q: How do you add and subtract complex numbers?

A: To add complex numbers, you add the real parts and the imaginary parts separately. For example, $(a+bi) + (c+di) = (a+c) + (b+d)i$. To subtract complex numbers, you subtract the real parts and the imaginary parts separately. For example, $(a+bi) - (c+di) = (a-c) + (b-d)i$.

Q: How do you multiply complex numbers?

A: To multiply complex numbers, you use the distributive property of multiplication over addition. For example, $(a+bi)(c+di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number $a+bi$ is $a-bi$. The conjugate of a complex number is used to simplify expressions and to eliminate imaginary parts.

Q: How do you divide complex numbers?

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

Q: What is the modulus of a complex number?

A: The modulus of a complex number $a+bi$ is $\sqrt{a^2+b^2}$. The modulus of a complex number is used to measure its distance from the origin in the complex plane.

Q: What is the argument of a complex number?

A: The argument of a complex number $a+bi$ is the angle between the positive real axis and the line segment joining the origin to the point $(a,b)$ in the complex plane. The argument of a complex number is used to determine its position in the complex plane.

Q: Can you give an example of a complex number?

A: Yes, an example of a complex number is $3+4i$. This complex number has a real part of 3 and an imaginary part of 4.

Q: Can you give an example of a real number?

A: Yes, an example of a real number is 5. This real number has no imaginary part.

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

A: A complex number is a mathematical concept that extends the real number system to include numbers with both real and imaginary parts. A vector is a mathematical concept that represents a quantity with both magnitude and direction. While complex numbers and vectors share some similarities, they are distinct mathematical concepts.

Q: Can you give an example of a vector?

A: Yes, an example of a vector is $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$. This vector has a magnitude of $\sqrt{3^2+4^2} = 5$ and a direction of $53.13^\circ$ counterclockwise from the positive $x$-axis.