Problem 39 - Precalculus : Building Concepts and Connections

Question

Find \(x+y, x-y, x y,\) and \(x / y\). $$x=-3+5 i ; y=2-3 i$$

Step-by-Step Solution

Answer

The solutions are \(x+y=-1+2i, x-y=-5+8i, x*y=9+19i, x/y=9/13 + 19/13i\).

Step 1: Addition

Add the real parts and the imaginary parts of \(x\) and \(y\) separately. So, \(x+y= (-3+2) + (5-3)i = -1+2i

Step 2: Subtraction

Subtract the real parts and the imaginary parts of \(x\) and \(y\) separately. So, \(x-y= (-3-2) + (5+3)i = -5+8i

Step 3: Multiplication

Here we apply the distributive law for multiplication: \(x*y = (-3+5i)(2-3i) = -6+9i+10i-15i^2 = -6+9i+10i+15 = 9+19i\ (since i^2=-1)

Step 4: Division

Firstly, let's calculate the modulus of \(y\): \(|y|^2= 2^2 + (-3)^2 = 13\). To divide \(x\) by \(y\), we multiply numerator and denominator by the conjugate of \(y\): \(x/y=(-3+5i)/(2-3i) * (2+3i)/(2+3i)=(-6 + 9i + 15 + 10i) / 13 = 9/13 + 19/13i

Key Concepts

Complex Number Addition

Complex number addition is a fundamental operation that involves combining the real and imaginary parts of two complex numbers separately. Imagine you're making a fruit salad, where each type of fruit represents either the real or imaginary component. Just as you would add apples to apples and bananas to bananas, in complex number addition, you add like parts together. In our exercise, we have two complex numbers, \(x = -3 + 5i\) and \(y = 2 - 3i\). To find \(x + y\), combine the real parts, \(-3\) and \(2\), and the imaginary parts, \(5i\) and \(-3i\), individually.
This results in \(x+y = (-3+2) + (5-3)i = -1 + 2i\). Thus, by simply adding each component, the addition of complex numbers can be easily accomplished.

Complex Number Subtraction

Subtracting complex numbers is like distributing portions of a pie, but in this case, you're dealing with the real and imaginary slices separately. Subtraction mirrors addition, but involves taking away the corresponding parts of one complex number from the other. Following our textbook example, given \(x = -3 + 5i\) and \(y = 2 - 3i\), the subtraction \(x - y\) requires subtracting the real part of \(y\) from the real part of \(x\), and the imaginary part of \(y\) from the imaginary part of \(x\).
Proceed with \(x-y = (-3-2) + (5+3)i = -5 + 8i\), and voila! The subtraction of complex numbers is accomplished with ease, maintaining a clear distinction between real and imaginary components.

Complex Number Multiplication

Multiplying complex numbers is akin to cross-breeding plants—you must consider every possible interaction. The distributive law is your tool: each part of one complex number must be multiplied with each part of the other. For the complex numbers \(x = -3 + 5i\) and \(y = 2 - 3i\), we apply this law: \(xy = (-3+5i)(2-3i)\).
Execute the multiplication step by step: first, multiply the real parts; second, the real part of one with the imaginary part of the other; and don't forget, \(i^2 = -1\).
So, \(xy = -6 + 9i + 10i - 15i^2 = -6 + 19i + 15 = 9 + 19i\). Complex number multiplication can feel like a dance of numbers, gracefully keeping track of the real and imaginary steps.

Complex Number Division

Division of complex numbers might seem daunting, like juggling while balancing on a ball, but it's manageable when you follow the steps. When we divide \(x = -3 + 5i\) by \(y = 2 - 3i\), it's like sharing a meal but needing to adjust the portions so everyone gets an equal share. We're looking to keep the division clean, without any imaginary numbers in the denominator. This is achieved by multiplying both numerator and denominator by the conjugate of the denominator.
The conjugate of \(y\) is \(2 + 3i\), and we calculate \(x/y = (-3+5i)/(2-3i) * (2+3i)/(2+3i)\). Simplify this to get \((9+19i) / 13 = 9/13 + 19/13i\). This strategy, using the conjugate, makes complex number division a straightforward, orderly process.