Complex Number Subtraction: Z1 - Z3 - Z2 Explained

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Hey guys! Let's dive into the world of complex numbers and tackle a subtraction problem. We're going to figure out the result of the operation z1 - z3 - z2, where:

  • z1 = 2 + 3i
  • z2 = -1 + 2i
  • z3 = 1 - i

We'll break down each step so it's super clear how to subtract complex numbers. Grab your pencils, and let's get started!

Understanding Complex Numbers

Before we jump into the subtraction, let's quickly recap what complex numbers are all about. A complex number is basically a number that can be expressed in the form a + bi, where:

  • 'a' is the real part
  • 'b' is the imaginary part
  • 'i' is the imaginary unit, defined as the square root of -1 (i.e., i² = -1)

So, a complex number has two parts: a real number and an imaginary number. Examples of complex numbers include 2 + 3i, -1 + 2i, and 1 – i, which are the ones we'll be working with today!

Complex numbers are used in tons of fields, like electrical engineering, quantum mechanics, and even applied math. They might seem a bit abstract at first, but they're incredibly useful for solving problems that regular real numbers just can't handle. Think of them as an extension of the number line, opening up a whole new dimension of possibilities!

Step-by-Step Subtraction of Complex Numbers

Okay, let's get to the main event: subtracting complex numbers. We need to calculate z1 - z3 - z2. Here’s how we’ll do it, step-by-step:

Step 1: Substitute the Values

First, we'll plug in the given values of z1, z2, and z3 into our expression:

z1 - z3 - z2 = (2 + 3i) - (1 - i) - (-1 + 2i)

Step 2: Distribute the Negative Signs

Next, we need to distribute the negative signs in front of the parentheses. Remember, subtracting a complex number is like adding its negative:

(2 + 3i) - (1 - i) - (-1 + 2i) = 2 + 3i - 1 + i + 1 - 2i

Pay close attention to the signs here. A negative sign in front of a parenthesis changes the sign of each term inside the parenthesis. For example, -(1 - i) becomes -1 + i, and -(-1 + 2i) becomes +1 - 2i.

Step 3: Combine the Real Parts

Now, let's gather all the real parts together. These are the numbers without the 'i' attached:

Real parts: 2 - 1 + 1 = 2

So, when we combine the real parts, we get 2. Easy peasy!

Step 4: Combine the Imaginary Parts

Next, we'll combine the imaginary parts. These are the terms with the 'i' attached:

Imaginary parts: 3i + i - 2i = (3 + 1 - 2)i = 2i

So, when we combine the imaginary parts, we get 2i.

Step 5: Write the Result in a + bi Form

Finally, we'll put the real and imaginary parts together to get our final answer in the form a + bi:

Result: 2 + 2i

So, z1 - z3 - z2 = 2 + 2i.

Detailed Explanation of Complex Number Subtraction

Let’s take a closer look at what we did. Subtracting complex numbers involves treating the real and imaginary parts separately. This is because real numbers and imaginary numbers are different types of entities – you can’t directly combine them.

When you subtract (or add) complex numbers, you’re essentially performing two separate operations at the same time:

  1. Subtracting (or adding) the real parts.
  2. Subtracting (or adding) the imaginary parts.

This is why we grouped the real parts together (2 - 1 + 1) and the imaginary parts together (3i + i - 2i). Think of it like combining like terms in algebra. You can only combine terms that have the same variable (or in this case, the same imaginary unit 'i').

Why Does This Work?

This method works because complex numbers are defined in a way that keeps the real and imaginary parts distinct. When we perform arithmetic operations on complex numbers, we maintain this separation. This is consistent with the properties of complex numbers and ensures that our results are mathematically sound.

For example, let’s say we have two complex numbers, a + bi and c + di. When we subtract them, we get:

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

Notice how we subtract the real parts (a - c) and the imaginary parts (b - d) separately. This is exactly what we did in our example problem.

Alternative Approach: Step-by-Step Pairwise Subtraction

Another way to approach this problem is to perform the subtraction in two steps:

  1. First, subtract z3 from z1.
  2. Then, subtract z2 from the result.

Let's try it out:

Step 1: Calculate z1 - z3

z1 - z3 = (2 + 3i) - (1 - i) = 2 + 3i - 1 + i = (2 - 1) + (3 + 1)i = 1 + 4i

Step 2: Calculate (z1 - z3) - z2

Now, we subtract z2 from the result we just obtained:

(1 + 4i) - (-1 + 2i) = 1 + 4i + 1 - 2i = (1 + 1) + (4 - 2)i = 2 + 2i

As you can see, we get the same result: 2 + 2i. This alternative approach can be helpful if you find it easier to break the problem down into smaller steps.

Common Mistakes to Avoid

When working with complex numbers, it’s easy to make a few common mistakes. Here are some things to watch out for:

  • Forgetting to Distribute the Negative Sign: This is a big one! Always make sure you distribute the negative sign correctly when subtracting complex numbers. For example, -(1 - i) is -1 + i, not -1 - i.
  • Combining Real and Imaginary Parts Incorrectly: Remember, you can only combine real parts with real parts and imaginary parts with imaginary parts. Don’t try to add a real number to an imaginary number directly.
  • Incorrectly Handling the Imaginary Unit 'i': Keep in mind that i² = -1. This is important when you’re multiplying complex numbers, but it doesn’t come into play when you’re just adding or subtracting them.
  • Sign Errors: Pay close attention to the signs of the real and imaginary parts. A simple sign error can throw off your entire calculation.

Double-checking your work can help you catch these mistakes before they become a problem.

Real-World Applications of Complex Number Subtraction

You might be wondering,