Complex Number Math: Addition & Subtraction

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the fascinating world of complex numbers, specifically tackling some addition and subtraction problems. You know, those numbers that have both a real and an imaginary part? They might seem a bit tricky at first, but trust me, once you get the hang of them, they're a blast to work with. We've got a set of problems here, and your mission, should you choose to accept it, is to match each problem with its correct answer. So, grab your favorite thinking cap, maybe a snack, and let's get started on this mathematical adventure!

Understanding Complex Numbers: A Quick Refresher

Before we jump into the problems, let's do a super quick rundown of what complex numbers are all about. A complex number is generally written in the form a + bi, where 'a' is the real part and 'b' is the imaginary part. The 'i' stands for the imaginary unit, which is defined as the square root of -1 (i.e., $i = \sqrt{-1}$). When we add or subtract complex numbers, it's kinda like combining like terms in algebra. We simply add or subtract the real parts together and the imaginary parts together. It's all about grouping what's similar! For example, to add $(a + bi)$ and $(c + di)$, you do $(a+c) + (b+d)i$. For subtraction, it's $(a + bi) - (c + di) = (a-c) + (b-d)i$. Pretty straightforward, right? We'll be using these exact rules to solve the problems ahead. So, keep this in mind as we go through each one. It’s the key to unlocking the correct pairings.

Problem Set: Let's Get Our Math On!

Alright, here's the lineup. We've got six problems that involve adding and subtracting complex numbers. Your job is to figure out which answer belongs to which problem. Remember the rules we just went over: add the real parts, add the imaginary parts; subtract the real parts, subtract the imaginary parts. Don't let those negative signs or the 'i' throw you off – they're just part of the game!

The Problems:

a. $1-3 i-(4-8 i)$

b. $8-7 i+(-2+6 i)$

c. $2+5 i-(-1+2 i)$

d. $-8+2 i+(-3+i)$

e. $-5+8 i+(7+2 i)$

f. $-6+6 i-(-1+3 i)$

The Answers:

1. $-5+3 i$
2. $2+10 i$
3. $-3+5 i$
4. $-11+3 i$
5. $2+10 i$

Solving the Mysteries: Step-by-Step

Now for the fun part – solving these bad boys! Let's go through each problem one by one and find its match. This is where you'll really see the addition and subtraction rules in action. Keep your eyes peeled and your calculators (or just your brains!) ready. We're going to break down each calculation, explaining the steps so you can follow along and, hopefully, nail the matching game.

Problem a: $1-3 i-(4-8 i)$

First up, we have subtraction. Remember to distribute that negative sign to both terms inside the second parenthesis. So, we have $(1-3i) - (4-8i)$. This becomes $1 - 3i - 4 + 8i$. Now, let's group the real parts and the imaginary parts: $(1 - 4) + (-3i + 8i)$. Simplifying this gives us $-3 + 5i$. So, problem 'a' matches with answer 3. Boom! One down, five to go.

Problem b: $8-7 i+(-2+6 i)$

This one is an addition. We just add the corresponding parts. So, $(8-7i) + (-2+6i)$. Grouping the real parts: $(8 + (-2))$. Grouping the imaginary parts: $(-7i + 6i)$. Adding them up gives us $6 + (-1i)$, which simplifies to $6 - i$. Uh oh, looking at the answers, I don't see a direct match for $6-i$. Let me recheck my calculation for 'b'. $(8 + (-2)) = 6$. And $(-7i + 6i) = -1i$. Okay, let me re-evaluate the provided answers. It seems there might be a typo in the original prompt, either in the problem or the answers. Let's proceed assuming the calculation is correct, and we'll address any discrepancies later. Correction: Upon re-examining the answers provided in the prompt, I notice that answer '2' is $2+10i$ and answer '5' is also $2+10i$. This indicates a possible error in the question's answer key. Let's recalculate problem 'b' very carefully. $8-7i + (-2+6i) = (8 + (-2)) + (-7i + 6i) = 6 - 1i$. My calculation remains $6-i$. It's possible the question meant to produce one of the given answers. For instance, if the problem was $8-7i + (-6+17i)$, it would result in $2+10i$. Or if it was $8-7i + (-6+17i)$, that would be $(8-6) + (-7+17)i = 2+10i$. Given the options, and the repeated answer '2' and '5', let's assume there's a typo in problem 'b' and it should lead to one of the answers. However, based on the literal problem as written, $6-i$ is the correct answer, which isn't listed. For the purpose of completing the exercise as intended by the prompt structure, I will assume a mistake in the problem statement for 'b' and see if other problems fit perfectly. Let's proceed with the remaining problems and revisit this if needed.

Self-correction: Okay, let's double-check everything very meticulously. It's possible I made a simple arithmetic error. For problem b: $8 - 7i + (-2 + 6i)$. Real parts: $8 + (-2) = 6$. Imaginary parts: $-7i + 6i = -1i$. Result: $6 - i$. Still $6-i$. I will proceed under the assumption that there might be a typo in the problem set or the answers provided and aim to match the other problems perfectly. If problem 'b' does not match any, I'll note that. For now, I'm marking 'b' as unmatched based on the given options.

Problem c: $2+5 i-(-1+2 i)$

Another subtraction problem, guys! Distribute the negative: $(2+5i) - (-1+2i)$ becomes $2 + 5i + 1 - 2i$. Combine the real parts: $(2+1)$. Combine the imaginary parts: $(5i - 2i)$. This gives us $3 + 3i$. Checking the answers again... hmm, $3+3i$ isn't listed either. This is getting interesting! Let me re-calculate 'c' one more time to be absolutely sure. $2+5i - (-1+2i) = 2+5i+1-2i = (2+1) + (5-2)i = 3+3i$. It seems like there might be a consistent issue with the provided answer key or the problems themselves. Let's pause and re-read the instructions.