Solving Complex Numbers (70 + 22i) - (71 + 70i) A Step-by-Step Guide

Hey guys! Let's dive into the fascinating world of complex numbers and break down this problem step by step: (70 + 22i) - (71 + 70i). Complex numbers might seem intimidating at first, but trust me, they're super cool once you get the hang of them. In this comprehensive guide, we'll not only solve the problem but also explore the fundamental concepts behind complex number arithmetic. So, buckle up and let's get started!

Understanding Complex Numbers

Before we jump into the solution, let's quickly recap what complex numbers are. A complex number is essentially a combination of a real number and an imaginary number. It's written in the form a + bi, where a represents the real part, b represents the imaginary part, and i is the imaginary unit, defined as the square root of -1. Think of i as a special number that allows us to work with the square roots of negative numbers. For instance, the number 5 + 3i is a complex number where 5 is the real part and 3 is the imaginary part. Similarly, in the expression (70 + 22i) - (71 + 70i), we're dealing with two complex numbers: 70 + 22i and 71 + 70i. Understanding this basic structure is crucial for performing operations like addition, subtraction, multiplication, and division with complex numbers.

Complex numbers aren't just abstract mathematical concepts; they have a wide range of applications in various fields, including electrical engineering, quantum mechanics, and fluid dynamics. In electrical engineering, complex numbers are used to analyze alternating current (AC) circuits, where the voltage and current oscillate sinusoidally. In quantum mechanics, complex numbers are fundamental to the wave function, which describes the probability amplitude of a quantum particle. In fluid dynamics, complex numbers can simplify the analysis of two-dimensional fluid flows. So, mastering complex number arithmetic is not only essential for solving mathematical problems but also for understanding many real-world phenomena.

When you first encounter complex numbers, you might wonder why they're called "imaginary." The term can be a bit misleading because these numbers are not just figments of our imagination. They're a legitimate extension of the real number system that allows us to solve equations that have no solutions within the realm of real numbers. For example, the equation x² + 1 = 0 has no real solutions because the square of any real number is non-negative. However, if we introduce the imaginary unit i, we can solve this equation by setting x = ±i. The name "imaginary" is historical, but it's important to recognize that complex numbers are a powerful tool for solving a wide variety of mathematical and scientific problems.

Breaking Down the Problem: (70 + 22i) - (71 + 70i)

Now that we have a good grasp of complex numbers, let's tackle the problem at hand: (70 + 22i) - (71 + 70i). The key to subtracting complex numbers is to treat the real and imaginary parts separately. Think of it like combining like terms in algebra. We'll subtract the real parts from each other and the imaginary parts from each other. This is because complex numbers follow the same algebraic rules as real numbers when it comes to addition and subtraction. To make things even clearer, let’s rewrite the expression by grouping the real and imaginary terms together. This will help us visualize the subtraction process and avoid any potential confusion.

So, let's rewrite the expression as follows:

(70 + 22i) - (71 + 70i) = (70 - 71) + (22i - 70i)

See how we've grouped the real parts (70 and 71) and the imaginary parts (22i and 70i)? This makes the subtraction process much more straightforward. Now, we can perform the subtraction for each group separately. Remember, when subtracting complex numbers, the distributive property applies, which means we're essentially subtracting each component of the second complex number from the corresponding component of the first complex number. This is analogous to subtracting binomials in algebra, where we distribute the negative sign across the parentheses.

This step-by-step approach not only simplifies the calculation but also reinforces the underlying principles of complex number arithmetic. By separating the real and imaginary parts, we can apply the familiar rules of real number arithmetic to each component. This makes the entire process more intuitive and less prone to errors. Plus, this method works for any subtraction problem involving complex numbers, regardless of how complicated the numbers may seem. So, by mastering this technique, you'll be well-equipped to tackle a wide range of complex number problems.

Performing the Subtraction

Okay, let's get our hands dirty and perform the subtraction. We've already grouped the real and imaginary parts: (70 - 71) + (22i - 70i). Now it's just a matter of doing the arithmetic. First, let's subtract the real parts: 70 - 71. This gives us -1. Easy peasy, right? Now, let's move on to the imaginary parts: 22i - 70i. Think of i as a variable, like x. So, we're essentially subtracting 70 of something from 22 of the same thing. This gives us -48i. Remember, the imaginary unit i is just a placeholder that indicates the imaginary part of the complex number. So, we treat it as a variable during the subtraction process.

Now, let's put the real and imaginary parts back together. We have -1 from the real part subtraction and -48i from the imaginary part subtraction. Combining these, we get -1 - 48i. And there you have it! That's our final answer. The result of (70 + 22i) - (71 + 70i) is -1 - 48i. See, complex number subtraction isn't so scary after all. It's just a matter of breaking it down into smaller steps and treating the real and imaginary parts separately.

To recap, we subtracted the real parts (70 and 71) to get -1, and we subtracted the imaginary parts (22i and 70i) to get -48i. Then, we combined these results to form the complex number -1 - 48i. This step-by-step approach is crucial for accuracy and helps to avoid common mistakes, especially when dealing with more complex expressions. By practicing this method, you'll become more confident and proficient in handling complex number operations.

Expressing the Answer in the Form (a + bi)

Our final answer is -1 - 48i. Now, the question asks us to express our answer in the form (a + bi). Take a look at our answer: -1 - 48i. Does it already look like (a + bi)? You bet it does! In this case, a is -1 and b is -48. So, we've successfully expressed our answer in the required form. Sometimes, the answer might need a little rearranging to fit the (a + bi) format, but in this case, we're good to go. This standard form is important because it allows us to easily compare and perform further operations on complex numbers. It's like having a common language for complex numbers.

Why is this (a + bi) form so important? Well, it's the standard way complex numbers are represented, making it easier to perform other operations like addition, multiplication, and division. It also helps in visualizing complex numbers on a complex plane, where the real part (a) is plotted on the horizontal axis and the imaginary part (b) is plotted on the vertical axis. This geometric representation of complex numbers is incredibly useful in various applications, such as signal processing and electrical engineering. So, always remember to express your final answer in the (a + bi) form to ensure clarity and consistency.

So, to reiterate, -1 - 48i is already in the form (a + bi), where a = -1 and b = -48. We've successfully completed all the steps and have our final answer. Awesome job, guys! This exercise demonstrates the simplicity of complex number subtraction when approached methodically. Remember, breaking down complex problems into manageable steps is the key to success in mathematics.

Final Answer: -1 - 48i

So, after carefully working through the problem (70 + 22i) - (71 + 70i), we've arrived at our final answer: -1 - 48i. We successfully subtracted the complex numbers by treating the real and imaginary parts separately, and we expressed our answer in the standard form (a + bi). Remember, the key to success with complex numbers is to break down the problem into smaller, manageable steps. First, we understood the basics of complex numbers, then we broke down the subtraction problem, performed the subtraction, and finally, expressed the answer in the correct format. By following this systematic approach, you can confidently tackle any complex number problem that comes your way.

This wasn't so bad, was it? Complex numbers might seem a bit daunting at first, but with a clear understanding of the fundamentals and a step-by-step approach, you can conquer them. Keep practicing, and you'll become a complex number whiz in no time! And remember, mathematics is like learning a language; the more you practice, the more fluent you become. So, keep exploring, keep learning, and keep having fun with math!

In conclusion, the solution to the problem (70 + 22i) - (71 + 70i) is indeed -1 - 48i. We've demonstrated how to perform complex number subtraction and how to express the result in the standard (a + bi) form. We hope this comprehensive guide has been helpful and has shed some light on the world of complex numbers. Keep up the great work, and remember to always approach math problems with a positive attitude and a willingness to learn. You've got this!