Simplifying Complex Numbers: Finding A + Bi For (1 + 2i) / (1 - I)

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Hey everyone, let's dive into the world of complex numbers! Today, we're going to figure out how to express the complex number (1 + 2i) / (1 - i) in the standard form a + bi. This form is super important because it helps us understand the real and imaginary parts of a complex number. Don't worry, it's not as scary as it sounds. We'll break it down step-by-step, making sure you understand every move.

Understanding Complex Numbers and the a + bi Form

So, what exactly are complex numbers? Well, they're numbers that have a real part and an imaginary part. They're written in the form a + bi, where:

  • a is the real part (a regular, everyday number).
  • b is the imaginary part (also a regular number), and it's multiplied by i.
  • i is the imaginary unit, defined as the square root of -1. Yes, you read that right! This allows us to deal with the square roots of negative numbers, which is pretty cool.

Our goal here is to take the complex number (1 + 2i) / (1 - i) and rewrite it so it looks like a + bi. This means we need to get rid of the division and have a single real number and a single imaginary number added together. Think of it like simplifying a fraction, but with complex numbers. The a + bi form is crucial because it lets us easily identify the real and imaginary components of the number. The real part, 'a', tells us where the complex number lies on the real number line, while the imaginary part, 'b', tells us its position relative to the imaginary axis. This is important for various calculations, including finding the magnitude and argument of the number, and understanding its geometric representation on the complex plane. Without converting to a + bi form, these operations would be significantly more complicated. Also, the a + bi form is the most common way to represent complex numbers, making it easy to compare and contrast different complex numbers, since they're all in a standard format. It's like having a universal language for these kinds of numbers, making it easy to communicate and share the results of any calculations involving complex numbers.

To get the (1 + 2i) / (1 - i) into this form, we're going to use a clever trick called multiplying by the conjugate. Let's get started!

Step-by-Step: Getting to a + bi

Here's the deal, guys. To simplify (1 + 2i) / (1 - i), we need to get rid of that complex number in the denominator. The trick is to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number a - bi is a + bi (you just change the sign of the imaginary part). In our case, the conjugate of (1 - i) is (1 + i). So we'll multiply both the top and bottom of our fraction by (1 + i). This is like multiplying by 1, so we don't change the value of the number, just its appearance.

Let's break down the math:

  1. Multiply the numerator and denominator by the conjugate:

    (1 + 2i) / (1 - i) * (1 + i) / (1 + i)

  2. Expand the numerator (top part):

    (1 + 2i) * (1 + i) = 1 * 1 + 1 * i + 2i * 1 + 2i * i = 1 + i + 2i + 2i²

    Remember that i² = -1. So, the numerator becomes:

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

  3. Expand the denominator (bottom part):

    (1 - i) * (1 + i) = 1 * 1 + 1 * i - i * 1 - i * i = 1 + i - i - i²

    Again, i² = -1. So, the denominator becomes:

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

  4. Put it all together:

    Our fraction now looks like this: (-1 + 3i) / 2

  5. Separate the real and imaginary parts:

    We can rewrite this as -1/2 + (3/2)i. This is now in the form a + bi!

So, after all that, we've found that (1 + 2i) / (1 - i) = -1/2 + (3/2)i. Boom! We did it! We have successfully simplified the complex fraction into the a + bi form, allowing us to easily identify both the real and imaginary components of the original expression. This process highlights the power of algebraic manipulation and understanding of basic complex number arithmetic. This manipulation is fundamental for more advanced topics in complex analysis, such as finding roots, solving equations, and understanding the geometrical interpretations of complex numbers.

The Final Answer and What It Means

So, the form a + bi of (1 + 2i) / (1 - i) is -1/2 + (3/2)i.

  • The real part (a) is -1/2.
  • The imaginary part (b) is 3/2.

This tells us that on the complex plane, this number is located at the coordinates (-1/2, 3/2). That's the real part, and the imaginary part, which tells us where it lies. It's a piece of cake, right? Knowing this makes a lot of other complex number calculations easier. For example, calculating the magnitude (or modulus) and the argument (angle) of the complex number, which can be easily derived from the a + bi format. Converting to a + bi gives us access to calculations involving various trigonometric functions, too, helping in plotting the complex number on an Argand diagram or complex plane. So, the a + bi form is essential for many purposes involving complex numbers. Using this format is important for complex number operations, making them easier to visualize and manipulate, and it also simplifies the process of representing and interpreting their values.

Why This Matters and Where to Go From Here

Understanding how to express complex numbers in the a + bi form is a fundamental skill. It's the basis for so many other concepts in math and physics, especially when dealing with:

  • Electrical Engineering: Complex numbers are used to analyze AC circuits.
  • Quantum Mechanics: They're essential for describing wave functions.
  • Signal Processing: Used to analyze and manipulate signals.

So, knowing how to do this is going to be helpful as you continue to learn. Keep practicing, and you'll become a pro in no time! The process we have used is fundamental to ensure that you are comfortable with the basics. Now that you've mastered this, you can start exploring more complex operations, like finding powers and roots of complex numbers, which build on the a + bi format. Remember, consistent practice is key. The more you work with complex numbers, the more comfortable and confident you'll become. You can also explore the geometrical representation of complex numbers on the complex plane, which can greatly improve your understanding of their behavior. Many online resources offer tutorials and practice problems, so keep exploring and keep learning! You are going to do great!

Hope this helps. Keep up the good work, and happy calculating!