Simplify Complex Number Expressions: Step-by-Step Guide

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Hey guys! Today, we're diving into the world of complex numbers. Complex numbers might seem a bit intimidating at first, but trust me, they're super manageable once you break them down. We're going to tackle an expression involving complex numbers and simplify it into the standard form of a + bi, where a and b are rational numbers. So, buckle up, and let's get started!

Understanding the Problem

Before we jump into the solution, let's quickly recap what complex numbers are all about. 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 (i = √-1). Operations with complex numbers involve treating i as a variable, but with the added rule that i² = -1. This property is key to simplifying expressions.

Our mission is to simplify the expression: $(6-4i)(6-4i) - 3(7-11i)$. This expression involves multiplication and subtraction of complex numbers. To simplify it, we'll first expand the product of the two complex numbers and then distribute the scalar multiplication. Finally, we'll combine like terms to get the expression in the desired a + bi form. It's all about careful calculation and a good understanding of how the imaginary unit i behaves!

Step-by-Step Solution

Let's break down the simplification process into manageable steps:

1. Expand the Product (6 - 4i)(6 - 4i)

First, we need to multiply the complex number (6 - 4i) by itself. We can use the FOIL method (First, Outer, Inner, Last) to ensure we multiply each term correctly:

  • First: 6 * 6 = 36
  • Outer: 6 * (-4i) = -24i
  • Inner: (-4i) * 6 = -24i
  • Last: (-4i) * (-4i) = 16i²

So, (6 - 4i)(6 - 4i) = 36 - 24i - 24i + 16i². Now, remember that i² = -1, so we can substitute that in:

36 - 24i - 24i + 16(-1) = 36 - 48i - 16

Combining the real parts (36 and -16), we get:

20 - 48i

2. Distribute the Scalar Multiplication 3(7 - 11i)

Next, we need to distribute the 3 across the complex number (7 - 11i):

3 * 7 = 21 3 * (-11i) = -33i

So, 3(7 - 11i) = 21 - 33i

3. Combine the Results

Now, we have two simplified expressions: 20 - 48i and 21 - 33i. The original expression was (6 - 4i)(6 - 4i) - 3(7 - 11i), so we need to subtract the second expression from the first:

(20 - 48i) - (21 - 33i)

To subtract complex numbers, we subtract the real parts and the imaginary parts separately:

(20 - 21) + (-48i - (-33i)) = (20 - 21) + (-48i + 33i)

This simplifies to:

-1 - 15i

Final Answer

Therefore, the expression (6 - 4i)(6 - 4i) - 3(7 - 11i) simplified to the form a + bi is:

−1−15i-1 - 15i

So, a = -1 and b = -15.

Key Takeaways

  • Complex numbers are in the form a + bi, where a is the real part and b is the imaginary part.
  • The imaginary unit i is defined as √-1, and i² = -1. This is the golden rule of complex number simplification.
  • Operations with complex numbers involve treating i as a variable, following algebraic rules, and using the property i² = -1 to further simplify.
  • FOIL method is helpful when multiplying two binomials, and the same applies to complex numbers in binomial form.
  • When subtracting complex numbers, subtract the real parts and the imaginary parts separately.

Common Mistakes to Avoid

  • Forgetting that i² = -1: This is the most common mistake. Always remember to substitute -1 for i² when simplifying.
  • Incorrectly distributing the negative sign: When subtracting complex numbers, make sure to distribute the negative sign to both the real and imaginary parts of the complex number being subtracted. For instance, -(a + bi) = -a - bi.
  • Combining real and imaginary parts incorrectly: You can only combine real parts with real parts and imaginary parts with imaginary parts. Don't mix them up!
  • Errors in basic arithmetic: Always double-check your addition, subtraction, multiplication, and division to avoid simple calculation errors.

Practice Problems

Want to test your understanding? Try simplifying these expressions:

  1. (3 + 2i)(1 - i)
  2. (5 - i) - 2(3 + 4i)
  3. (2 + i)² + (2 - i)²

Simplifying these expressions will give you solid practice in manipulating complex numbers. Remember to apply the steps we discussed earlier and pay close attention to the properties of the imaginary unit i.

Why are Complex Numbers Important?

You might be wondering, “Okay, I can simplify these expressions, but what's the point?” Well, complex numbers aren't just abstract mathematical concepts. They have many real-world applications in various fields, including:

  • Electrical Engineering: Complex numbers are used extensively in AC circuit analysis.
  • Quantum Mechanics: They are fundamental to describing the behavior of particles at the quantum level.
  • Fluid Dynamics: Complex potentials are used to solve 2D fluid flow problems.
  • Signal Processing: Complex numbers are used to analyze and process signals.
  • Mathematics: They are essential in many areas of mathematics, such as fractal geometry and number theory.

Conclusion

So, there you have it! We've successfully simplified a complex number expression and learned the key concepts and techniques involved. Remember to practice regularly and pay attention to the details. Complex numbers might seem complex (pun intended!), but with a solid understanding and careful approach, you'll be able to tackle any expression with confidence. Keep practicing, and you'll become a complex number whiz in no time!

If you have any questions or want to discuss this further, feel free to leave a comment below. Happy simplifying, guys!