Simplify Complex Numbers: Z = 1/(5+4i) - 1/(5-4i)

by Andrew McMorgan 50 views

Hey guys! Ever get stumped by complex numbers, especially when you're trying to express them with a real denominator? It can seem a bit daunting at first, but trust me, it's totally manageable once you break it down. Today, we're tackling a specific problem: expressing the value of z = 1/(5 + 4i) - 1/(5 - 4i) with a real denominator. This is a super common type of question in mathematics, and understanding how to handle it will boost your confidence in dealing with complex fractions. We'll walk through each step, making sure you guys grasp the concepts without feeling overwhelmed. Let's dive in and demystify this complex calculation!

Understanding Complex Numbers and Denominators

Alright, so before we jump into solving for z, let's quickly chat about what we're dealing with. Complex numbers, as you know, have a real part and an imaginary part. They're usually written in the form a + bi, where a is the real part and b is the imaginary part, and i is the imaginary unit (the square root of -1). Now, when we have a complex number in the denominator of a fraction, like 1/(5 + 4i), it's often our goal to get rid of that imaginary part in the denominator. This process is called rationalizing the denominator, and it's crucial for simplifying expressions and making them easier to work with. Think of it like tidying up your room – you want everything in its proper place, and a real denominator is considered the 'proper place' in many mathematical contexts. The reason we want a real denominator is that it allows us to easily separate the real and imaginary parts of the resulting complex number. When you have a complex denominator, it's hard to tell at a glance what the real and imaginary components of the entire fraction are. But once it's rationalized, it's all neat and tidy, usually in the form c + di, where c and d are real numbers. This is the standard form we aim for.

To rationalize a complex denominator of the form a + bi, we multiply both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of a + bi is a - bi. Why does this work? Because when you multiply a complex number by its conjugate, the imaginary parts cancel out, leaving you with a purely real number. Let's see this in action: (a + bi) * (a - bi) = a*a - a*bi + bi*a - bi*bi = a^2 - abi + abi - b^2*i^2. Since i squared (i^2) is equal to -1, this becomes a^2 - b^2*(-1) = a^2 + b^2. And voilà! a^2 + b^2 is a real number. This fundamental property is the key to simplifying our expression for z. So, in our problem, the denominators are (5 + 4i) and (5 - 4i). Notice that these two are already complex conjugates of each other! This is a helpful simplification for our specific problem, but the general principle of using the conjugate still applies.

Step-by-Step Solution for z

Okay, team, let's get down to business and solve for z. Our expression is z = 1/(5 + 4i) - 1/(5 - 4i). The first thing we need to do is find a common denominator so we can combine these two fractions. Since the denominators are (5 + 4i) and (5 - 4i), the least common denominator is simply their product: (5 + 4i) * (5 - 4i). This is convenient because, as we just discussed, multiplying a complex number by its conjugate results in a real number.

So, let's calculate that denominator first:

(5 + 4i) * (5 - 4i) = 5*5 - 5*(4i) + (4i)*5 - (4i)*(4i) = 25 - 20i + 20i - 16i^2

Remember that i^2 = -1, so:

= 25 - 16*(-1) = 25 + 16 = 41

Boom! We've successfully rationalized the denominator, and it's a nice, clean 41. This is exactly what we wanted. Now, let's apply this common denominator to both fractions in our expression for z.

For the first fraction, 1/(5 + 4i), to get the common denominator (5 + 4i)*(5 - 4i), we need to multiply the numerator and denominator by (5 - 4i):

1/(5 + 4i) = (1 * (5 - 4i)) / ((5 + 4i) * (5 - 4i)) = (5 - 4i) / 41

For the second fraction, 1/(5 - 4i), to get the common denominator (5 + 4i)*(5 - 4i), we need to multiply the numerator and denominator by (5 + 4i):

1/(5 - 4i) = (1 * (5 + 4i)) / ((5 - 4i) * (5 + 4i)) = (5 + 4i) / 41

Now that both fractions have the same denominator, we can subtract them:

z = (5 - 4i) / 41 - (5 + 4i) / 41

When subtracting fractions with the same denominator, you just subtract the numerators and keep the denominator the same:

z = ((5 - 4i) - (5 + 4i)) / 41

Let's simplify the numerator. Be careful with the minus sign – it applies to both terms in the second parenthesis:

Numerator = 5 - 4i - 5 - 4i

Combine the real parts (5 - 5) and the imaginary parts (-4i - 4i):

Real parts: 5 - 5 = 0 Imaginary parts: -4i - 4i = -8i

So, the numerator simplifies to 0 - 8i, which is just -8i.

Now, substitute this back into our expression for z:

z = -8i / 41

And there you have it! We have successfully expressed z with a real denominator. The result is -8i/41.

Expressing z in Standard Form (a + bi)

So, we found that z = -8i / 41. This is a perfectly valid answer, and it does have a real denominator. However, in mathematics, we often like to express complex numbers in the standard form a + bi, where a is the real part and b is the imaginary part. Our current result, -8i/41, can be rewritten to fit this standard form.

Think of -8i/41 as 0 + (-8/41)i. Here, the real part (a) is 0, and the imaginary part (b) is -8/41. This makes it super clear what the real and imaginary components are. So, in standard form, z = 0 - (8/41)i.

Let's just recap what we did. We started with z = 1/(5 + 4i) - 1/(5 - 4i). The first major step was to find a common denominator, which we correctly identified as the product of the two complex denominators, (5 + 4i)(5 - 4i). This product conveniently yielded a real number, 41, because (5 + 4i) and (5 - 4i) are complex conjugates. We then rewrote each fraction with this common denominator:

1/(5 + 4i) became (5 - 4i)/41 1/(5 - 4i) became (5 + 4i)/41

Subtracting these fractions gave us:

z = (5 - 4i)/41 - (5 + 4i)/41 z = (5 - 4i - 5 - 4i)/41 z = (-8i)/41

Finally, to express this in the standard a + bi form, we simply separate the real and imaginary parts:

z = 0 - (8/41)i

This shows that the real part of z is 0, and the imaginary part is -8/41. It's always good practice to present your final answer in its simplest and most conventional form, and for complex numbers, that's usually the a + bi format. This breakdown should make it crystal clear how to tackle similar problems involving complex fractions and rationalizing denominators. Keep practicing, guys, and these concepts will become second nature!

Key Takeaways and Common Pitfalls

Alright, let's round this up with some key takeaways and a heads-up on common mistakes, so you guys can crush these problems every time. The main goal when dealing with expressions like z = 1/(5 + 4i) - 1/(5 - 4i) is to simplify it into the standard form a + bi where the denominator is real. The magic tool we used here is the complex conjugate. Remember, for any complex number a + bi, its conjugate is a - bi. Multiplying a complex number by its conjugate always results in a real number (a^2 + b^2), which is exactly what we need to get rid of imaginary numbers in the denominator.

In our specific problem, the denominators were already conjugates of each other. This made finding the common denominator straightforward: (5 + 4i) * (5 - 4i) = 5^2 + 4^2 = 25 + 16 = 41. So, the common denominator was 41. We then adjusted the numerators of each fraction to match this common denominator, which involved multiplying the first fraction's numerator by (5 - 4i) and the second fraction's numerator by (5 + 4i). After subtraction, we got z = -8i / 41. This is the value of z with a real denominator. Expressed in standard a + bi form, it's 0 - (8/41)i.

Now, for those common pitfalls to watch out for:

  1. Sign Errors During Subtraction: This is a big one, especially when subtracting complex numbers. When you subtract (5 + 4i) from (5 - 4i), it becomes 5 - 4i - 5 - 4i. Many people forget to distribute the minus sign to both the real and imaginary parts of the second complex number, mistakenly writing 5 - 4i - 5 + 4i. Always double-check your signs when dealing with subtraction, especially with parentheses involved.

  2. Forgetting i^2 = -1: When you multiply conjugates, you get terms like -16i^2. If you forget that i^2 equals -1, your calculation will be wrong. a^2 - b^2*i^2 becomes a^2 - b^2*(-1), which simplifies to a^2 + b^2. Missing this -1 substitution is a classic mistake.

  3. Not Finding a Common Denominator Correctly: While in this problem the denominators were conjugates, sometimes you'll have fractions like 1/(a+bi) and 1/(c+di). In such cases, the common denominator is (a+bi)(c+di). You need to make sure you multiply the numerator of the first fraction by (c+di) and the numerator of the second by (a+bi).

  4. Errors in Multiplication: Expanding (5 + 4i)(5 - 4i) requires careful application of the FOIL method (First, Outer, Inner, Last) or recognizing it as a difference of squares. Ensure each term is multiplied correctly and that terms cancel out as expected.

  5. Final Simplification: Make sure your final answer is in the simplest form. If you get something like z = (0 - 8i) / 41, it's best to simplify it to z = -8i/41 or z = 0 - (8/41)i.

By keeping these points in mind and practicing consistently, you guys will become real pros at simplifying complex numbers. It's all about methodical steps and paying attention to the details. Keep up the great work, and don't hesitate to tackle more problems like this one!