Find The Complex Number With Absolute Value 5

Hey guys! Let's dive into the fascinating world of complex numbers and tackle a question that often pops up in mathematics: Which complex number has an absolute value of 5? This question isn't just about crunching numbers; it’s about understanding the fundamental properties of complex numbers and how they behave in the complex plane. So, grab your thinking caps, and let's embark on this mathematical journey together!

Before we jump into the solutions, let's rewind a bit and understand what complex numbers and their absolute values are all about. A complex number, in its simplest form, is a number that can be expressed as a + bi, where a and b are real numbers, and i is the imaginary unit. Remember, i is defined as the square root of -1, a concept that extends our number system beyond the real numbers. The real part of the complex number is a, and the imaginary part is b. Think of complex numbers as living in a two-dimensional plane, where the x-axis represents the real part, and the y-axis represents the imaginary part. This plane is aptly named the complex plane. Now, what about the absolute value? The absolute value of a complex number, also known as its modulus, is the distance from the origin (0,0) to the point representing the complex number in the complex plane. It's essentially the length of the vector connecting the origin to the complex number. We calculate it using a formula derived from the Pythagorean theorem: |a + bi| = √(a² + b²). This formula is your key to unlocking the solution to our question and many others involving complex numbers.

Decoding Absolute Value: A Step-by-Step Guide

To truly master this concept, let's break down the formula and see why it works. Imagine plotting the complex number a + bi on the complex plane. You move a units along the real axis and b units along the imaginary axis. This gives you a point (a, b). The line connecting this point to the origin forms the hypotenuse of a right-angled triangle. The sides of this triangle have lengths a and b. Using the Pythagorean theorem (a² + b² = c²), we can find the length of the hypotenuse, which is the absolute value of the complex number. Hence, |a + bi| = √(a² + b²). This visual representation makes the concept of absolute value much more intuitive. It's not just a formula; it's a distance, a magnitude, a measure of how far a complex number is from the heart of the complex plane. Now, with this understanding firmly in place, we're ready to tackle the question at hand and find the complex number with an absolute value of 5.

Cracking the Code: Finding the Complex Number with Absolute Value 5

Now, let's get back to the original question. We need to find a complex number from the given options that has an absolute value of 5. We’ll use the formula we just discussed: |a + bi| = √(a² + b²). Our goal is to find a pair of a and b such that √(a² + b²) = 5. This means we need to find a and b where a² + b² = 25. This equation should ring a bell – it's the equation of a circle with a radius of 5 centered at the origin in the complex plane! Any complex number that lies on this circle will have an absolute value of 5. Now, let's test the given options one by one to see which one fits the bill.

Option A: -3 + 4i – A Perfect Match

Let’s start with option A: -3 + 4i. Here, a = -3 and b = 4. Plugging these values into our formula, we get: |-3 + 4i| = √((-3)² + (4)²) = √(9 + 16) = √25 = 5. Bingo! This complex number has an absolute value of 5. It’s like hitting the jackpot on our first try. But, just to be thorough and ensure we understand the process, let’s examine the other options as well.

Option B: 2 + 3i – Close, But No Cigar

Next up is option B: 2 + 3i. Here, a = 2 and b = 3. Calculating the absolute value, we get: |2 + 3i| = √(2² + 3²) = √(4 + 9) = √13. This value is not equal to 5, so option B is not the answer. It’s a good reminder that not all complex numbers will have the absolute value we’re looking for. The absolute value depends on the specific values of the real and imaginary parts.

Option C: 7 - 2i – Off the Mark

Moving on to option C: 7 - 2i. In this case, a = 7 and b = -2. Let's calculate: |7 - 2i| = √(7² + (-2)²) = √(49 + 4) = √53. This is definitely not 5, so option C is also incorrect. It's important to note the negative sign in front of the 2; we square it, so it becomes positive, but it still affects the overall calculation.

Option D: 9 + 4i – Not Even Close

Finally, let's consider option D: 9 + 4i. Here, a = 9 and b = 4. The absolute value is: |9 + 4i| = √(9² + 4²) = √(81 + 16) = √97. This value is significantly larger than 5, so option D is not the correct answer. By now, we've thoroughly examined all the options, and it's clear that only option A satisfies the condition of having an absolute value of 5.

The Verdict: Option A is the Winner

After meticulously analyzing each option, we've confidently arrived at the answer. Option A, -3 + 4i, is the complex number with an absolute value of 5. We achieved this by understanding the concept of absolute value, applying the formula |a + bi| = √(a² + b²), and systematically testing each option. This problem highlights the beauty of complex numbers and how they can be visualized and manipulated using geometric principles. Remember, mathematics is not just about finding the right answer; it’s about understanding the underlying concepts and the process of problem-solving. So, keep exploring, keep questioning, and keep delving deeper into the fascinating world of mathematics!

So, to wrap it up, the correct answer is A. -3 + 4i.

Understanding complex numbers is a cornerstone of advanced mathematics, and grasping the concept of absolute value is crucial. By practicing these types of problems, you'll not only sharpen your calculation skills but also deepen your understanding of the complex plane and its properties. Remember, the key is to break down complex concepts into smaller, manageable steps. Visualize the problem, apply the relevant formulas, and work systematically through the options. And most importantly, don't be afraid to make mistakes – they are valuable learning opportunities! Keep exploring the world of mathematics, and you'll discover the beauty and elegance hidden within its intricate structures. Who knows what fascinating mathematical puzzles you'll solve next?