Solving Complex Number Equations A Step By Step Guide

Sam Evans · · 3 min read

Table Of Content

  1. Understanding the Problem
  2. Solving the Equation
  3. Evaluating the Options
    1. Option A: 12(4+i)=2+12i\frac{1}{2}(4+i)=2+\frac{1}{2} i21​(4+i)=2+21​i
    2. Option B: 2(4+i)=2+12i2(4+i)=2+\frac{1}{2} i2(4+i)=2+21​i
    3. Option C: −2(4+i)=2+12i-2(4+i)=2+\frac{1}{2} i−2(4+i)=2+21​i
    4. Option D: −12(4+i)=2+12i-\frac{1}{2}(4+i)=2+\frac{1}{2} i−21​(4+i)=2+21​i
  4. Final Answer and Conclusion

Hey there, math enthusiasts! Ever stumbled upon a problem that seems like a complex maze at first glance? Well, today, we're diving into a fascinating problem involving complex numbers and real number multiplication. This isn't just about crunching numbers; it's about understanding how these mathematical entities interact. So, buckle up as we break down this problem step by step, making it as clear as a sunny day. Let's get started, guys!

Understanding the Problem

The problem at hand presents us with a scenario where the complex number 4+i4 + i is multiplied by a real number, resulting in another complex number, 2+12i2 + \frac{1}{2}i. Our mission, should we choose to accept it, is to identify the correct equation that represents this mathematical relationship. Seems straightforward, right? But let's not jump to conclusions just yet. We need to unpack what this means in the language of mathematics.

Complex numbers, as you might already know, are numbers that can be expressed in the form a+bia + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of -1. The real part is 'a', and the imaginary part is 'b'. In our case, 4+i4 + i has a real part of 4 and an imaginary part of 1, while 2+12i2 + \frac{1}{2}i has a real part of 2 and an imaginary part of 12\frac{1}{2}.

The heart of the problem lies in figuring out what real number, when multiplied by 4+i4 + i, gives us 2+12i2 + \frac{1}{2}i. This is a classic example of how real numbers can scale complex numbers, affecting both their real and imaginary components. It's like stretching or shrinking a vector in a 2D plane, where the real part corresponds to the x-coordinate and the imaginary part corresponds to the y-coordinate. Finding this real number is akin to finding the scaling factor that transforms one vector into another.

To approach this problem, we'll use a bit of algebraic thinking. We'll represent the unknown real number with a variable, let's say 'x', and set up an equation. This equation will express the multiplication of the complex number 4+i4 + i by 'x' and equate it to the resulting complex number 2+12i2 + \frac{1}{2}i. Once we have this equation, we can solve for 'x'. This involves distributing 'x' across the complex number and then comparing the real and imaginary parts on both sides of the equation.

The options provided give us a head start by suggesting specific values for this real number. However, it's crucial to verify each option rigorously. We can't just pick one based on a hunch; we need to ensure it satisfies the conditions of the problem. This is where our algebraic skills come into play. We'll substitute each real number into the equation and check if it holds true. This process of verification is not only essential for solving the problem at hand but also for building confidence in our problem-solving abilities.

So, in essence, we're not just looking for an answer; we're embarking on a journey of mathematical exploration. We're diving into the realm of complex numbers, real number multiplication, and algebraic problem-solving. By the end of this, we'll not only have the solution but also a deeper understanding of the concepts involved. Let's dive deeper into the heart of this problem, guys!

Solving the Equation

Now, let's roll up our sleeves and get our hands dirty with the nitty-gritty of solving this equation. As we discussed earlier, the core of the problem is to find the real number that, when multiplied by 4+i4 + i, yields 2+12i2 + \frac{1}{2}i. To do this, we'll start by representing the unknown real number with a variable. Let's use 'x' for clarity. This allows us to express the problem as an equation:

x(4+i)=2+12ix(4 + i) = 2 + \frac{1}{2}i

This equation is the cornerstone of our solution. It encapsulates the entire problem in a concise mathematical statement. The next step is to distribute 'x' across the complex number 4+i4 + i. Remember, when we multiply a complex number by a real number, we multiply both the real and imaginary parts by that real number. So, when we distribute 'x', we get:

4x+xi=2+12i4x + xi = 2 + \frac{1}{2}i

Now, we have an equation with complex numbers on both sides. For these two complex numbers to be equal, their real parts must be equal, and their imaginary parts must be equal. This gives us two separate equations:

  • Real parts: 4x=24x = 2
  • Imaginary parts: x=12x = \frac{1}{2}

Notice that we have 'x' in both equations, which is fantastic because it provides a consistency check. If the value of 'x' we find from the real parts equation is the same as the value of 'x' from the imaginary parts equation, we're on the right track. Let's solve the first equation for 'x':

4x=24x = 2

Divide both sides by 4:

x=24=12x = \frac{2}{4} = \frac{1}{2}

Lo and behold! The value of 'x' we obtained from the real parts equation is the same as the value of 'x' from the imaginary parts equation. This confirms that our approach is correct and that x=12x = \frac{1}{2} is indeed the real number we're looking for.

Now that we've found the value of 'x', the final step is to match this value with one of the options provided in the problem. This is where our detective work pays off. We'll go through each option and see which one correctly represents the equation with x=12x = \frac{1}{2}. This isn't just about finding the correct answer; it's about making sure we understand why that answer is correct. It's about solidifying our understanding of the relationship between complex numbers, real number multiplication, and algebraic equations.

So, let's keep our focus sharp and our minds engaged as we navigate through the final steps of this problem. We're not just solving an equation; we're unraveling a mathematical puzzle. Let's dive into those options, guys!

Evaluating the Options

Alright, let's put on our detective hats and evaluate the options provided in the problem. We've already determined that the real number we're looking for is 12\frac{1}{2}. Now, we need to find the equation that correctly represents the multiplication of 4+i4 + i by 12\frac{1}{2} resulting in 2+12i2 + \frac{1}{2}i. This is where careful substitution and verification come into play.

The options given are:

A. 12(4+i)=2+12i\frac{1}{2}(4+i)=2+\frac{1}{2} i B. 2(4+i)=2+12i2(4+i)=2+\frac{1}{2} i C. −2(4+i)=2+12i-2(4+i)=2+\frac{1}{2} i D. −12(4+i)=2+12i-\frac{1}{2}(4+i)=2+\frac{1}{2} i

Let's take each option one by one and see if it holds true. This is a systematic process, ensuring we don't miss anything. We'll distribute the real number on the left side of the equation and then compare the result with the complex number on the right side.

Option A: 12(4+i)=2+12i\frac{1}{2}(4+i)=2+\frac{1}{2} i

Distribute 12\frac{1}{2} across (4+i)(4 + i):

12∗4+12∗i=2+12i\frac{1}{2} * 4 + \frac{1}{2} * i = 2 + \frac{1}{2}i

2+12i=2+12i2 + \frac{1}{2}i = 2 + \frac{1}{2}i

This equation holds true! The left side is indeed equal to the right side. So, option A looks promising. But let's not jump the gun just yet. We need to check the other options to be absolutely sure.

Option B: 2(4+i)=2+12i2(4+i)=2+\frac{1}{2} i

Distribute 2 across (4+i)(4 + i):

2∗4+2∗i=2+12i2 * 4 + 2 * i = 2 + \frac{1}{2}i

8+2i=2+12i8 + 2i = 2 + \frac{1}{2}i

This equation is not true. The real and imaginary parts on the left side are different from those on the right side. So, we can confidently rule out option B.

Option C: −2(4+i)=2+12i-2(4+i)=2+\frac{1}{2} i

Distribute -2 across (4+i)(4 + i):

−2∗4+(−2)∗i=2+12i-2 * 4 + (-2) * i = 2 + \frac{1}{2}i

−8−2i=2+12i-8 - 2i = 2 + \frac{1}{2}i

This equation is also not true. The signs and magnitudes of the real and imaginary parts don't match up. So, option C is not the correct answer.

Option D: −12(4+i)=2+12i-\frac{1}{2}(4+i)=2+\frac{1}{2} i

Distribute −12-\frac{1}{2} across (4+i)(4 + i):

−12∗4+(−12)∗i=2+12i-\frac{1}{2} * 4 + (-\frac{1}{2}) * i = 2 + \frac{1}{2}i

−2−12i=2+12i-2 - \frac{1}{2}i = 2 + \frac{1}{2}i

This equation is not true either. The real parts have opposite signs, so this option is incorrect.

After carefully evaluating all the options, we've found that only option A holds true. This reinforces the importance of verifying each option, even when one looks promising at first glance. This meticulous approach is what sets apart a good problem solver from a great one. So, let's celebrate our success in identifying the correct equation. We've not only found the answer but also reinforced our understanding of how real numbers interact with complex numbers through multiplication. Great job, guys!

Final Answer and Conclusion

After our thorough investigation, we've arrived at the final answer. We meticulously analyzed the problem, set up the equation, solved for the unknown real number, and rigorously evaluated each option. The journey through this complex number problem has been both enlightening and rewarding. So, let's state our conclusion with confidence and clarity.

The correct equation that represents the multiplication of the complex number 4+i4 + i by a real number to yield 2+12i2 + \frac{1}{2}i is:

A. 12(4+i)=2+12i\frac{1}{2}(4+i)=2+\frac{1}{2} i

This equation perfectly encapsulates the relationship described in the problem. When we multiply (4+i)(4 + i) by 12\frac{1}{2}, we indeed obtain 2+12i2 + \frac{1}{2}i. This confirms our algebraic manipulations and our understanding of complex number multiplication.

Throughout this problem-solving process, we've not only honed our skills in algebra and complex number arithmetic but also reinforced the importance of careful and methodical problem-solving. We've seen how a systematic approach, involving setting up equations, solving for unknowns, and verifying solutions, can lead us to the correct answer with confidence.

This problem serves as a great example of how mathematical concepts intertwine. We've touched upon complex numbers, real numbers, multiplication, and algebraic equations. Each of these concepts plays a crucial role in the solution, highlighting the interconnectedness of mathematics. It's like a beautiful tapestry, where each thread contributes to the overall pattern.

So, what's the takeaway from this? It's not just about finding the right answer; it's about the journey we take to get there. It's about the skills we develop, the concepts we understand, and the confidence we build along the way. Each problem we solve is a stepping stone to becoming more proficient and confident mathematicians. So, let's continue to embrace these challenges, dive into the world of numbers and equations, and unravel the mysteries of mathematics. You guys are doing awesome!

In conclusion, remember that mathematics is not just about formulas and calculations; it's about logical thinking, problem-solving, and the joy of discovery. Keep exploring, keep questioning, and keep learning. The world of mathematics is vast and fascinating, and there's always something new to discover. Until next time, keep those mathematical gears turning!

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