Simplifying Complex Fractions Express In A + Bi Form

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Hey guys! Ever stumbled upon a complex fraction and felt a little lost? Don't worry, we've all been there. Today, we're going to break down how to simplify a complex fraction and express it in the standard form of a complex number, which is $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit ($i = \sqrt{-1}$). Our specific mission? To simplify $\frac{5}{2+i}$. Buckle up, because we're about to dive into the fascinating world of complex numbers!

Understanding Complex Numbers and the Need for Simplification

Before we jump into the simplification process, let's quickly recap what complex numbers are. A complex number is essentially a number that can be expressed in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part. The imaginary part is multiplied by $i$, which, as we mentioned, is the square root of -1. Now, having a complex number in the denominator of a fraction isn't considered simplified. It's like having a radical in the denominator – we want to get rid of it! This is where the concept of the conjugate comes in handy.

The main goal here is to express the given complex number, $\frac{5}{2+i}$, in its simplest form, which adheres to the standard $a + bi$ format. This involves eliminating the imaginary part from the denominator. Why? Because it makes it easier to perform other operations with the complex number and to compare it with other complex numbers. Think of it as tidying up your room – everything has its place, and in the world of complex numbers, that place is the $a + bi$ form. Now, why can't we just leave it as $\frac{5}{2+i}$? Well, it's not mathematically "wrong," but it's not considered standard practice. It's like writing a sentence without proper punctuation – you can technically understand it, but it's not as clear and concise as it could be. In the context of more advanced mathematical operations and comparisons, having the complex number in the $a + bi$ form simplifies things significantly. It allows for easier addition, subtraction, multiplication, and division of complex numbers. Moreover, when dealing with complex functions and mappings, the $a + bi$ representation provides a clearer picture of the number's position in the complex plane, which is a graphical representation of complex numbers. So, by simplifying to the $a + bi$ form, we're not just being mathematically neat; we're setting ourselves up for success in more complex calculations and analyses. This is why this step is important and necessary.

The Conjugate: Our Secret Weapon

The conjugate of a complex number is formed by simply changing the sign of the imaginary part. So, if we have a complex number $a + bi$, its conjugate is $a - bi$. The magic happens when we multiply a complex number by its conjugate. Let's see why:

(a+bi)(a−bi)=a2−abi+abi−(bi)2=a2−b2i2{(a + bi)(a - bi) = a^2 - abi + abi - (bi)^2 = a^2 - b^2i^2}

Since $i^2 = -1$, we get:

a2−b2(−1)=a2+b2{a^2 - b^2(-1) = a^2 + b^2}

Notice that the result is a real number! This is the key to eliminating the imaginary part from the denominator. For our specific problem, the conjugate of $2 + i$ is $2 - i$. We're going to use this to our advantage.

This seemingly simple trick is the foundation for simplifying complex fractions. By multiplying a complex number by its conjugate, we transform it into a real number. This is incredibly useful because it allows us to eliminate the imaginary part from the denominator of a fraction, which is precisely what we need to do to express the complex number in the standard $a + bi$ form. But why does this work? Well, when we multiply a complex number by its conjugate, the imaginary terms cancel each other out, leaving us with only real terms. This is because the multiplication process creates both a positive and a negative imaginary term, which perfectly balance each other out. The result is a sum of squares, which is always a real number. This elegant property of complex conjugates makes them an indispensable tool in complex number arithmetic. They not only help us simplify fractions but also play a crucial role in solving equations, finding roots of polynomials, and exploring various other concepts in complex analysis. In essence, the conjugate is our secret weapon, a powerful technique that allows us to navigate the world of complex numbers with ease and precision. Without understanding the conjugate, simplifying complex fractions would be a much more challenging, if not impossible, task.

Step-by-Step Simplification of $\frac{5}{2+i}\

Okay, let's get our hands dirty and simplify $\frac{5}{2+i}$. Here's the breakdown:

  1. Identify the conjugate: The conjugate of $2 + i$ is $2 - i$.

  2. Multiply both the numerator and denominator by the conjugate:

    52+i×2−i2−i{\frac{5}{2+i} \times \frac{2-i}{2-i}}

    Remember, multiplying by $\frac{2-i}{2-i}$ is the same as multiplying by 1, so we're not changing the value of the fraction, just its form.

  3. Expand the numerator and the denominator:

    • Numerator: $5(2 - i) = 10 - 5i$
    • Denominator: $(2 + i)(2 - i) = 2^2 - (i)^2 = 4 - (-1) = 5$

  4. Rewrite the fraction:

    10−5i5{\frac{10 - 5i}{5}}

  5. Separate the real and imaginary parts:

    105−5i5{\frac{10}{5} - \frac{5i}{5}}

  6. Simplify:

    2−i{2 - i}

And there you have it! We've successfully simplified $\frac{5}{2+i}$ to $2 - i$, which is in the desired $a + bi$ form.

Each of these steps is crucial for a successful simplification. The first step, identifying the conjugate, sets the stage for the entire process. It's like finding the right key for a lock – without it, you can't proceed. The second step, multiplying both numerator and denominator by the conjugate, is where the magic happens. This step leverages the property of conjugates to eliminate the imaginary part from the denominator. It's important to multiply both the numerator and denominator to maintain the value of the fraction. Think of it like scaling a recipe – if you double the ingredients, you need to double everything, not just some parts. The third step, expanding the numerator and denominator, involves applying the distributive property and the difference of squares formula. This step requires careful attention to detail to avoid errors in arithmetic. The fourth step, rewriting the fraction, is a simple but necessary step to consolidate the results of the expansion. The fifth step, separating the real and imaginary parts, prepares the fraction for the final simplification. It's like sorting your laundry before washing it – separating the colors makes the process more efficient. The final step, simplifying, is where we reduce the fraction to its simplest form. This often involves dividing both the real and imaginary parts by a common factor. This step is the culmination of all the previous steps, resulting in the desired $a + bi$ form. By meticulously following these steps, anyone can confidently simplify complex fractions and express them in the standard form. It's a systematic approach that demystifies complex numbers and makes them more accessible.

Final Answer

So, $\frac{5}{2+i}$ simplified to the form $a + bi$ is $\bf{2 - i}$. Pretty neat, huh? This skill is super useful in various areas of math and engineering, so keep practicing!