Problem 27 - College Algebra

Question

In Exercises \(21-28,\) divide and express the result in standard form. $$\frac{2+3 i}{2+i}$$

Step-by-Step Solution

Answer

The result in standard form is \(1/5 + (4/5)i\).

Step 1: Identify the Conjugate of the Denominator

The conjugate of a complex number \(a+b i\) is \(a-b i\). Therefore, for the complex number \(2+i\), its conjugate is \(2-i\).

Step 2: Multiply by the Conjugate

Multiply the numerator and the denominator by the conjugate of the denominator. \((2+3 i)(2-i) / ((2+i)(2-i))\). This leads to a fraction where the numerator is a complex number in standard form and the denominator is a real number.

Step 3: Simplify the Expression

Simplify the numerator and the denominator separately. The denominator simplifies to \((2+i)(2-i) = 4 - (-1) = 5\). The numerator simplifies to \((2+3 i)(2-i) = 4 - 2i + 6i - 3 = 1 + 4i\). So we get \(1 + 4i / 5\).

Step 4: Write the Result in Standard Form

The standard form of a complex number is \(a + bi\), where \(a, b\) are real numbers. So to make the fraction conform to this form, individual real and imaginary parts must be divided by the denominator. We get \((1/5 + (4/5)i)\).

Key Concepts

Standard Form of a Complex Number

The standard form of a complex number is perhaps the most fundamental concept when dealing with complex numbers. In mathematics, it's essential to represent complex numbers in a consistent format for clarity and ease of use. The standard form is expressed as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part, with both \(a\) and \(b\) being real numbers and \(i\) representing the imaginary unit, satisfying the equation \(i^2 = -1\).

When you come across a complex number, it is crucial to identify its real and imaginary components. This is because all arithmetic operations, including addition, subtraction, multiplication, and division, rely on manipulating these two parts separately. For instance, the number \(2+3i\) contains a real part of 2 and an imaginary part of 3i. Whenever you're dividing complex numbers, like in the exercise \(\frac{2+3i}{2+i}\), the goal is to express the final answer in this standard form, making it straightforward to understand and use in further calculations.

Conjugate of a Complex Number

Understanding the conjugate of a complex number is crucial for various operations, including division. The conjugate is simply the mirror image of a complex number, reflecting across the real axis on the complex plane. It is obtained by changing the sign of the imaginary part. So for a complex number \(a + bi\), the conjugate would be \(a - bi\).

Why is the conjugate so important? When you multiply a complex number by its conjugate, the result is always a real number, specifically, the sum of the squares of the real and imaginary parts: \((a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2\). This property is particularly useful because it allows us to simplify expressions and resolve divisions of complex numbers by turning the denominator into a real number, as we see in the division exercise. By multiplying both the numerator and denominator of \(\frac{2+3i}{2+i}\) by the conjugate of the denominator \(2-i\), we simplify the division to a more manageable expression.

Simplifying Complex Expressions

Simplifying complex expressions is the process of reducing them to the simplest form possible, often to a standard form so that they are easily interpretable and useful for further operations. Multiplication and division with complex numbers aren't as straightforward as with real numbers because of the presence of the imaginary unit \(i\). You can think of simplifying complex expressions as a way of 'tidying up' the math, making sure there's nothing more to combine or reduce.

The steps to simplify a complex expression, especially with division, include identifying the conjugate, multiplying by it, and then distributing and combining like terms. The aim is to eliminate the imaginary unit from the denominator to prevent division by a complex number. After multiplication and combining like terms, the resulting complex number can often be simplified to its standard form. In our exercise, following these steps transforms the complex division into \(\frac{1 + 4i}{5}\), which can be further simplified to yield the standard form \(\frac{1}{5} + \frac{4}{5}i\). Each component — the real part \(\frac{1}{5}\) and the imaginary part \(\frac{4}{5}i\) — is a fraction on its own, showcasing the simplified expression distinctly.