In Exercises 21–28, divide and express the result in standard form.


2+3i / 2+i

Verified step by step guidance

1. Step 1: Identify the complex numbers in the division problem. Here, the numerator is \(2 + 3i\) and the denominator is \(2 + i\).
2. Step 2: To divide complex numbers, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 + i\) is \(2 - i\).
3. Step 3: Multiply the numerator \((2 + 3i)\) by the conjugate of the denominator \((2 - i)\). Use the distributive property: \((2 + 3i)(2 - i)\).
4. Step 4: Multiply the denominator \((2 + i)\) by its conjugate \((2 - i)\). This results in a difference of squares: \((2 + i)(2 - i) = 2^2 - i^2\).
5. Step 5: Simplify the expression obtained in Step 3 and Step 4. Express the result in standard form \(a + bi\), where \(a\) and \(b\) are real numbers.

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part and b is the imaginary part. In the given expression, 2 + 3i and 2 + i are both complex numbers. Understanding how to manipulate these numbers is essential for performing operations such as addition, subtraction, multiplication, and division.

Division of Complex Numbers

Dividing complex numbers involves multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi. This process eliminates the imaginary part in the denominator, allowing the result to be expressed in standard form, which is a + bi, where both a and b are real numbers.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where a and b are real numbers. This format is crucial for clarity and consistency in mathematical communication. When performing operations with complex numbers, such as division, the goal is to simplify the result into this standard form to easily interpret and utilize the values in further calculations.