In Exercises 21–28, divide and express the result in standard form. (2 + 3i)/(2 + i)

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Identify the problem: You are dividing two complex numbers, \( \frac{2 + 3i}{2 + i} \). To simplify, we need to eliminate the imaginary part from the denominator by multiplying both numerator and denominator by the conjugate of the denominator.

Find the conjugate of the denominator \( 2 + i \). The conjugate is \( 2 - i \). Multiply both numerator and denominator by \( 2 - i \): \( \frac{(2 + 3i)(2 - i)}{(2 + i)(2 - i)} \).

Simplify the denominator using the difference of squares formula: \( (2 + i)(2 - i) = 2^2 - i^2 = 4 - (-1) = 5 \). The denominator becomes 5.

Expand the numerator using the distributive property: \( (2 + 3i)(2 - i) = 2(2) + 2(-i) + 3i(2) + 3i(-i) = 4 - 2i + 6i - 3i^2 \). Simplify the terms: \( 4 + 4i - 3(-1) = 4 + 4i + 3 = 7 + 4i \).

Combine the results: The simplified expression is \( \frac{7 + 4i}{5} \). To express in standard form, divide each term in the numerator by the denominator: \( \frac{7}{5} + \frac{4}{5}i \).

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주요 개념

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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 coefficient of the imaginary unit i (where i² = -1). Understanding complex 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.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers. In this form, a represents the real part and b represents the imaginary part. Expressing a complex number in standard form is crucial for clarity and further mathematical operations, making it easier to interpret and manipulate.