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Ch. 1 - Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 1 - Equations and InequalitiesProblema 27
Capitolo 2, Problema 27

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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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.
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Dividing Complex Numbers

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.
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Dividing Complex Numbers

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.
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Multiplying Complex Numbers
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