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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
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Tutti i libri di testoBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 5.27
Capitolo 5, Problema 5.27

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


2+3i / 2+i

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Identify the given complex division problem: \(\frac{2+3i}{2+i}\), where \(i\) is the imaginary unit with the property \(i^2 = -1\).
To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \(2+i\) is \(2 - i\).
Multiply numerator and denominator by the conjugate: \(\frac{2+3i}{2+i} \times \frac{2 - i}{2 - i} = \frac{(2+3i)(2 - i)}{(2+i)(2 - i)}\).
Expand both numerator and denominator using the distributive property (FOIL method): - Numerator: \((2)(2) + (2)(-i) + (3i)(2) + (3i)(-i)\) - Denominator: \((2)(2) + (2)(-i) + (i)(2) + (i)(-i)\).
Simplify the expressions by combining like terms and using \(i^2 = -1\), then write the result in the form $a + bi$, where \(a\) and \(b\) are real numbers.

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Complex Number Division

Dividing complex numbers involves expressing the quotient in a form that separates real and imaginary parts. This is typically done by multiplying numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator.
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Dividing Complex Numbers

Complex Conjugate

The complex conjugate of a number a + bi is a - bi. Multiplying a complex number by its conjugate results in a real number, specifically a^2 + b^2, which helps simplify division by removing the imaginary component from the denominator.
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Complex Conjugates

Standard Form of a Complex Number

The standard form of a complex number is a + bi, where a is the real part and b is the imaginary part. Expressing results in this form makes it easier to interpret and use complex numbers in further calculations.
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Complex Numbers In Polar Form
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