Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problema 2

Capitolo 5, Problema 2

In Exercises 1–3, perform the indicated operations and write the result in standard form. 5 / 2−i

Guida verificata passo dopo passo

Identify the given expression: \( \frac{5}{2 - i} \). The goal is to write this expression in standard form, which means expressing it as \( a + bi \), where \( a \) and \( b \) are real numbers.

To eliminate the imaginary unit \( i \) from the denominator, multiply both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of \( 2 - i \) is \( 2 + i \). So multiply by \( \frac{2 + i}{2 + i} \).

Perform the multiplication in the numerator: \( 5 \times (2 + i) = 10 + 5i \).

Perform the multiplication in the denominator using the difference of squares formula: \( (2 - i)(2 + i) = 2^2 - (i)^2 = 4 - (-1) = 4 + 1 = 5 \).

Now, write the expression as \( \frac{10 + 5i}{5} \). Then, separate the real and imaginary parts by dividing both terms in the numerator by 5: \( \frac{10}{5} + \frac{5i}{5} = 2 + i \), which is the standard form.

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Complex Numbers and Standard Form

Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. Writing a complex number in standard form means expressing it explicitly as a sum of its real and imaginary components.

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Percorso guidato

04:47

Complex Numbers In Polar Form

Division of Complex Numbers

Dividing complex numbers involves multiplying the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator. This process simplifies the expression into standard form.

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Percorso guidato

4:22

Dividing Complex Numbers

Complex Conjugate

The complex conjugate of a number a + bi is a - bi. Multiplying by the conjugate removes the imaginary part from the denominator, making it a real number, which is essential for simplifying complex fractions.

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Percorso guidato

5:33

Complex Conjugates

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