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

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Identify the complex number in the denominator: \(3 - i\).

To divide by a complex number, multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of \(3 - i\) is \(3 + i\).

Multiply numerator and denominator by \(3 + i\): \(\frac{2}{3 - i} \times \frac{3 + i}{3 + i} = \frac{2(3 + i)}{(3 - i)(3 + i)}\).

Expand the numerator: \(2(3 + i) = 6 + 2i\).

Expand the denominator using the difference of squares formula: \((3 - i)(3 + i) = 3^2 - (i)^2 = 9 - (-1) = 9 + 1 = 10\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Complex Number Standard Form

A complex number is expressed in standard form as a + bi, where a is the real part and b is the imaginary part. Writing results in this form helps clearly separate the real and imaginary components, making it easier to interpret and use in further calculations.

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 converts the division into a simpler form that can be expressed as a standard complex number.

Complex Conjugate

The complex conjugate of a number a + bi is a - bi. Multiplying by the conjugate removes the imaginary part from the denominator because (a + bi)(a - bi) equals a² + b², a real number. This technique is essential for simplifying complex fractions.