Find each quotient. Write answers in standard form. (1-3i) / (1+i)

검증된 단계별 안내

Identify the problem as dividing two complex numbers: \(\frac{1 - 3i}{1 + i}\).

To simplify, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \(1 + i\) is \(1 - i\), so multiply both numerator and denominator by \(1 - i\):

\[\frac{1 - 3i}{1 + i} \times \frac{1 - i}{1 - i}\]

Use the distributive property (FOIL) to expand both numerator and denominator:

Numerator: \((1 - 3i)(1 - i) = 1 \cdot 1 - 1 \cdot i - 3i \cdot 1 + (-3i)(-i)\)

Denominator: \((1 + i)(1 - i) = 1 \cdot 1 - 1 \cdot i + i \cdot 1 - i \cdot i\)

Simplify both expressions by combining like terms and using \(i^2 = -1\).

Finally, write the result in standard form \(a + bi\) by separating the real and imaginary parts.

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주요 개념

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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 answers in standard form means presenting the result explicitly as a sum of a real number and an imaginary number, such as x + yi.

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 to standard form.

Complex Conjugate

The complex conjugate of a number a + bi is a - bi. Multiplying by the conjugate removes the imaginary part in the denominator because (a + bi)(a - bi) equals a² + b², a real number, facilitating division.