In Exercises 1–8, add or subtract as indicated and write the result in standard form. (3 + 2i) − (5 − 7i)

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Identify the problem as the subtraction of two complex numbers: \((3 + 2i) - (5 - 7i)\).

Recall that to subtract complex numbers, subtract their real parts and their imaginary parts separately.

Subtract the real parts: \(3 - 5\).

Subtract the imaginary parts: \(2i - (-7i)\), which simplifies to \(2i + 7i\).

Combine the results to write the answer in standard form \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of \(i\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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. The standard form means writing the result explicitly as a sum of a real number and an imaginary number.

Addition and Subtraction of Complex Numbers

To add or subtract complex numbers, combine their real parts and their imaginary parts separately. For example, (a + bi) − (c + di) = (a − c) + (b − d)i.

Imaginary Unit i and Its Properties

The imaginary unit i is defined as the square root of -1, with the property i² = -1. Understanding this helps in simplifying expressions involving imaginary parts.

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