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

Verified step by step guidance

1. Identify the problem as the subtraction of two complex numbers: \((3 + 2i) - (5 - 7i)\).
2. Rewrite the expression by distributing the subtraction sign to the second complex number: \((3 + 2i) - 5 + 7i\).
3. Group the real parts together and the imaginary parts together: \((3 - 5) + (2i + 7i)\).
4. Perform the subtraction and addition separately for the real and imaginary parts: calculate \$3 - 5\( and \)2i + 7i$.
5. Write the final result in standard form \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary part.

Key Concepts

Complex Numbers

Complex numbers are numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit with the property i² = -1. They combine real and imaginary parts and are used to represent quantities that cannot be expressed on the real number line alone.

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. This operation follows the same rules as combining like terms in algebra.

Standard Form of a Complex Number

The standard form of a complex number is written as a + bi, where a is the real part and b is the coefficient of the imaginary part. Writing the result in standard form means expressing the answer clearly with the real and imaginary parts separated.