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 real and imaginary parts of each complex number: \((3 + 2i)\) and \((5 - 7i)\).
2. Subtract the real parts: \(3 - 5\).
3. Subtract the imaginary parts: \(2i - (-7i)\).
4. Simplify the subtraction of the real parts to get a new real part.
5. Simplify the subtraction of the imaginary parts to get a new imaginary part.

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. Understanding complex numbers is essential for performing operations such as addition and subtraction.

Addition and Subtraction of Complex Numbers

To add or subtract complex numbers, combine their real parts and their imaginary parts separately. For example, when adding (3 + 2i) and (5 - 7i), you would add 3 and 5 to get 8, and 2i and -7i to get -5i, resulting in the complex number 8 - 5i.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where 'a' and 'b' are real numbers. This format is important for clarity and consistency in mathematical communication, allowing for easier interpretation and manipulation of complex numbers in various mathematical contexts.