In Exercises 1–10, perform the indicated operations and write the result in standard form. (8 − 3i) − (17 − 7i)

Verified step by step guidance

1. Identify the expression to simplify: \((8 - 3i) - (17 - 7i)\).
2. Distribute the negative sign across the second set of parentheses: \(8 - 3i - 17 + 7i\).
3. Combine the real parts: \(8 - 17\).
4. Combine the imaginary parts: \(-3i + 7i\).
5. Write the result in standard form \(a + bi\), where \(a\) is the real part and \(b\) is the 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, subtraction, multiplication, and division.

Operations with Complex Numbers

To perform operations with complex numbers, such as addition and subtraction, you combine the real parts and the imaginary parts separately. For example, in the expression (a + bi) - (c + di), the result is (a - c) + (b - d)i. This concept is crucial for simplifying expressions involving complex numbers.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where 'a' and 'b' are real numbers. When performing operations on complex numbers, the result should be expressed in this form to clearly distinguish between the real and imaginary components. This clarity is important for further mathematical analysis and applications.