In Exercises 1–8, add or subtract as indicated and write the result in standard form.(7 + 2i) + (1 − 4i)

Verified step by step guidance

1. Identify the real and imaginary parts of each complex number: \((7 + 2i)\) and \((1 - 4i)\).
2. Add the real parts together: \(7 + 1\).
3. Add the imaginary parts together: \(2i - 4i\).
4. Combine the results from the previous steps to form a new complex number.
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 and subtraction.

Addition of Complex Numbers

To add complex numbers, you combine their real parts and their imaginary parts separately. For example, when adding (7 + 2i) and (1 - 4i), you add 7 and 1 to get 8, and 2i and -4i to get -2i, resulting in the sum 8 - 2i. This process highlights the importance of treating real and imaginary components distinctly.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where 'a' and 'b' are real numbers. Writing complex numbers in this form is crucial for clarity and consistency in mathematical communication. The result of operations on complex numbers should always be expressed in this standard form to ensure it is easily understood and correctly interpreted.