Question

Perform the indicated operation and write the result in standard form. $$(3+2 i)-(6+13 i)$$

Short answer

The result of the operation \((3+2i) - (6+13i)\) is \(-3 - 11i\).

Step-by-step solution

Identify the real and imaginary parts

The first number is \(3+2i\), where \(3\) is the real part and \(2i\) is the imaginary part. The second number is \(6+13i\), where \(6\) is the real part and \(13i\) is the imaginary part.

Perform the subtraction

Subtract the real parts from each other: \(3 - 6 = -3\). This is the real part of the answer. Then subtract the imaginary parts from each other: \(2i - 13i = -11i\). This is the imaginary part of the answer.

Write the answer in standard form

Combine the real part and the imaginary part to write the answer: \(-3 - 11i\).

Key concepts

Standard Form in Complex Numbers

Complex numbers are a sort of extension of the real number system that include solutions to certain algebraic equations that no real number can solve, such as the equation x^2 + 1 = 0. These are written in what we call the 'standard form' which looks like a + bi. In this expression, a and b are real numbers, and i is the imaginary unit, defined as the square root of -1. The real number a represents the 'real part' and bi represents the 'imaginary part' of the complex number.

To manipulate complex numbers, whether it's for addition, subtraction, multiplication, or division, using this standard form makes the arithmetic much clearer and easier to understand. For instance, in our exercise (3+2i)-(6+13i), both numbers are already presented in the standard form, which allows us to identify and operate on the real and imaginary parts separately.

Real and Imaginary Parts

In order to effectively work with complex numbers, it's crucial to distinguish between their real and imaginary parts. This is because each part behaves differently under mathematical operations. The 'real part' is the component without the imaginary unit i, while the 'imaginary part' includes the imaginary unit. In the context of our exercise, the real part of the first complex number 3+2i is 3, and its imaginary part is 2i. Similarly, the second complex number 6+13i has a real part 6, and an imaginary part 13i.

By separating these parts, we can handle them using the ordinary rules of algebra applied to real numbers, before merging them back into a complex number in standard form as the final step of our operation.

Complex Number Arithmetic

Complex number arithmetic might seem daunting at first, but once you understand the concept of separating real and imaginary parts, it becomes straightforward. Let's use the subtraction operation, as seen in our exercise, to illustrate this.

When subtracting complex numbers, you subtract the real part of one from the real part of the other, and do the same with the imaginary parts. So for the subtraction (3+2i)-(6+13i), respectively subtract 6 from 3 to get -3, and 13i from 2i to get -11i. Then, you simply combine these results to express the final answer in standard form, which here is -3 - 11i.

This example shows that each part is dealt with on its own, following the basic principles of subtraction - a reassuringly familiar process for any student comfortable with elementary arithmetic. Remember that i is treated as a constant; its presence helps in indicating the imaginary part and doesn't affect the subtraction operation itself.