Question

Combine and simplify. $$(a-3 i)+(a+5 i)$$

Short answer

2a + 2i

Step-by-step solution

Group Real and Imaginary Terms

Combine the like terms by grouping the real parts and the imaginary parts of the complex numbers separately. In this case, group the real parts 'a' and 'a', and the imaginary parts '-3i' and '5i'.

Combine Like Terms

Add the real parts together and the imaginary parts together. The real parts add up to '2a' (since a + a = 2a) and the imaginary parts add up to '2i' (since -3i + 5i = 2i).

Write the Simplified Expression

Express the result as a single complex number by writing the combined real part and the combined imaginary part together, resulting in '2a + 2i'.

Key concepts

Complex Numbers

At the heart of algebra and geometry lies the fascinating concept of complex numbers. These numbers are an expansion of the real numbers and include all possible numbers that can be expressed in the form of a + bi, where a and b are real numbers, and i is the imaginary unit satisfying i^2 = -1. This seems tough, doesn't it? Well, think of i as a little twist in our number system, allowing us to solve equations that don’t have solutions within the real numbers, such as x^2 = -1.

Using complex numbers, we can represent two dimensions on a plane: the real part, a, is on the horizontal axis, while the imaginary part, bi, is on the vertical axis. This way, every point on the plane corresponds to one unique complex number. The point of their combination? It's like bringing together the best of both dimensions, creating a full spectrum of possibilities for calculation and representation in various fields, especially in engineering and physics.

Real and Imaginary Parts

When dealing with complex numbers, it's crucial to understand the difference between the real and the imaginary parts. In a complex number of the form a + bi, a is the real part, and bi is the imaginary part. Real numbers are the ones we use in everyday life, like 3, -5, or 0.42, and they represent a point on a straight line. The imaginary part, however, involves the mind-bending i, the square root of -1.

Why Separate Them?

Separating the real and imaginary parts has a clear advantage: it simplifies the arithmetic of complex numbers. Just as you wouldn't mix up apples and oranges, you shouldn't mix real numbers with imaginary ones when you're adding or subtracting. Keep the real parts with the real and the imaginary parts with the imaginary, and you will navigate through problems with complex numbers with much more ease. In our exercise, identifying a as the real part and -3i and 5i as the imaginary parts sets the stage for a simplified solution.

Combining Like Terms

Now, let's get into the heart of algebra - combining like terms. This is where we streamline expressions so they’re easier to work with. Just like organizing a toolbox, combining like terms in algebra means putting similar items together. In the context of complex numbers, like terms are the real parts that can be added together, and the imaginary parts that can also be combined.

How to Combine Them?

Simply by adding or subtracting them! If you're dealing with a - 3i and a + 5i, you add a to a to get 2a, and -3i to 5i to get 2i. Remember, you cannot combine a real number with an imaginary number - they are as different as distances and temperatures.

So, why is this so fundamental? Well, combining like terms not only cleans up expressions and makes them easier to understand but also sets the stage for further manipulation, such as solving equations. It's an essential tool for anyone delving into algebra!