Problem 5 - Algebra and Trigonometry Real Mathematics, Real People

Question

What is the additive inverse of the complex number \(2-4 i ?\)

Step-by-Step Solution

Answer

The additive inverse of the complex number \(2-4i\) is \(-2+4i\).

Step 1: Identify the complex number

For this particular exercise, the complex number given to us is \(2-4i\). This is of the form \(a+bi\) where \(a = 2\) and \(b = -4\).

Step 2: Find the additive inverse

The additive inverse of a complex number \(a+bi\) is given by \(-a-bi\). Applying this rule to our complex number, we get \(-2+4i\) as the additive inverse.

Key Concepts

Complex Numbers

Complex numbers might seem intimidating at first, but they are quite straightforward. You can think of them as an extension of the real numbers. Each complex number consists of two parts: a real part and an imaginary part. They are usually written in the form of \(a + bi\), where \(a\) represents the real part, and \(bi\) stands for the imaginary part.

A complex number like \(2 - 4i\) can be visualized as a point on a two-dimensional plane. The horizontal axis is for the real part, and the vertical axis is for the imaginary part. This representation helps in understanding the behavior of complex numbers when you add, subtract, multiply, or find inverses of them.

This brings us to the additive inverse. The additive inverse of a complex number \(a+bi\) is simply \(-a-bi\). It's what you add to the original number to get zero. For example, for the complex number \(2 - 4i\), the additive inverse is \(-2 + 4i\). Just imagine flipping the sign of both parts!

Imaginary Unit

The imaginary unit is typically represented as \(i\). It forms the foundation of constructing imaginary numbers. In mathematical terms, \(i\) is defined as the square root of \(-1\).

That might sound strange because there's no real number whose square is negative. That's where the power of \(i\) comes into play. It allows mathematicians to extend the number system to solve equations that have no real solution, such as \(x^2 + 1 = 0\).

When dealing with complex numbers, \(i\) is used to represent the imaginary part. For instance, in \(2 - 4i\), the \(-4i\) includes the imaginary unit, making it clear that \(-4\) is the coefficient of the imaginary part. Mastering how to manipulate and understand \(i\) will open up a new dimension to solve mathematical problems.

Algebraic Expressions

Algebraic expressions involving complex numbers work similarly to those with just real numbers. The key difference lies in handling the imaginary unit \(i\). You will often see expressions like \(a + bi\) where you need to apply basic algebraic operations.

Combining LIKE terms
Remember the basics: only combine similar terms. This means you combine real numbers with real numbers and imaginary parts with imaginary parts. For example, \( (3 + 2i) + (4 - 5i) = 7 - 3i \). You add the real parts \(3 + 4\) and the imaginary parts \(+2i - 5i\).

Using the distributive property
When multiplying complex numbers, expand like regular binomials while respecting \(i^2 = -1\). For example, \((1 + i)(2 + 3i)\) becomes \(1(2) + 1(3i) + i(2) + i(3i)\), which simplifies to \(2 + 3i + 2i + 3(-1)\), and finally \(-1 + 5i\).

Algebra with complex numbers is not difficult, but it requires paying attention to \(i\) and applying traditional algebra rules carefully. Once you get the hang of it, you'll find manipulating complex expressions to be a breeze.