Understanding Complex Numbers Made Easy

Complex numbers are an important part of math, especially when you reach Year 9.

When we talk about multiplying complex numbers, there are a few main ideas to grasp.

The most important ones are:

1. The distributive property
2. The special rule for the imaginary unit $i$, where $i^2 = -1$.

Let’s break down how to multiply complex numbers in a way that’s easy to understand.

What Are Complex Numbers?

Complex numbers are written in the form $a + bi$, where:

• $a$ is the real part
• $b$ is the imaginary part
• $i$ is the imaginary unit, which means $i^2 = -1$

For instance, $3 + 4i$ is a complex number. Here, $3$ is the real part, and $4$ is the imaginary part.

How to Multiply Complex Numbers

When you want to multiply two complex numbers, like $(a + bi)$ and $(c + di)$, you can use something called the distributive property. This is often shown as the FOIL method, which stands for First, Outer, Inner, Last.

Here’s how to do it step by step:

1. Identify the Parts:

   • First number: $a + bi$
   • Second number: $c + di$

2. Use the Distributive Property: You can write it out like this:

   $(a + bi)(c + di) = a \cdot c + a \cdot (di) + (bi) \cdot c + (bi) \cdot (di)$

3. Calculate Each Piece:

   • First: $a \cdot c$
   • Outer: $a \cdot (di) = adi$
   • Inner: $(bi) \cdot c = bci$
   • Last: $(bi) \cdot (di) = b \cdot d \cdot i^2 = bd(-1) = -bd$

4. Combine Like Terms: Now you put all the pieces together: $ac + adi + bci - bd = (ac - bd) + (ad + bc)i$

So, the product of the complex numbers $(a + bi)$ and $(c + di)$ can be simplified to: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

An Example of Multiplying Complex Numbers

Let’s look at a specific example:

Multiply $(2 + 3i)$ and $(4 + i)$.

1. Identify the parts:

   • $a = 2$, $b = 3$, $c = 4$, $d = 1$

2. Use the distributive property: $(2 + 3i)(4 + i) = 2 \cdot 4 + 2 \cdot i + 3i \cdot 4 + 3i \cdot i$

3. Calculate each piece:

   • First: $2 \cdot 4 = 8$
   • Outer: $2 \cdot i = 2i$
   • Inner: $3i \cdot 4 = 12i$
   • Last: $3i \cdot i = 3i^2 = 3(-1) = -3$

4. Combine the terms: $8 + 2i + 12i - 3 = (8 - 3) + (2 + 12)i = 5 + 14i$

Conclusion

The main ideas for multiplying complex numbers are to understand what they are, use the distributive property, and remember that $i^2 = -1$.

If you master these steps, you’ll be able to confidently work on problems with complex numbers. This knowledge is a key part of your math learning in Year 9!