Steps to Multiply Complex Numbers

Multiplying complex numbers might seem tricky at first, but it’s straightforward once you understand the process. Complex numbers have the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit with the property $i^2 = -1$. Let’s break down the steps to multiply two complex numbers.

Step-by-Step Process

1. Write Down the Numbers
   Suppose you want to multiply two complex numbers: $(a + bi)$ and $(c + di)$. For example, let’s take $(3 + 2i)$ and $(1 + 4i)$

2. Use the Distributive Property
   Next, we’ll distribute each term in the first complex number by each term in the second complex number, similar to how you would use the distributive property in algebra:

   $(a + bi)(c + di) = a times c + a times di + bi times c + bi times di$

   Using our example:

   $(3 + 2i)(1 + 4i) = 3 times 1 + 3 times 4i + 2i times 1 + 2i times 4i$

3. Perform the Multiplications
   Now, multiply each pair of terms:

   $3 times 1 = 3$

   $3 times 4i = 12i$

   $2i times 1 = 2i$

   $2i times 4i = 8i^2$

4. Combine Like Terms
   Combine the like terms (real parts with real parts, imaginary parts with imaginary parts):

   $3 + 12i + 2i + 8i^2$

5. Simplify Using $i^2 = -1$
   Recall that $i^2 = -1$. So, replace $i^2$ with $-1$ in the expression:

   $8i^2 = 8 times -1 = -8$

   Now, substitute back into the expression:

   $3 + 12i + 2i – 8$

6. Final Simplification
   Combine the real parts and the imaginary parts:

   $(3 – 8) + (12i + 2i) = -5 + 14i$

   So, $(3 + 2i)(1 + 4i) = -5 + 14i$

Example Problem

Let’s try another example: $(2 + 3i)(4 – i)$

1. Distribute the terms:

$(2 + 3i)(4 – i) = 2 times 4 + 2 times -i + 3i times 4 + 3i times -i$

2. Perform the multiplications:

$2 times 4 = 8$

$2 times -i = -2i$

$3i times 4 = 12i$

$3i times -i = -3i^2$

3. Combine like terms:

$8 – 2i + 12i – 3i^2$

4. Simplify using $i^2 = -1$:

$-3i^2 = -3 times -1 = 3$

5. Substitute back in:

$8 – 2i + 12i + 3$

6. Final simplification:

$(8 + 3) + (-2i + 12i) = 11 + 10i$

Thus, $(2 + 3i)(4 – i) = 11 + 10i$

Conclusion

Multiplying complex numbers involves using the distributive property, combining like terms, and simplifying using $i^2 = -1$. With practice, this process becomes second nature, allowing you to handle complex numbers with ease.

Citations

1. 1. Khan Academy – Multiplying complex numbers
2. 2. Math is Fun – Complex Numbers
3. 3. Purplemath – Multiplying Complex Numbers