What is the result of combining complex numbers?

Understanding how to combine complex numbers is a fundamental skill in algebra and higher mathematics. Complex numbers are numbers that have both a real part and an imaginary part. They are usually written in the form $a + bi$, where $a$ is the real part, and $bi$ is the imaginary part.

Adding Complex Numbers

To add two complex numbers, you simply add their corresponding real parts and their corresponding imaginary parts. For example, let’s add the complex numbers $3 + 4i$ and $1 + 2i$:

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

First, add the real parts:

$3 + 1 = 4$

Then, add the imaginary parts:

$4i + 2i = 6i$

So, the result of the addition is:

$4 + 6i$

Subtracting Complex Numbers

Subtracting complex numbers follows a similar process. You subtract the real parts and the imaginary parts separately. Let’s subtract $1 + 2i$ from $3 + 4i$:

$(3 + 4i) – (1 + 2i)$

First, subtract the real parts:

$3 – 1 = 2$

Then, subtract the imaginary parts:

$4i – 2i = 2i$

So, the result of the subtraction is:

$2 + 2i$

Multiplying Complex Numbers

Multiplying complex numbers is a bit more involved because you need to use the distributive property (also known as the FOIL method for binomials). Let’s multiply $3 + 4i$ and $1 + 2i$:

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

Use the distributive property:

$3 times 1 + 3 times 2i + 4i times 1 + 4i times 2i$

This simplifies to:

$3 + 6i + 4i + 8i^2$

Since $i^2 = -1$, we can replace $8i^2$ with $-8$:

$3 + 6i + 4i – 8$

Combine like terms:

$-5 + 10i$

Dividing Complex Numbers

Dividing complex numbers involves multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number $a + bi$ is $a – bi$. Let’s divide $3 + 4i$ by $1 + 2i$:

$frac{3 + 4i}{1 + 2i}$

Multiply the numerator and the denominator by the conjugate of the denominator:

$frac{(3 + 4i)(1 – 2i)}{(1 + 2i)(1 – 2i)}$

Use the distributive property:

$frac{3 – 6i + 4i – 8i^2}{1 – 4i^2}$

Simplify the numerator and the denominator (remember $i^2 = -1$):

$frac{3 – 6i + 4i + 8}{1 + 4}$

Combine like terms:

$frac{11 – 2i}{5}$

Separate into real and imaginary parts:

$frac{11}{5} – frac{2}{5}i$

Conclusion

Combining complex numbers involves straightforward operations on their real and imaginary parts. Whether you’re adding, subtracting, multiplying, or dividing, understanding these steps will help you manage complex numbers effectively.

Citations

1. 1. Khan Academy – Complex Numbers
2. 2. Math is Fun – Complex Numbers
3. 3. MIT OpenCourseWare – Complex Numbers