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.