Recently I was talking to my teacher about complex and imaginary numbers and he told me basically that $i$ is a complex number; its real part is just 0. However, this has made me wonder; if you can see $i$ as a complex number because you could argue its real part is 0, how can you differentiate between complex numbers and imaginary numbers?
[Math] Difference between imaginary and complex numbers
complex numbers
Related Solutions
Ideally what you want to have at the end is a nice complex number of the form : $$z = x+iy$$ Unfortunately as you have shown, complex numbers aren't always written this way... Being able to write them under an exponential form is a great way to evaluate them for a few reasons.
The first reason I see is that it is very easy to take their conjugate form, simply change the sign of the argument.
Another good reason is that $re^{i\theta} = r\cos(\theta) + ir\sin(\theta)$ making it easy to transform into a nice form.
They are very easy to multiply and divide with each another.
One problem though : you can't add them simply.
So when evaluating horrific complex numbers here are my advice:
- If you need to add or subtract complex numbers of the form $x + iy$, then just use the normal formula.
- If you have expression you want to add but aren't of the form $x+iy$, you should normally be able to write them as a polar form and then rewrite them as $x+iy$ using Euler's formula.
- If you want to divide, multiply or take a power of two complex numbers, always use exponential forms, it is the easiest way.
With all of this you should (slowly) be able to evaluate complex numbers written using elementary operations. As you can see, the exponential form is used very often except when adding or subtracting. In the end you should always be able to write it under the form $x+iy$ where the imaginary and real parts are explicit.
Let $z = a + bi$ be a complex number. The real part of $z$ is $a$. The imaginary of $z$ is $b$. Thus, the real part of $5 + 3i$ is $5$ and its imaginary part is $3$. Note that the real and imaginary parts of a complex number correspond to its coordinates on the complex plane.
An imaginary number is a complex number of the form $z = 0 + bi = bi$. An example of an imaginary number is $2i$.
Note that an imaginary number is a complex number whose real part is equal to $0$, while a real number is a complex number whose imaginary part is equal to $0$.
Best Answer
Every complex number can be written as $z=a+bi$, where $a,b\in \mathbb{R}$ (real numbers). The number $a$ is called real part of $z$ and the number $b$ is the imaginary part of $z$.
If the real part is zero then we call $z=bi$ as pure imaginary complex number.
Here is a diagram to show the inclusions: