Complex numbers

[boneill3]

Hi guys just wondering

is the the complex number 3i expressed as a+bi

equal to (0+3i)

also is -2

the same as (-2+0i)

Thanks
Brendan

[MaximilianMayrhofer]

yep.

[NickThissen]

Draw both 'variations' in the complex plane (y-axis: imaginery axis, x-axis: real axis) and you will see that they are ofcourse the same.

You could even say that every number is a complex number, just without a complex part!

[MaximilianMayrhofer]

You could probably say it because it's already an accepted mathematical fact. The set of complex numbers, C, includes every single possible number. It is the global set. The set of real numbers, R, is a perfect subset of C.

[boneill3]

Thanks for your help guys
regards
brendan

[jemidiah]

You could probably say it because it's already an accepted mathematical fact. The set of complex numbers, C, includes every single possible number. It is the global set. The set of real numbers, R, is a perfect subset of C.

In all technicality, R is not quite a subset of C--instead "R+0i = {a+0i | a in R}" is a subset of C, and R is isomorphic to R+0i.

I always find these sort of things interesting which is why I mention it

[Lerroy_Jenkins]

Hey guys, what is a complex number?

[NickThissen]

Hey guys, what is a complex number?

The complex numbers are an extension of the real numbers. Real numbers are the numbers you come across usually, numbers like 1, 2, 3 or numbers like 5.281985...

The real numbers can be extended to the complex numbers by using the imaginery unit i, which is defined as i² = -1.
You will notice that any real number multiplied by itself (squared) is always positive, this is what makes i so special.
You may now also notice that the square root of a negative number is no longer impossible. In the real numbers, the square root of a negative number does not exist, since no real number squared is negative. In the complex numbers however, you can define the square root of a negative (real) number.
For example, the square root of -9 = 3i.

Complex numbers can be written in the form z = x + iy, where x and y are real numbers. All complex numbers have both an imaginery part (y) and a real part (x) (although sometimes one or both may be zero).

You can visualize a complex number by first visualizing the 'real number line'. The real number line is just a line on which all real numbers are shown, ranging from -infinity through zero, to +infinity.
The complex number is the extension of this line, to the complex plane. Imagine a normal graph with the x and y axis. The x-axis is now the real number line, and the y-axis is the imaginery axis.

(Source: Wikipedia)

Complex numbers are used in a variety of different situations but you usually encounter them in wave-like situations. For example, an AC (changing) current or voltage can be described using complex numbers, which allow for more easy calculations.

I believe that the 'discovery' of complex numbers occured when some mathematician (Euler maybe?) tried finding the roots of third degree polynomials (cubics). While the three roots were all real, if you wanted to calculate them you had to take the square root of a negative number, which was up until that moment not possible.

You may find it weird that we can simply 'define' a complex number and the imaginery unit, but it's not actually weird at all. Back in the old days, the only numbers that existed were natural numbers (0, 1, 2, 3, 4...). There were no negative numbers and no 'comma-numbers' (3.19858...)!

Here's the wikipedia site you may find interesting, although it is probably a bit too complicated for you if you don't know what a complex number is.
http://en.wikipedia.org/wiki/Complex_numbers

[Lerroy_Jenkins]

Well, that made almost 100% Sense lol. I have always been good at numbers, but I did not study maths further at college.

Thank you for taking the time to explain that to me.