As I mentioned many posts before, maths is a language – a much more elegant and concise language when taking about maths stuff. This will be very apparent when I talk about the expressions in the clock I just bought for my office:
Each one of the maths expressions on the clock equal the number corresponding to its position on the clock. So I will be talking about each of these.
By far, the expression corresponding to “1” on the clock, illustrates the conciseness of maths expressions:\[
This expression makes use of more mathematical knowledge than any of the other expressions. It uses the maths constants e and 𝜋, and uses radians, trigonometric (or circular) functions, exponentials, and Euler’s identity. The two parts of this expression are e𝜋i and cos 𝜋. Let me first talk about cos 𝜋.
Now I have talked about trig functions before, but I mainly talked about sine of an angle (abbreviated “sin”). The cosine of an angle (abbreviated “cos”) is similar. You can take the cosine of an angle on your calculator, but you need to tell the calculator whether you are measuring angles in “degrees” or “radians”. In this expression, we are taking the cosine of 𝜋 radians. If your calculator is in radians mode and if you have ‘𝜋’ button, taking the cosine of 𝜋 will show -1. If you do not have a 𝜋 button, take the cosine of 3.14179 and you will get an answer approximately equal to -1. So, cos 𝜋 = -1.
Now the e𝜋i part requires a bit more of an explanation. The numbers e and 𝜋 have been discussed before. They are both irrational numbers which means that you cannot write them down exactly using numbers. We just agree that the symbol e is exactly e and the symbol 𝜋 is exactly 𝜋. If you have an ex button on your calculator, you can get an approximate value for e by typing 1, then the ex key. I have talked about e before. The letter e is for Euler who did a lot of work with this constant. So what is i?
The number i is not a real number. It is actually called an imaginary number. It is formally defined in terms of its square:
i2 = -1
which means that i is the square root of -1. I’m not making this up.
Since in the realm of real numbers, you know that you cannot take the square root of a negative number. So defining i is creating a whole new realm of numbers called complex numbers. Actually, complex numbers are the union of two realms: the reals and the imaginaries. There is a lot more to say about this but this will have to be a topic for a future post.
So what is e𝜋i where i is in the exponent? There is an identity (a maths rule) called Euler’s identity that explains what e𝜋i is equal to:
e𝜋i + 1 = 0
So e𝜋i must equal -1 for Euler’s identity to work. Again, explaining anything more is the topic of several future posts.
So if e𝜋i = -1 and cos 𝜋 = -1, then
e𝜋icos 𝜋 = (-1)(-1) = 1 which is the 1 o’clock spot on the clock.
I haven’t even scratched the surface of explaining this expression, but it has already filled up this post. The other expressions will not take as long to explain.