Yes, I do realize this node has an odd structure. This is because it used to be "The four most important numbers" and it had two or three other posts. When I posted mine (origanally all this but the first paragraph), I simply explained why I believed i should be the fifth most important number. Being the lazy-ass that I am, I decided to just put a first paragraph in as an introduction, since by this time, the other posts were nuked. So in conclusion, please forgive the odd structure of this node.

On my way back from a Mu Alpha Theta competition, a fellow student of mine asked me what were the five most important numbers. I sat and thought about it for a few minutes and responded, "Pi, e, 0, 1 and infinity". He then explained that infinity is not a number. I then sat back into my seat and rethought. Hmmm, what's another— and then it popped into my mind...

Many consider i (the square root of -1) to be another extremely important number. Why is it important, you ask? My guess is its impact on mathematics. If you think about it, the discovery of i created a whole new field of mathematics, possibly even eventually leading toward some of the non-real mathematics that we do today, like fuzzy logic (like when something can be both true and false!). All of these five numbers (0, 1, e, π and i) can be found in the equation "ei + 1 = 0". This is Euler's Relation.
If you don't understand why this works, here's an explanation:
eia = cos(a) + i *sin(a)
ei = cos(π) + i *sin(π)
ei = -1 + i *0
ei = -1 + 0
ei + 1 = 0
note: A more complete and better explained proof can be found at Euler Formula

As for why the other numbers are important, here's my take:
0; for the longest time, people didn't have the concept of nothing. Believe it or not, the idea of zero wasn't thought up until the dark ages! One of my favorate math jokes was from a movie when one guy says to the other "I've discovered it!" and the other guy goes "What?" and the first guy responds, "Oh, nothing, nothing." Zero is a lot like sex. Before you know about it, it isn't a necessity at all. Once you find about about it, it's really cool and useful. Zero also has some very odd properties. Like you can multiply with any real number to produce itself! It is also neither positive nor negitive. It isn't even real, but it isn't imaginary, either! It's just... nothing! It's the bulls-eye on the Cartesian coordinate system (also know as the origin) and on the complex plane.
1; this is another number with odd properties. Like whenever you multiply it by another real number, you always get that number! It is also the only number with only one factor: itself, which means one is neither prime (having only two divisors, 1 and itself) or composite (having more than two divisors) note: 1 is not prime has a very good discussion on this. It is, in one word, "unity". Sing it! "One... is the loneliest number..."
π; (~3.14) this constant gave birth an easy (well, not THAT easy since you can't precisely calculate π) way to calculate areas of round-shaped objects, like circles,elipses, spheres, etc. Instead of having to look at really small portions of the shape and approximating those values, all one would need to do (for a circle/ellipse) is measure the major axis and minor axis, multiply those together and then multiply by π. π defined as the ratio between the circumference and diameter of a circle.
e; (~2.71) this constant, created by Euler (and named after him, to boot), is defined as
 lim  (1+1/x)x.
This number is very important because it can be used in almost every branch of mathematics. For example, it can be used to calculate continuously compounded interest, which banks use.
for more information on why these numbers are so special, visit 0, 1, e, pi, i

Hey? Why isn't positive and negative infinity on the list of the the 5 most important numbers?!
#1, because positive and negative infinity are the same thing, because if you keep going in one direction, you'll eventually converge with all other directions.
#2, because infinity is not a number, it's a concept.

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