Let's work our minds together, it'll be fun. So :
Let's consider the set of all the cardinals of definite setsi, we'll call it the set of natural numbers and note it ℕ.
We can now define the function addition, which takes two elements out of ℕ and adds them together. For instance :
- 2 + 4 = 6
Addition happens to be a commutative operation, which means it yields the same result irrespective of the order of the two elements :
- 4 + 2 = 6
Since in the lovely field of mathematics there exists no entropy and thus no timeii any function can be reversed, which is to say we can define the function subtraction, which given a pair of numbers yields that third number which at some point in the indefinite past was added to the second to yield the first. Like so :
- 6 - 4 = 2
2 being that number which was sometime added to 4 in order to yield 6. Unlike addition, however, subtraction is not commutative. That is to say,
- 4 - 6 = -2
-2 is clearly not 2, and what's worse, -2 is not actually in ℕ, as there may exist an empty bag in which if you put two eggs you'll get a bag with two eggs in it, and there may also exist a bag with two eggs in it from which you can thus extract two eggs rendering it empty. However there exists no bag in which you can insert two eggs in order to thus render it empty, nor an empty bag from which you could pull forth two eggsiii. Which means that by logically extending the original addition function through constructing its reverse, we've also forced an expansion of our original set of cardinals. And so ℤ is borniv.
ℤ is simply ℕ with its evil mirror -ℕ, which is to say 1 and -1, 2 and -2, 56 and -56 and so on. Now on this ℤ we can define the function multiplication, which takes two elements and multiplies them together. Like so :
- 2 × 4 = 8
Much like the earlier addition, multiplication is also commutative, which is to say
- 4 × 2 = 8
And much like in the case of addition an inverse could be constructed, which is to say a function which given two numbers would find the third which multiplied with the second would yield the first. Like this :
- 8 ÷ 4 = 2
This would be called division, and much like subtraction, it is not commutative. To wit :
- 4 ÷ 8 = 1/2
1/2, or 0.5 as it is otherwise known, is a fraction, which is to say not an integer, which is to say not Zahlen and thus not in ℤ. So yet again we've ended up extending our original set of "numbers" to include even more types of numbers to satisfy the ever growing demands of our synthetic constructs which at the time seemed logical and down the road ended up spelling trouble. And thus ℚ is born.v
ℚ is the set of rational numbers, which is to say all natural ℕs, its evil mirror the un-natural -ℕ as well as all un-integers ie the fractions, much to teenagers' delight.
ℚ of course allows the following abstraction be introduced, that the function power is defined, which simply multiplies its first element with itself as many times as the second counts. This function can well be defined on ℤ, because :
- 4 2 = 16
- 4 -2 = 1/16vi
So now we're stuck with a function defined on ℚ with elements from ℤ ? What nonsese is this, seriously. It should properly be expanded so it takes elements from ℚ and gives results in ℚ, if indeed ℚ is to hold its rank of truely and definitively the last and ultimate all-numbers concentration camp. (Javohl!)
What could a fraction possibly mean should it happen in the 2nd spot however ? Quite obviously... the reverse function of this raising to the power. Thus
- 4 1/2 = the square root of 4 = that number which multiplied with itself twice yields 4.
And now... suddenly we've lost commutativity, because
- 2 1/4 = √
God help us, because the square root of 2 is not a fraction, nor could it be expressed as a finite decimal notation. It's only 1.414213562 because your computer runs out of decimals there, but with a better one you could see it's really endless.
So now it seems we've managed to expand ℚ again. Fine, but let this be for the last time : we'll call it ℝ for real numbers and let it include nonsense like the square root of two, with the strong implication behind the name being, of course, that we are finally done : really and definitively 4ever&ever, these are the numbers and we're done with the nonsense.
Except... what do you suppose would be the square root of -1 ? Clearly a negative multiplied with itself could only yield a positive, and thus the square root of -1 could never yield a number, even in ℝ. Which means ℝ just got expanded again, we'll call the √beauty i - short for imaginary, 'cause by now we've definitely run ourselves outside of reality - and the new set ℂ, for complex, because well... it's complicated.
ℂ will thus include all natural and all unnatural numbers, as well as the uninteger, irrational and imaginary. All ℂ numbers will be noted as a+ib, where a is the real part and ib the imaginary part, so that all ℝ numbers are simply ℂ numbers with a convenient b = 0 tail.
And now we're done, right ? Wrong, of course. The reason we're not done is expressed by Godel, we'll never be done, we can not be done. We can't ever have a set of numbers and a set of operations so that we include everything. Just how it goes.
This completes the first part of our miracle.
Every circle has a radius.
Every circle also has a circumference.
The ratio of a circle's radius and a circle's circumference will always be π, which is an irrational number, much like the square root of 2.
This completes the 2nd part of our miracle.
Suppose you have a Bitcoin on deposit somewhere, with a merchant of spotless reputation and infinite credibility, who pays you 100% per year. If the interest is credited at the end of the year, clearly after one year you will have two Bitcoins. What happens however if the interest is computed more often ?
If you get the interest calculated each 6 months, you will end up with 50% and 50% yielding 2.25 Bitcoin at the end of the year. If if is calculated each quarter you will end up with 2.44140625 Bitcoin.
And if it is calculated every millionth of a millionth of a millionth of a ... microsecond ? Why, you will end up with e Bitcoins. That's right, e, Euler's constant, it's the limit of f(x) = (1 + 1/x)x when x goes to infinity. An irrational number, just like π and i.
This completes the third and last part of our miracle, we are now ready to proceed. Behold :
e π × i = -1
That's right baby : the piece-of-shit tail-of-the-devil problem-number we ended up straddled with while doing set theory, the essence of the circle and the essence of usury all come together, as if by magic, into one breathtakingly simple formula.
Why is this ? Why should this be ? What in the world and all the stars could these three things possibly have to do with each other ? And nevertheles... there they are.
So now : this is why educated gentlemen such as yourselves have no need, no use and no interest for pedestrian nonsense a la "intelligent design", and this is why we don't go to church, that wanna-be thought simulacrum built entirely in the manner of a ten year old wearing his father's pistol belt.
We don't need to, we got all the miracles we could possibly need right here. Enjoy!———
- The particular form of the definition is important for reasons. The meaning is as follows : any collection of anything is a set. A definite set is any collection which you could - in principle - produce for my inspection. In this sense the fingers on your hand are a definite set of fingers, the pockets on your jacket a definite set of pockets and the stars on the sky a definite set of stars, whereas your future jobs aren't a definite set of jobs, all the potatoes you could eat aren't a definite set of potatoes and so on. A cardinal is simply the count, for instance the cardinal of the set of the fingers on your hand is - in most cases - five. [↩]
- Because time is strictly the direction in which entropy flows, much like "down" is, as far as rubber ducklings in a tub are concerned, the direction in which the water sucks them. [↩]
- Ah, if only, how popular frittata'd be. [↩]
- ℤ stads for Zahlen, which in German means number, because Frege and the gang did a lot of work on the topic. [↩]
- It so happens that while ℕ and ℤ both have the same cardinal, which is to say contain the same count of elements,
ℚℝ is actually larger. This is a topic of much advanced studying we won't really be covering here. [↩]
- This is why your height, say 187 centimeters, could be expressed as 1.86 × 10-3 kilometers just as well as 1.86 × 103 milimeters. [↩]