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November 2, 2016  

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Imaginary Numbers: What the heck is it really?? How can a number be imaginary?? Well friends, let me tell you, on this Halloween Night, they are real. There are INDEED numbers which are considered imaginary. They have very special properties which do not exactly line up with what one might consider the "conventional" theory of mathematics, but is now so embedded in it, that it matches theory to a T. Quantum Mechanics cannot be described without imaginary numbers. So what are they?? Well, imagine this. What is a square root? A square root is a number which, when multiplied by itself, equals another number, it's square. So, the square root of 4 is 2. 2 multiplied by 2 is 4. The square root of 16 is 4. 4x4 = 16. Numbers whose square roots are a whole number are referred to as perfect squares. Now, let's consider this. Consider negative four. -4 times -4 = 16. So, the square root of 16 can be either positive or negative four. For the most part we forget the negative, since it's usually most practical to use the positive number. However, it does lead to a complex situation, there are no square roots for negative numbers??

That's kind of a pain for lots of calculations, and actually limits the boundaries of physics and mathematics. So, they came up with a solution. It's an imaginary number, called i. i stands for imaginary. Now, if you square i, you get negative 1, the square root of -1 is i. This allows us to have the square root of a negative number, which happens from time to time in calculations. What does that mean in reality? Well, there are what are known as real numbers, any number, positive or negative with any number of decimal points, finite or infinite. Then we have imagiary numbers, which gives us literally infinitely more numbers. It's also possible to have a 2-D plot of numbers, real on the so-called x-axis, and imaginary numbers on the y-axis, which means you can now plot a combination of these numbers. So there you go for Halloween, some spooky imaginary numbers. 

Next up: multiple infinities. Infinity is the biggest thing ever right? Wrong. Turns out, there are different infinities, each bigger than the next. This was in the mix for hundreds of years, but was finally set in place by Georg Cantor, in the late 1800's. So can it be? Well, all of these talks of multiple infinities starts in a field of mathematics called Set Theory. I actually took a Set Theory course in college, just to understand how this whole multiple infinities thing works. Let me tell you, while being very, very exciting, at the same time, it is very very complicated, and tedious. So tedious and literally insane that its father, the aforementioned Georg Cantor, went insane several times, spending much of his later life in insane asylums. He also had many detractors, including the incredible Henri Poincare, who said that his contributions were a disease infecting the discipline of mathematics. Unfortunately, Cantor turned out to be right. I think Cantor and Kurt Godel, with his incompleteness theorem, were like the two most famous hackers of mathematics. They just take this wonderfully, painstakingly logical structure, hack inside, and just bring it to the ground, with the implications of what they discovered using the rules of mathematics. 

So let's first simply consider integers, i.e. whole numbers, no decimals, negative and positive, so in the positive direction we have 1,2,3,4.... on and on, and in the negative direction we have -1, -2, -3, -4...and on it goes. So there are an infinite number of integers, right? However, we have developed a system where theoretically, given enough time and resources, you could count them all right? We know how to order them, and how to count them. This is what is referred to countably infinite. You know that after 100,000, the next number is 100,001. And on and on. Now next, we need to consider all the numbers which have decimals. Even just 1.1, 1.2, 1.11, 1.12, numbers which are referred to as rational numbers, as in having a repeatability to their decimals, feels a whole lot bigger a whole lot faster. Now let's consider irrational numbers, like pi, and e, and any other weird number which repeats on and on in no pattern whatsoever. These are referred to uncountably infinite, meaning that there is literally no way to count them all. And, if you took each number like this, and put them inside of a bracket, like [1.2123124124124, 223.2342938414234....,43.1234124,....] this is referred to as a SET of numbers. Now, the SET of irrational numbers simply between 0 and 1, is more vast than all the countable integers, and this is how we arrive at different levels of infinity. So, every time you think you have a bead on the universe, it will throw you a curveball and send you flying out in some other direction believe me. 
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