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Show Info: Learn about or get an edge on your introductory physics course with this podcast!
Show Info: Learn about or get an edge on your introductory physics course with this podcast!
For transcript, links and pictures, check out pwnphysics.blogspot.com 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