yall wanna hear something cool and completely unrelated to this blog, okay theres all these different sets of numbers right, and all these sets of numbers are infinite
So the number sets kinda look like this,
a [all the real numbers]
a + bi [the complex numbers which include i the imaginary number which is really just another way of looking at how algebra can function (we can now take the square root of negative numbers), which is a different take on algebra where we simply had the set of real numbers]
and then you get
a + bi + cj + dk [quaternions, which is the next step up from the complex numbers in algebra, and are a new set of numbers that have a new way of looking at algebra, and are also infinite]
So about the complex numbers, thats numbers that include i, the ~imaginary number~ which is a terrible name for it, it’s not imaginary, just a completely different bit of algebra that doesnt work with what was previously thought to be algebraic laws and rules. Because before we had i you couldn’t take the square root of a negative number, because thats just not how the real numbers worked. It changed how algebra functioned with this new set of numbers.
You can just change the set of rules for math for it to work, by creating something new, as long as it all logically follows.
Anyway, once you get past the complex numbers you hit this cool thing called quaternions, which is another set of numbers, except in this set of numbers, which are also infinite, they don’t follow all of the rules of algebra we were previously taught to believe, in this set of numbers we don’t have the commutativity property,
[commutativity property is where a*b = c, and b*a = c]
without this it means if you multiply numbers together together in different orders, you get different answers. Which isn’t how any of the other previous sets of numbers work in algebra.
in regular algebra which works with all those numbers up to the complex numbers: 2*3 = 6, and 3*2 = 6
In quaternions: j*k = i, but k*j = -i
So with quaternions
i^2 = -1
j^2 = -1
k^2 = -1
But you know whats wild about this, in this set of numbers:
i*j*k = -1
And the order does matter in this set of numbers.
because in this set of algebra
i*j = k BUT heres the wild part j*i = -k
we can see it again:
j*k = i, BUT k*j = -i
When you multiply j*k you get i, when you multiply k*j you get -i, which isn’t how any of the other number sets behind this one in algebra work, isn’t that wild?
There’s a part of math where basic definitions of algebra don’t exist for these types of numbers to exist, and the further you keep going into new numbers, the more fundamental rules of algebra you start losing for the number sets to exist.
and this guy just came up with it walking across a bridge