As with imaginary numbers, the concept of time dilation and the implications of applying the theories of relativity are difficult to grasp for our reptilian brains.
However, We rely upon such slippery concepts every day in order for our satnavs to function, the maths works whether it makes intellectual sense to us or not.
And what the hell is anti-matter anyway? Well, it is the stuff that makes a PET scanner work.
In short, don’t worry about it. If you can’t visualise “i” in a way that makes sense to you then that puts you firmly alongside the majority of human beings. Even those who use “i” on a daily basis don’t necessarily intuitively grasp the concept but they do know how to use it for practical purposes.
Except that it originally was intended as a judgement-when Rene Descartes first heard of the concept, he thought it was so preposterous that he coined the term “imaginary” to mock the entire notion.
Piggybacking on engineer_comp_geek’s superb post …
You may see outfits selling power factor conversion devices for the home. They’re simultaneously valid science, an engineered solution to a engineering non-problem, and pure commercial snake oil.
I thought it was clear that I was saying it IS useful in doing some calculations. Mainly, it comes does to Euler’s Identity. As noted earlier, this comes in handy when analyzing wave mechanics, and so physicists used this in certain QM formulations.
The weird thing is, I’ve come to accept how i can be used to model rotation, but I still have very little intuitive concept of what this has to do with the square root of -1. The best I can come up with (and it’s probably wrong) is that if you’re trying to solve for the roots of a parabola that doesn’t intersect the x axis, you’ll get imaginary or complex answers, which correspond to rotating the parabola 90 degrees and finding where that parabola intersects the axis.
You’re not so much rotating the parabola as you are adding another dimension to the graph. When you plot a function on the cartesian plane, you’re only seeing the real-number outputs of the function on the Y axis. But that’s only part of the story. The function may have complex outputs as well, but you can’t see them unless you plot the function in more dimensions.
Take the parabola y = x[sup]2[/sup] + 2, which has a y-intercept of 2 and therefore does not touch the x-axis. So we might conclude from the graph that it has no roots, because the curve never reaches zero. But it actually does have two roots at i√2 and -i√2. If you substitute those for x, you get y = 0.
Cool video. I had somehow never heard of quaternions before (or if I had, I just never bothered to look up what they were.) That Numberphile guy just makes everything in math seem so fun and interesting. Love his enthusiasm and just pure joy at explaining mathematical concepts.
By the same token, it makes sense to say that you own 6i apples if you possess 6 apples each of which has been rotated exactly 90 degrees around its stem from whatever position it was in when you received it.
When I was a kid, I actually thought they were called “irrational” because they were these crazy numbers that caused ridiculous run on non-repeating decimals.
What a lot of these responses are skirting is the ontological status of any number. You can have 3 apples or even square root of three apples, but you cannot have 3 or square root of 3 or square root of -3. So what is a number. One answer is that it is an agreed upon fiction. And along with the agreement comes agreed upon rules for manipulating them. And one of the rules we agree on is that i*i = -1. Using these rules we solve real-world problems. And has been pointed out the quaternions give a good handle on the group of symmetries of a sphere, the key to rotation in 3 dimensions.
Seriously tho - I think this kind of discussion, a bit of minor philosophizing about what a number really is, is one of the huge stumbling blocks in math education that trips a lot of people up. It’s a topic that should be addressed early on, while still learning basic arithmetic: All this stuff is made up. But we make you learn it because it’s useful.
Quaternions aren’t unique in this, though. The culprit here is actually Euler angles, which are commonly converted to rotation matrices and/or quaternions, thus not avoiding the problem at all. Euler angles are really easy to work in, and tend to be the “default” for representing rotations unless you anticipate gimbal lock being an issue.
