Imaginary numbers are best understood as symbolizing rotation. If we’re imagining a number line here, “looking back from infinity” - at a scale where Grahams number looks like the mass of an atom expressed in kilograms, i would not be in that infinite set of numbers, it would be a point above that line and creating a perpendicular plane to it.
I hate the term “imaginary” because it’s misleading. Most high school algebra teachers don’t understand what they are either, so people learn about these things called “imaginary” numbers, never learn any applications with them, hopefully graph them at best, and then move on understanding nothing new about math.
Students also tend to get really confused about it as possibly a variable, (it’s really annoying with in second year algebra courses, where e and logs also show up). We say “ah yeah, if you get a negative sign, just pull it out as an i and don’t worry about it. or just say no real solutions.”
I think in precalculus at least, something like this is not too hard to show and explain to a student. This would be a fine “final” thing to end the typical high school math career on - showing how all of the different concepts you’ve explored come together.
The list of things I shouldn’t do, but do regardless, stretches past infinity.
Imaginary numbers are best understood as symbolizing rotation. If we’re imagining a number line here, “looking back from infinity” - at a scale where Grahams number looks like the mass of an atom expressed in kilograms, i would not be in that infinite set of numbers, it would be a point above that line and creating a perpendicular plane to it.
I hate the term “imaginary” because it’s misleading. Most high school algebra teachers don’t understand what they are either, so people learn about these things called “imaginary” numbers, never learn any applications with them, hopefully graph them at best, and then move on understanding nothing new about math.
Students also tend to get really confused about it as possibly a variable, (it’s really annoying with in second year algebra courses, where e and logs also show up). We say “ah yeah, if you get a negative sign, just pull it out as an i and don’t worry about it. or just say no real solutions.”
I agree. Although better and more illustrative videos have since been made on YouTube, my favorite introduction to the square root of negative one is Chapter 22 of the Feynman Lectures on Physics Vol. I
Feynman has so much great stuff to plagiarize.
I think in precalculus at least, something like this is not too hard to show and explain to a student. This would be a fine “final” thing to end the typical high school math career on - showing how all of the different concepts you’ve explored come together.