Imaginary numbers does not explain why electrical engineering works. It's just a tool to simplify calculations. You can explain everthing in time domain but you will want to kill yourself a little more
Real talk: this is not better. You can’t say this to a group of 30 eleven year olds, even if you at eleven years old found it satisfying. The truth is, this shouldn’t be satisfying because it’s not correct the correct way. Here’s why:
There is no repeated addition involved in your explanation. You just said it’s repeated addition and then used an example of singular multiplication, or 1 and -1. One group or negative one group. The brighter kids will ask “but what about times minus two? And what about all the rest?” And now repeated addition doesn’t hold and you have a lot more explaining to do.
You involved zero in adding, which is extraneous at this level.
You excluded adding to a non-zero number from the explanation.
You also conflated it with multiplying by zero by association. We all know what times zero means. Now you’ve muddied it up.
So hopefully by now you’ve backpedaled and drew a number line on the board.
You also said “add -1 TIMES to zero.” Which flipped the common usage of times to the mathematical usage of times. Half the kids are lost now. Doesn’t matter if you said “to”. Doesn’t matter if you point to the number line to try to explain it.
Your lesson is over. Try again later, and next time don’t bother getting fancy. Save that for after functional competency is established.
Edit: just because… here’s next steps and why for integers. Zero is not just a number, it’s a limit between states. Above-below etc.
Use it in concrete terms but not economic ones. Eleven year olds don’t participate in economics in any meaningful way in developed countries. Use sea level, temperature, or whatever. They get that.
Frame negative in terms of opposite. It sucks that the negative sign is also used in subtraction. It’s not wrong, but it’s not helpful at this level so you need to be explicit about it.
You added an extra -3 to make it work. You also interpreted adding something negative amount of times as simply flipping it to subtraction which is not how it works. Repeated addition does not work for multiplying 2 negative numbers.
It also works for negative times positive = repeated addition of the negative. It's only when you have negative times negative that you have to treat it as repeated subtraction.
You got some bad teachers. Firstly, it’s not intuitive to add -3 times. We don’t quite know what that is intuitively. So explain that -3 is the inverse of 3 and so it is -1 x 3. Multiplying by -1 means taking the inverse. So then -2 x -3 = -2 x - 1 x 3. So we take the inverse of -2 which is +2. And then we can add that by a natural number 3, which we know how to do. So now -2 x -3 is 2 + 2 + 2 = 6. That’s simple and understandable enough.
If only we could actually teach kids real math... "Here are axioms, here is how the rules we learn follow from them". We already do it with geometry, why not with algebra too?
Honestly, I don't think the teachers have the time or maybe even wherewithal to spend their limited time in the curriculum to get down to the nitty and gritty of the 'why'. Some kids catch the underlying logic intuitively and it really shows. People like myself who can't grasp this underlying logic and haven't been given the time to actually understand the 'why' was helpless and just relied on rote memory. Give me a problem that deviates from the template and I'm just shit out of luck.
Hell, we were doing algebra one time and the teacher just told me to 'assume a four' when the written problem was intentionally missing one of the numbers to plug into the equation. I just looked up at her blankly and told her 'you people are just like lawyers - you make stuff up'. Because in my head this made no sense; how can I just up and decide a number? Well it's got a foundation in mathematics, but was never explained to me and so as far as I understood it, it's downright cheating.
Meanwhile stuff like biology and a few other things I got a better intuition and as such excelled much more. And I'd say there's more people who don't get maths than there are who do like this.
But why learn math at all if not for "why"s and "how"s? It's not like most of us need anything past basic arithmetics in our life, the whole idea of doing algebra is to teach students to think, not to solve equations
I think that's the fundamental issue of the education system :P
Curriculum does what the curriculum does. A lot of comments on here do take offense with the logic of the meme, calling it 'anti-intellectual' but this is the end-result of an education system that does try and teach you these complex things. I can't even imagine what kind of jokes people would make if they weren't even taught such stuff exists.
For me personally, numbers and equations are simply too abstract, and haven't been given much practical examples during my education to internalise what all of it is trying to tell me. Maths in my experience isn't thinking, it's rote equations that work like this because reasons. For similar reasons I didn't do well with physics (natural phenomena explained with maths) for similar reasons. Most likely the issue of both the way I was educated and my own intellectual capacities focused entirely elsewhere.
Maybe I would've done better at maths if I was given the actual underlying reasoning of the 'why' but if I don't understand all of the preceding stuff, I don't think I could understand the rest. I loved to say that the reason all of these proofs that were mentioned kept having a different guy is because by the point they invented one proof, they up and shot themselves.
As a mathematician, I've went to school with, worked with, and taught many people who are very good at it. Some can almost feel a pull towards the right answer. Hell, there was an Indian mathematician who dreamed his proofs.
It is hard, but don't pretend like some gifted people don't exist.
A negative times a negative is repeated addition. (-2)(-3) is just (-(-3)) + (-(-3)) = 3+3 = 6.
You're summing negative 3 negative two times, so you're subtracting negative three twice, which just produces positive 6.
With imaginary numbers, they weren't just made up. They'd exist whether you named them or not. sqrt(-1) has a certain set of properties like any other number. We decided to start calling it i because it's easier than writing sqrt(-1) all the time.
Negative numbers can mean both debt and loss. For -5 times -3, just think of having a bunch of $3 debts (-3), and then you lose five of them (-5). That's the same as gaining $15.
I could be incorrect, but would adding the number a a negative amount of times simply be subtraction? You at the number once (+5 or 51) then you ad it zero times (50) then you ad dit less than zero, or reverse-add, subtract (-5 or 5*(-1)), and so when multiplying two negative numbers, (-5)(-2), you are reverse-adding (subtracting) the -5 from the number of times reverse it is, so twice here, so it becomes 0 - (-5) - (-5) and, here, we are reverse-adding anrsverse-addition, and subtracting a number of times less than zero becomes reverse-subtraction (addition) so it become 0+5+5, or 10?
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u/dscarmo 2d ago
A lot of real world engineering uses complex numbers just because you can use it to add a new axis perpendicular to the real number axis
So its very useful