THE EVIL MIRROR SOLUTION, WITH A GOATEE 
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We’re not defective. We just operate differently.
Take art, for instance. There may be some painters who are slaves to math, and use it as a primary tool in their work, a la Thomas Eakins or Euan Uglow, but there are also many who utilize intuitive thinking, and can compress 3-dimensional objects into a 2-dimensional plane with a grace and exactitude that is extraordinary.
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she reminds me of a bangeldeshi librarian christel who was softly spoken
she could take a picture cause you knows pretty as sh1t
her French mother hounded her like the queen in the mickey
mouse item 
Meanwhile, in Australia…
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You should sell Merchandise with this quote, I’d buy it.
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Ah. That’s because you don’t prove 1+1=2 : it’s more of a definition.
To put it in a way what won’t make mathematicians throw up in their mouth a bit, let us define an initial Z (which will turn out to be zero) in a set I (for integers) and say that every number in set I has a unique number that comes after it, and that number is not one of the ones that goes before it. So there is a successor to Z, Zs which we commonly call 1, and a successor to that Zss which we commonly call 2, and so on. And all of number theory descends from that.
However, if you wade through Principia Mathematica, you end up with the lurking suspicion that you haven’t actually proved counting and maths, but have ended up counting the number of letter ‘s’ after a Z instead, so your typography is doing the counting instead of you. And the actual proofs you are looking for are still just out of reach.
Or as Neumann put it, much better than I could…
Young man, in mathematics you don’t understand things. You just get used to them.
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And there was me reading the post, thinking ‘I might find out what i is and why anyone gives a hoot about its square root’
Nah. None of that. Straight in with the higher math and no background for the non-math/lay person. Lost me at happy friends, smiley faces and “x=iy” and I gave up at sine and cosine - I know they exist and was taught it once, but no idea today what they are, how they are used/calculated etc. Gave up, realising I should have known that if I didn’t know what i was to start with I should never have got sucked in by that headline. ;-(
(But to be clear, I am not disappointed. I blame myself.) 
Okay, I’ll bite…
Suppose you put your counting numbers in a horizontal line 0,1,2,3… You can add numbers by adding lengths. If you subtract numbers you can find new ‘negative’ numbers, which can’t be used for counting in the ordinary way, but seem to exist just the same.
Multiplying by a number is like scaling up. Multiplying by 2 is doubling the distance between the number and zero. Multiplying by -2 is doubling the distance between the number and zero and flipping it over. And if you do this twice then you get to 4, so the square root of 4 can be 2 or -2. But we can’t come up with a simple scaling that gives a negative number if you do it twice.
And now here’s the beautiful bit which she skipped. Negating our number was flipping it over, but it could also have been rotating it by 180 degrees. Suppose we imagine an extra sort of number that went vertically instead of horizontally. Let’s call it ‘i’. Multiplying by ‘i’ gets you from 1 to i, which is like rotating anticlockwise through 90 degrees. Multiplying it again is like rotating again anticlockwise through 90 degrees, getting you to -1. So i is like the square root of -1. And -i, which is rotating the other way by 90 degrees is the other square root. And all of adding and multiplying just goes on working with these new numbers.
You have acquired a new skill: complex math! You go up a hat size!
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Thanks for trying - truly appreciated.
But…
I see no practical difference. I guess the key word there is ‘practical’ seeing as you then move to the imaginary realm.
Which has too many possibly pithy responses.
Why?
Suppose we don’t.
I mean, suppose we imagine all sorts of things, they’re still imaginary things.
Yeah, I know I’m just yanking your chain a bit and I do appreciate your effort, but you’re talking to someone who got the highest grade in GCE Maths exam and the lowest in GCE Advanced Maths.
“dy over dx, sir? Can’t we just cancel out the d’s? Isn’t that how it’s supposed to work?”
‘Sir’ never once explained any practical use for this esoterica so when I was faced with “a frog jumps half the width of a road and then half the remaining width in each successive jump…calculate …” I used normal maths to approximate some sort of answer as an alternative to leaving the answer space blank.
I have long been resigned to the fact that complex math was not for me and my hats fitted just fine. Thanks (again) but sorry.
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What a lovely video. Nice and easy to understand. Well, relatively easy to understand compared to other explanations of the same concepts.
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As someone else who’s job is applied mathematics, I still rely on mental “short-cuts” to do mental arithmetic, short-cuts that are frowned on teaching in the public education arena. Otherwise I need a calculator. I’m not saying everyone is geared to learn advanced math, but math is beautiful and it irks me how our K-through-12 system more often than not scares students away from it.
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Came for Tom Leher’s New Math. Leaving satisfied.
Though if I may offer an embedded link that plays without going over to the U-tubes…
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Well, yes, they are called imaginary numbers. But negative numbers are imaginary too: you can’t actually have minus one apples. Even owing someone an apple isn’t the same. You can have plus one apples, but even then the one is just an imaginary thing. All numbers are imaginary. But the good numbers are useful.
What’s it for? Well, if you need a formula for, say cos(3x) I can write it straight out as cos(x)^3 - sin(x)^2.cos(x) because that’s the real part of (cos(x) + i.sin(x))^3. Simples.
Yep. Most people can get by without ever having to do that. But it is still beautiful.
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I never understood anything above basic math – I memorized it long enough to pass the test. Then, I forgot it.
LOL - that was several levels above any “what’s it for” question! I might interested in your ability to write the same thing two or three other different ways using keyboard characters that have no meaning to me in this context IF I even knew what the thing was in the first place. Probably best you do not try to explain cos and sin to me because it really would not be a good use of your time. 
(I appreciate that I really should really not be on this thread at all. I’m not safe to be let out, on some daystopics.)
Beauty is in the eye of the operator (?)
Simples to you too!
Sadly my test required its real-world application. Even memorising it was not enough.
Hmmm. Is this new? When I was in school I recall estimating to be a skill we were specifically taught, usually right alongside the more exact stuff pretty much the whole way through. Maybe the ubiquitousness of calculators and computers has caught up with us?
