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Welcome to ANI In the Air, Wondrous Wednesday, where I talk about something wondrous.
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Today I'm going to talk about a kind of a guess and check method for solving just about anything in mathematics.
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So, I think it's pretty wondrous.
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I also wonder why it's not taught, but that was mostly yesterday's talk.
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And tomorrow I'll have some Tip of the Week revolving mathematics.
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But today I'm going to talk about essentially how to solve any equation you like.
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Well, that's a bit of a stretch.
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But this is the seed of the idea that works for a heck of a lot of things and is something that people actually use.
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Although, yeah, I don't actually. It's like, you know, maybe engineers get taught this.
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I don't know. Somebody does because they use it, but it's not any, you know.
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It's it's I just not really taught the way that it gets used.
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I'm really confused by the whole thing, how it's not. But whatever.
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So this is somewhat related to what I was talking about yesterday.
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But now I'm going to actually go into the mathematics about the detail and kind of demonstrate the different kind of techniques.
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So let's imagine you're putting in a pool and it's going to be a dumb pool because it's got to be a math problem.
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So there is basically you don't know how wide the pool is and the length of the pool.
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Well, you want three yards for the for the shallow end, and then you want the deep end to be the same length as the width of the pool.
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So you kind of have like the square where it's the deep end.
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Then you've got this potentially rectangle thing with one of the dimensions being three.
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And then the other one is this unknown thing, X. All right.
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And for some reason, you want the the surface area of the pool to be 50 square yards.
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All right. So this is my best attempt at actually coming up with something like of a physical thing that might actually, you know, exist.
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And I mean, no one actually, you know, would want this, but, you know, it's a stretch to come up with simple problems that can explain technique and somehow be relevant.
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So I just, you know. All right. So how do you solve this?
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Well, first of all, you need to write down and down some kind of equation how to compute this thing.
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Essentially, this is, you know, length times width is area.
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So the width was our unknown, which we'll call X, and the length was three plus X.
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The three. I mean, it's all in yards. So three is in yards and you multiply them.
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So it's X times X plus three. And you want that to be 50. So that's your equation.
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You can write that down if you want. Now, this is an example of something called a quadratic equation.
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And there's a couple of ways of doing it.
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So if you're familiar with the completing the square method, which is what all the cool cats do, it's actually not too much algebra to get to an acceptable, precisely perfectly exact answer.
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Essentially, what you do is, well, the X factor is zero when X is zero, and the other one is zero at minus three, because minus three plus three is zero.
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And then you look at the halfway point, minus 1.5, and that's going to be where this quadratic thing has its kind of minimum.
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And so we can write the thing as X minus 1.5 squared. And then you just need to figure out what that constant term is.
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So you kind of square it out and I guess you do have to, well, yeah, you square it out and you get, you know, whatever you get, you know.
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Sorry, X plus 1.5 squared. That's why it was not working. X squared plus 3X. And then the square of 1.5 is, well, 15 squared is 225, so that's 2.25.
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And that's what you have to actually subtract in order to make everything balance, and that's supposed to be equal to 50.
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And more or less, X plus 1.5 squared is equal to 52.25.
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So you square root it, so you take the square root, however you compute that, it's a thing, and you subtract the 1.5 and you get minus 1.5 plus or minus the square root of 52.25.
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All right, and then in math class you'd say, "Oh, I'm done." Now, you probably didn't follow anything of what I said. I barely followed it.
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And I was writing it. You kind of have to write it down. And, you know, it's like the answer, minus 1.5 plus or minus the square root of 52.25 is like, "What the heck is that?"
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You still have to go to the calculator and plug it in, and it's like, "What a complicated form." Now, and there was some sort of weird thing, I was mumbling about zeros and what, like it's just some kind of weird knowledge thing, right?
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Great. So that was, that was actually kind of cool when you kind of learn about it, but it's a bunch of gobbledygook.
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Good. All right, next up. Let's make, put this in standard quadratic form. So we multiply the x times the x plus 3 to get x squared plus 3x.
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That's supposed to be equal to 50, so that means you can subtract 50, put it on the other side.
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So you got a minus 50 there, and it's equal to 0. And now we use the good old quadratic formula, which when you have ax squared plus bx plus c equal to 0, it is minus b plus or minus the square root of b squared minus 4sc, all of that over 2.
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So, and there's actually a better way of writing all this, but moving on. Minus 3 plus or minus the square root of b squared, b was 3, so that's 9, and then minus 4ac, so 4 times a, a is 1, and then times c, which is 50, minus 50.
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So it's plus 200, all over 2. So it's minus 3 plus or minus the square root of 209, all over 2.
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Great. So that's another precise exact answer. I have no idea what it is.
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So, you know, again, you go to calculator, you got to approximate the square root, you know, but we call it done. We call it done in the math world. That's crazy.
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All right. Now, let's get into the real stuff. So hopefully, up to this point, if I haven't lost you and you're still listening, which is, you know, fantastic on you.
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But anyway, you know, hopefully this next bit will make some sense, because you know what? I'm just going to say it. I'm going to guess. I'm going to guess. That's what I'm going to do.
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So I got x times x plus 3, and I want that to be 50. All right. Okay. So what could we do?
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Well, I don't know. You know, if you take 5, and I want to get to something like 50. I mean, if I want to get to 50, and there's a 5 in there.
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So maybe 5 in there. So if x is 5, and then 5 plus 3 is 8. So what's that? So 8 times 5 is 40. All right. So that's not 50, but it's close, right?
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And I'm going to want something larger. So I'm going to go with 6. So what's 6? Well, 6 plus 3 is 9, and 6 times 9 is 54.
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So that's -- so if my unknown was 6 yards, then one side would be 6, and the other side would be 9, and I'd have 54 square yards as my surface area of the pool, right?
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So it's between -- so I've got 40 and 54. I want 50. Neither of them are correct, but I'm kind of there.
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Now, the idea of this technique -- I mean, those are just guessing. Guess whatever you want. Fine. I happen to write problems down that I feel like I can guess pretty closely, which I do.
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So that's just on me. But you guess whatever you want to guess. Doesn't matter. Anyway, so how do we get to 50?
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Well, so going from 5 to 6 was a step size of 1, right? But going from 40 to 54 is a step size of 14.
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So I'm just going to say, you know, the 50 and the 40 and 54, I'm going to say that's the Y values. That's how we talk in math. And we've got the unknown, the X values, so the change in X was 1, and the change in Y was 14.
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Now, I want -- if I'm starting at the 6, which was 54 was the value of the area, and I want to get to 50, well, how far do I want to go back?
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I want to lower X, make it smaller. I don't want to go down to 5, because I was 40. And it kind of feels like -- well, I want to go down by 4, and that's kind of a third of 14.
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More or less, not exactly, but whatever, I don't care. And just to make my life, you know, somewhat easy, I'm just going to use .3, 3/10. That's not a third. A third is .3333333, whatever, but I don't care.
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This is all just easy stuff, easy guessing, whatever. So, going down from 6 by .3 is 5.7. And then adding 3 to that is 8.7.
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So, to compute the area of this thing, I'm going to want to do 5.7 times 8.7, and you do it. It's just a computation one can do.
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That's actually one that's fairly easy to do by hand, but you can also use a calculator as you see fit.
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And when I computed out, I got 49.59, which is pretty good. I would certainly call it a day on that one, 5.7 yards. Done, right? Finito, with all the square root stuff.
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But let's say you want to go just a little bit more. All right, well, okay, so I want to go a little bit more.
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So, I went down, my change in x was .3, and my change in y was, well, I went down basically 4.6.
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So, a step size of .3 led to a change in 4.6. Now, I want to go up from the 49.59 to get to 50, so I want to add in .4.
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Well, if I went down by 4.6 with a step size of .3, then if I divide by that step size change of 4.6 by 10, it's .46, close enough.
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And then I change my x correspondingly by dividing by 10, so it's .03.
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So, now I'm going to try 5.73. I'm adding because I was a little under the 50, so I want to go get a little larger.
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And if you do 5.73 x 8.73, you get 50.0229. Pretty good, huh? I would certainly call it a day at that point.
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And, you know, as you do this, and you can keep going, I think the, I don't know if I did it again. Did I do it again?
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I don't know. I thought I did, but maybe I called it a day. Or maybe I just did it on the calculator.
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But you could continue on in this fashion. I think you had to go down the other way for the next step.
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It might be a .0015 change, so it's like, you know, 5.7285, and you multiply that by 8.7285, and you get like 50.0020's and something.
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And so you can just keep on going. You can be as precise as you like. It just keeps honing on the answer.
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And keep in mind that there was really no algebra, right? I just wrote down how to compute it.
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If I have the value of the length, or the unknown value, I can multiply those numbers together, and I get my area, and that's all I cared about.
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And then it's just like kind of figuring out where my next step is. And the beautiful thing about this is it's self-correcting.
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I mean, I'm a little sloppy, I'm making a little bit of mistakes, whatever. The next time around, if you do that one right, you'll go back to where it is.
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And if you do it wrong, you'll probably get farther away, and you'll be like, "Wait, what happened?"
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Whereas, think about that with the quadratic formula, where it's like minus 1.5 plus or minus the square root of 209 over, you know, all over, well, square root of 209 over 2.
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It's like, wow, there's a lot of things I could have done wrong. Lots of signs going on. There's actually two of these solutions, plus or minus.
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I don't have that problem with this other method, because I'm guessing in the area of the thing that, of the answer that actually makes sense.
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I'm not, the other one is, you know, in the quadratic formula, you get a negative answer, which doesn't make any sense, so you discard it, but you have to kind of consciously understand that.
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This is just like, oh, I'm just guessing and checking, and I'm getting closer and closer, and boom, you're done. Done!
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That, people, is a wonderful thing.
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All right, so, as I said, this is a technique that is actually quite robust, solves all equations.
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Completing the square in the quadratic formula, not so much. It's really, they're quite limited.
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Well, to quadratics, in fact.
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Now, let's say we did a, let's say we had a cubic. This is one where you're multiplying an unknown value times itself, a total of three factors.
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So, if you had ten, for example, ten cubed is a thousand, three zeros.
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All right, so, let's go back to this, kind of this pool problem, and say, why are we talking about surfacing?
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Why don't we talk about depth? Well, to talk about depth, you need to have the depth.
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Now, at that point, you might have multiple variables or whatever going on, and this technique generalizes to that, but that's way outside the scope of this podcast, or even my brain span right now.
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So, instead, let's do this. The shallow end will have a depth of one yard.
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So, the volume from the shallow end will be, you take the area, three times x, and then you multiply it by the depth of one, so it's just three x.
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So, that actually hasn't really changed much, but the depth of the deep end, I'm going to say, is going to be x itself.
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So, instead of having a nice, just a square area of it, it's actually going to be a cube of water, so it's x cubed.
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This is the stupidest pool design I've ever thought of, and I've thought of some stupid pool designs, but that's what I'm going with.
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And now, we want volume, so it's, I've got yards, so cubic yards, and I did this ahead of time, so I looked up kind of what a standard pool size might be, and came up with 30,000 gallons on the bigger side of things.
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Not an Olympic-sized pool, that was, like, way bigger, but converting that to cubic yards is about 150, like 148 or something, around and up to 150.
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You can tell that I'm just fundamentally an imprecise kind of guy.
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All right. So, we've got, so the thing we need to solve is now 3x plus x cubed equals 150. Great.
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Well, so how do we do this? Now, there is a cubic formula, like the quadratic formula. I recommend you look it up. Really look it up.
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So, cubic formula. It's on Wikipedia. Tell you, and this is some great backstory on that one.
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1500s, some kind of nose fight. It's pretty crazy, but if you look at the formula, you'll be like, "What is this?"
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That's an impressive piece of work. Not terribly useful. Some people find it useful, I'm sure.
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So, again, we just guess and check. So, let's try x equals 10. I don't know. Why not?
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And so you put in 10 for x, and you get the first term being 30, and the second term, the x cubed term, is 1,000. 10 times 10 times 10.
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So, you get 10,030. Great. Now, I want 150. So, that's just terrible.
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Well, let's try 5. If I worked before, why not work now? Well, 5 times 3 is 15, and then 5 cubed is 125.
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So, 15 plus 125 is 140. Pretty good. And then x equals 6. So, 140 is a little low. You do the 6, and you'll get 234. Hopefully.
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Alrighty. Moving on. Oh, yeah. So, the change in our little x step was 1, just like how we did it before. And the change in y.
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Well, the difference between 140 and 234 is, well, it's almost 100. It's 6 off, so it's like, you know, what is it, 94.
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But let's just say 100, because what the heck. And we want to go from 140 to 150. So, that's, we want to go up by 10, and that's a tenth of the change before.
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Right? So, 4, we went up 100, and so now we're going to go up just a tenth of that. So, we add a tenth to our 5. 5.1.
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And you plug that in, and you get 147.95 as the answer, or roughly 148. So, actually, in terms of converting from the 30,000 gallons, which was my original thing, it's pretty much spot on.
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But, moving on. Let's say that you wanted to do a little bit more. You want to get to that two, extra two cubic yards.
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And what do you do? Well, again, so, we went from 140 to 148, so that was a change in 8 of the value. And we did that with a step size of .1.
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And now we want to add another two to get to 150. And that's a quarter of that previous step size. Because, yeah, 8 divided by 4 is 2.
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And so, a quarter of a tenth is, you know, 1/40, which converts to, or you can view it as a quarter, which is .25 and then divided by 10, which is .025. That's the easier way.
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And so we add .025 to the 5.1. We do 5.125, plug that into that formula above, and we get 149.986, which is pretty darn close on the nose.
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Now, just think about that. I hope you followed me on these guess and checks. But, just, there was, like, no algebra thing. It's just compute the values and figure out the changes and then do this thing.
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And it's just completely, you know, guess and check, basically, with an educated direction and size for that changes of the guess.
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And that is fantastic. And it works for any kind of equation you're trying to solve. You take a guess, ideally a good guess. That can be a little hard, particularly for the more complicated functions. You really kind of need to have some sense of where you're at.
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But that's all you got to do. You just guess and do a couple guesses, figure out where you want to go from where you've been, and then that would be it.
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So, yeah, it's kind of based on Newton's method, kind of more like it's called the secant method if you want to look it up.
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And, you know, like, I just explained it. I can, you know, like, understanding why it's true, that can be complicated. And, of course, it's not true.
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There are examples where this doesn't work. And it's also what exactly is true, right? You have to even formulate that. It's about, you know, the error.
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The difference from where you've been to where you are getting smaller and kind of saying how much smaller. And all of that's hard, you know, complicated.
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But just the basic sort of tool of, you know, like, you know, you've got a hammer and you're going to whack the nail. You don't really care what the physics of a metal nail going into wood is and your hammer coming down and the forces and all that.
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You just want to take your hammer and smash the nail, right? This is the hammer method. Call that the hammer method of mathematics is you just smash that thing down, get an answer.
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Don't stare at it without any idea of how to proceed. You try something, right? And that's fantastic. That is wondrous that it works and works so well and blows my mind that it's not really taught.
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I mean, I know why it's not taught. It's, you know, it's a little bit of black magic, although all of, I mean, completing the square and quadratic formula, that just seems like black magic to people, even though it can be explained, it still seems like black magic.
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So that can't really be the answer. It could be that people don't understand or know it, but it also is just kind of like, it's messy and you can't say somebody's wrong, right? It's, you know, I mean, all right, you can say somebody's wrong.
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I mean, if the answer is like around 5.125 and you say it's 57,000, well, you're wrong, right? But if you say it's 5.133 and stop there, then what's wrong with that, right?
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So it's a complicated thing to actually kind of grade for, which is probably why it doesn't appear in anything. But man, it is a beautiful, wondrous, absolutely fantastic thing in math.
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It's just the best, the best. And it's a great metaphor for living life, you know, just trying some stuff and you, you know, and you can see kids playing with it too, like 20 questions. That's what this is, right?
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You take a couple of guesses and now you've got some direction and kind of do that. The game Warmer and Colder, that's also this, like this is so human. This is the human hammer method of mathematics, the HHM.
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If you can't tell, I really love, I love this method. All right, well, I've kind of droned on way too long. And if you have followed me, my goodness, that's amazing. Congratulations.
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I'm sure you are not listening to this, but man, if you are, kudos to the in the airheads. Love it. Have a good one. See you when I see you.