Math as a Hobby – Ep. 3 Addition Exercises (w/ Fibonacci, Sum of Odds, Matrices)

[00:00:00]Hey, what's up, YouTube? Another day practicing addition. I promise next week we will be moving on to multiplication.

[00:00:09]Uh so, yeah, I'm excited. I've been thinking about the structure of this show, of the series, and uh I got a feedback from a few people uh that episode two was a little too much all in one episode. So, I think that they're right. I'm starting to think I'm going to aim for one theory video per week, and then if possible, another video um on practice. So, today will be an addition practice episode, and we will be working multiple little exercises or demos.

[00:00:45]Um so, we will be working the sums of integers, evens, odds. We'll be uh talking about the Fibonacci numbers again.

[00:00:54]Uh we'll be talking about the sums of Fibonacci numbers and get some addition practice using the Fibonacci numbers.

[00:01:01]Uh and then I'll introduce matrices and addition of matrices. And then finally, um we will do a proof from Lang. Um that one proves uh cancellation law for algebra. So, uh let's go. I'm excited.

[00:01:17]Let's go. So, first uh I wanted to do uh sums.

[00:01:23]Let's see. Oops. All right. Sums of integers, [snorts] evens, and odds. So, I feel pretty passionate that we should give uh students a way to make up your own problems. Man, like make up your own problems and just play with numbers. Like, it's so much more fun to uh to work your own problems that you've made up as a way to practice and get better. So, we saw last time um that that there is a special formula that Gauss found. But, if you if your purpose is practicing addition, which was the point of the teacher's, you know, assignment, practice it. You know, 1 + 2 + 3 uh 3 + 3 is 6. So, this is 6 + 4. 6 + 4 is 10.

[00:02:23]Um 1 + dot dot dot but I'm going to go and show the previous step and then the next step. So, we already saw that one uh up to four is 10. So, then 10 + 5 is 15.

[00:02:37]Uh 1 + dot dot dot 5 + 6 is 21.

[00:02:44]Right? So, you could you could do the sequence that way or you could do it as one uh 1 + 2 is 3. 3 + 3 is 6. 6 + 4 is 10.

[00:03:00]10 + 5 is 15. 15 + 6 is 21. 21 + 7 is 28. 28 uh + 8 is 36.

[00:03:12]On and on.

[00:03:14]Um one thing that we noticed last time was that Gauss had a very special formula um that we noticed that these numbers always match this pattern.

[00:03:28]So, like one through the sum of one up to five matches the pattern 5 * 6 / 2. Is that true? 6 / 2 is 3. 3 * 5 is 15 and 15 was indeed the last uh result.

[00:03:49]So, what happens if we add six to that, can we reconfigure this at all? So, six is uh 12/12 over two, right? Um and then now 5 * 6 is uh 30.

[00:04:12]30 + 12 / 2 Um 30 + 12 is uh 42 / 2 and then 42 uh let's see. We're looking for 6 * 7 and yes, 6 7 / 2.

[00:04:31]Cool.

[00:04:32]So, it seems like this pattern worked again and we could uh show it. Oh, you know what? We would have had a much a much nicer proof this way. Here's a nicer proof.

[00:04:45]Better proof. Better proof.

[00:04:47]Uh you should always look for an improved proof uh when you're when you're working these.

[00:04:53]So, um all right. Let's do this again.

[00:04:57]5 * 6 / 2 + uh we want to convert to have the same uh denominator. Like a least common denominator would be two.

[00:05:08]Um instead of actually multiplying 6 * 2, I'm just going to do 2 * 6.

[00:05:15]And now when we add the numerators, 5 * 6 + 2 * 6 all divided by two keeping the six factored in on in this right um fraction Now now it's easy to factor it out so that we have 6 * 5 + 2 / 2 and that is 6 7 / 2.

[00:05:47]Heck, yeah. All right, so there's just another way to see that yet again, um, we do have the pattern, um, holding up.

[00:05:59]So, cool. That was one little practice rep. I wanted to see what if we twisted this?

[00:06:10]What if we change this a bit?

[00:06:15]Um what if I maybe added every other number?

[00:06:31]So, what if I add the odd numbers? 1 3 5 7 9 uh, 11, etc., etc. Okay, so maybe that's our last one that we'll do.

[00:06:47]Uh, let's try one is clearly just one. 1 + 3 is 4. 1 + 3 + 5, well, that would be 4 + 5 is 9.

[00:07:00]1 + 3 + 5 + 7 would be 9 + 7 uh, which is 16.

[00:07:10]1 3 5 uh, oops.

[00:07:17]What am I doing?

[00:07:18]Oh, yeah, yeah, uh, 1 3 5 7, uh, 9 Okay, so again we are seeing parentheses in the previous answer to make these easier for ourselves, and 1 up to 7 is 16. 16 + 9 is 25.

[00:07:39]Okay, and then all of this is 25. 25 + 11 25 + 11 is 36.

[00:07:49]Yeah, wait a minute. Do those numbers seem kind of interesting at all to anybody out there? Like, what? Wait a minute. Wait a minute.

[00:07:59]Do Do those sound familiar to anybody else? 1 4 9 16 25 36 One is one squared. Two squared is four. Three squared is nine. Four squared is 16.

[00:08:16]Five squared is 25. And six squared is 36.

[00:08:24]So, the sum of the first n odd numbers We'll see this soon that an odd number can be written as 2n - 1.

[00:08:39]Uh the sum of the first n odd numbers is n squared.

[00:08:50]We will prove this formally in the future. And we will use an argument that's very similar to how I structured this example.

[00:08:59]Uh and we will learn that later. It's something called induction if you want to look ahead. But, this pattern always holds. Um 36 + 13 is 49. 49 + 15 is uh 64, right? And it just continues on and on.

[00:09:20]So, try to do these in your head when you're trying to fall asleep. Um It's It's a great way to fall asleep.

[00:09:27]Okay. Uh these These are some interesting addition problems, I think.

[00:09:32]Like, interesting little uh pure math math facts. Like, wait a minute. there's some kind of structure in there. That's That's kind of cool, I think.

[00:09:41]So, I think that's all I want to do here. Um it turns out there's not I don't think that there's something interesting with the sum of the even numbers. So, we'll just do the odds.

[00:09:52]Next, we've got the Fibonacci numbers, my favorite, dude. There's just so much with them. Um Okay, and they were originally an exercise for pure math practice. They actually were invented, apparently, to help uh demo the benefits of the Arabic numeral system as opposed to the Roman numeral system, which is really cool. So, Fibonacci numbers.

[00:10:21]Uh for those who don't remember, you start with two ones, and then you add the last two numbers. So, 2 + 1 is 3, 3 + 2 is 5, 5 + 3 is 8, 8 + 5 is 13, 13 + 8 21, 21 + 13 is 34, 34 + 21 is 55, 55 + 34 89 89 + 55 144, um 144 + 89 I'm adding 100 and then subtracting 10 to get 233.

[00:11:00]Um 377.

[00:11:03]Uh 610.

[00:11:06]987, and that's all the three-digit numbers.

[00:11:11]I think that'll be enough for us right now.

[00:11:13]Uh so, these These are terrific, dude. Like, I really think just set a 5-minute timer on your phone or with a delightful sand timer. Uh and just see how how far can you go with these things? Um Um, at some point they get hard to do in your head for anybody. Like it's a great way to just exercise addition practice again and make up some problems for yourself.

[00:11:42]Like this is the theme I want to be conveying right now. Just make up your own problems sometimes. Like we can make up our own problems with the Fibonacci numbers, too.

[00:11:54]Okay. So, we um Recently I did like a 5-minute I want to see I want to I want to hear in the comment section. Try it yourself.

[00:12:05]I want to get a 5-minute leaderboard.

[00:12:07]There was somebody who got to like 30-something million, uh, which is crazy.

[00:12:13]Uh, they got a like a few more than I did. Leaderboard.

[00:12:17]Um Let's see. So, in a 5-minute leaderboard, my personal best right now uh, was a couple of days ago. Where did that notebook go?

[00:12:28]Oh, well. I'll I'll show it from my other lecture notes. I I abandoned this uh this slide deck.

[00:12:36]I tried to I tried to do a whole thing on the Fibonacci numbers. It was just a little overkill. Here, this one.

[00:12:47]Recently I hit um I got to the 23rd Fibonacci number doing it all on paper entirely mental math.

[00:13:04]Uh, and I felt good about it. We got to 28,657.

[00:13:09]I'm sure if I practice it more and track it in a spreadsheet or in a you know, like just get some graph paper and track them like on this was on the 13th or something and I would write down n is 23. I got to the 23rd number.

[00:13:26]Um, let's make up a new game.

[00:13:28]Uh, let's copy these.

[00:13:32]We saw something interesting before when we added numbers together. What What if we added the Fibonacci numbers together?

[00:13:41]Does anything come up?

[00:13:43]I don't know.

[00:13:45]Uh, 1 + 1 is 2.

[00:13:49]Uh, 1 + 1 + 2 is 2 + 2 is 4.

[00:13:56]1 + 1 + 2 + 3 4 + 3 Um, Okay, so here, this is an opportunity for mental math. You can try to do it in your head, right? 1 + 1 is 2, 4, 7, uh, 12, 20, uh, 33, uh, 33, uh, 54, right? 54, 89, Oh, wait, no.

[00:14:33]88, 54, 88, so you could you can try to to do that in your head. That's one way to stretch and and try to to um, you know, to exercise this stuff.

[00:14:51]You could write it down as you go. 1 + 1 is 2, uh, + 2 is 4, + 3 is 7, + 5 is 12, + 8 is 20, uh, + 13 33, uh, 33 + 21 is 54, um, + 34, 80, 88, uh, and then we get 143 on and on. Um plus 100 minus 11 would be 200 and uh 32.

[00:15:34]Okay.

[00:15:35]That was fun. That was fun. We could keep going. You could track that in 5 minutes, see how far you get.

[00:15:42]I don't think anything too I I'm not noticing anything too special here. But last time when we counted every other one, we got something interesting. What happens What if we add every other one?

[00:15:55]So, this time I'm just going to write down every other number in the list. 1 2 5 13 21 34 um 89 233 uh 610 And then those are all of the numbers in the previous list. Or like I I was just skip skipping and just writing down every other one in a new list.

[00:16:32]Right?

[00:16:34]Okay, let's try it.

[00:16:36]If we add um one and Uh 1 + 2 is 3 + 5 is 8 + 13 is 21 + 34 is 55 + 89 is 144 + 233 is 377 + 610 is 987.

[00:17:10]Do those sound familiar? Huh?

[00:17:14]Those are the missing numbers that we didn't circle down, aren't they?

[00:17:21]Like those are exactly the missing numbers, especially if I included this one right here.

[00:17:28]Those are all the missing numbers from before. So, to be completely clear, what I was doing was 1 1 + 2 1 + 2 + 5 1 + 2 + 5 + 13 right?

[00:17:49]1 + 2 is 3 1 + 3 + 5 is 8 8 + 13 is 21 and just keep going.

[00:17:59]Um so, so, uh looks like looks like the sum of every other Fibonacci number is a Fibonacci number.

[00:18:31]That's pretty cool, dude. This is This is really cool if you ask me. Like I didn't know this one until a year ago or two. And that's I don't know, that's just delightful. It's just like, oh, that's that's neat. Like that's that's kind of cool. They're popping up again.

[00:18:48]They've got all this structure in them.

[00:18:50]Um try to convince yourself why. Try to think of a reason why. Can you prove it for yourself?

[00:18:56]Uh you can do it in the way that I was alluding to before. You might be able to do it a different way. Um what if we use the other approach?

[00:19:07]So, in this if I copy that, paste, uh if I added the ones that were not circled, 1 + 3 is 4. 1 + 3 + 8 is 4 + 8 is 12.

[00:19:33]1 + 3 + 8 + 21 21 + 12 is uh 33.

[00:19:44]1 + 3 + 8 + 21 + 55 33 + 55 is 88.

[00:19:55]Um there might be something interesting about this, but I don't see anything obviously interesting right now. And maybe there is, but um I mean, there must be some kind of structure generally in these in these types of number facts, but this one doesn't pop out the the Fibonacci numbers like the other one did.

[00:20:20]Mystery, right? Uh someday we will prove these things. We don't just have to believe them on evidence or the fact that it's worked so far.

[00:20:31]So, now we've gotten a little demo about the Fibonacci numbers, sums of Fibonacci numbers. I want to show you next matrices.

[00:20:46]Matrices, I think are a really great way, especially if you combine with your favorite numbers, ways to practice a lot of addition.

[00:20:55]Um I've said it in a couple of videos before. I think that matrices are the most important one of the most important mathematical objects today. Like they are so important. They're used in uh pretty much any any electronics. Like the reason GPUs are so expensive is because matrices. Like matrix multiplication is expensive. Uh you can parallelize things.

[00:21:27]Like this is a That's very vague and I don't really want to introduce uh matrix multiplication today.

[00:21:35]Um but what I'm trying to get at is you can collect a a collection of numbers in a rectangle.

[00:21:43]Um so 1 2 3 4 5 6 7 8.

[00:21:50]If I let the first one be this matrix 1 2 3 4 and B is 5 6 7 8, then I can ask what is A + B.

[00:22:00]A + B is 1 2 3 4 plus 5 6 7 8.

[00:22:10]And here the rule is that you will add the corresponding number.

[00:22:18]Uh so in the top left, we will get six.

[00:22:23]I'll highlight that one and then the others I'll leave alone. 2 + 6 is 8. 3 + 7 is 10. 4 + 8 is 12. Hey, look at that.

[00:22:33]We got even numbers between six and 12.

[00:22:38]Um we can do the same thing without a square matrix. 1 2 3 4 5 6 plus 7 8 9 10 11 12.

[00:22:56]Uh 1 + 7 is 8. 2 + 8 is 10. 3 + 9 is 12.

[00:23:04]Uh 4 + 10 is 14. 5 + 11 is 16. 6 + 12 is 18. And we got the same pattern again.

[00:23:16]Look at that. Isn't that nice?

[00:23:18]Just I didn't even expect that one to happen, but again, we counted by twos on all the entries. That's That's kind of cool.

[00:23:27]Um you can make up your own problems. Uh 1 1 2 3 5 8 uh 13 21 34.

[00:23:38]Now this is a 3 by 3 matrix.

[00:23:43]Uh with three rows and three columns.

[00:23:48]Three rows three columns.

[00:23:53]So as long as the other matrix that you add it to has the same number of rows and columns three rows and three columns, then addition is defined on two matrices.

[00:24:09]Um so we could keep going with the Fibonacci numbers and get some big ones.

[00:24:14]Let's see.

[00:24:15]I I remember them.

[00:24:17]Uh 55 89 144 uh 233 uh 377 um 377 610 uh 987 1597 2,584.

[00:24:41]Yep.

[00:24:42]2,584.

[00:24:45]Wait, 987.

[00:24:52]610 + 987 Yeah, that is right. 2,584.

[00:24:59]Okay, so you could add those together.

[00:25:03]The first entry would be 56, the second entry would be 90, uh the next entry would be 146.

[00:25:13]Uh 236.

[00:25:19]Um so just some ideas. Like this is a way that you can create some of your own addition problems, um instead of like trying to buy like a child a childish book of like addition problems, which is kind of strange as an adult. Um so so this could be a way that you generate some. You could just roll dice, you could ask for random numbers, you know, through Excel or something.

[00:25:48]Um but here's a here's a way, just like pick some of your favorite numbers, pick sequences like the prime numbers. If you remember those, uh 2 3 5 7 11 13, uh Um 23.

[00:26:08]Right?

[00:26:09]Uh what's the next prime after 23?

[00:26:13]Uh 25 7 29.

[00:26:17]Uh 31.

[00:26:19]37.

[00:26:23]30 No, 41.

[00:26:28]43.

[00:26:30]Uh 40 seven? Is 47 prime? I think it I think it is.

[00:26:38]Um yes, 49 51 is not, 53 is.

[00:26:47]57 is.

[00:26:50]And 59 is, I believe. I believe those are all right. And then, you know, add them together. Just add them together.

[00:27:00]Uh cool.

[00:27:02]Finally, I think that I think that's pretty good in terms of just trying to motivate, um introducing I just wanted to show matrices, just show some notation for matrices.

[00:27:15]Usually, we use capital letters to represent a block of numbers, a matrix.

[00:27:22]Um and you'll notice that because we're just adding the elements within, like the same elements, um well, you can convince yourself that A + B = B + A and that this matrix addition is commutative.

[00:27:46]You can add uh from either direction.

[00:27:50]So, on this one, I could have done 5 + 1 and 6 + 2 and 7 + 3, 8 + 4 instead of 4 + 8, etc. Uh cool. So, here's commutativity coming back again.

[00:28:05]Commutative t. Awesome.

[00:28:10]Well, thank you. Let's uh let's do one problem from Lang. Let's prove the cancellation law.

[00:28:19]I haven't looked at this one in a while.

[00:28:21]I think that I'll be able to do it. I hope so.

[00:28:25]Um so, this is section Uh 22.

[00:28:34]Prove the cancellation law, cancellation law for addition.

[00:28:51]So, this is something that we've taken for granted, but we need to we need we want to prove this.

[00:28:59]If a + b = a + c, then uh b = c.

[00:29:16]Cool.

[00:29:21]Very quickly, one last thing I wanted to say.

[00:29:24]Actually, I'm so sorry. One last thing I wanted to say about uh matrices is you might have heard before that the Fibonacci numbers are connected to the golden ratio.

[00:29:46]Um personally, I have never cared.

[00:29:49]>> [laughter] >> Like personally speaking, just for me, I never cared about that at all with uh when I heard about the Fibonacci numbers. And people just like I I never cared at all about that.

[00:30:02]But really, not that long ago, just like a year ago or something, pretty recently, I found out that it's because they are related to this matrix.

[00:30:17]You can You can show that this matrix will generate all of the Fibonacci numbers. And it's such a simple little matrix, 1 1 1 0.

[00:30:28]And then you apply linear algebra, which is the entire study of matrices.

[00:30:37]If you use linear algebra, then you can prove this fact is true.

[00:30:44]Um so I never cared about the fact that the Fibonacci numbers are connected to the golden ratio, but once I learned this proof that it like why they're connected to the golden ratio, I thought that that was incredibly beautiful. Like I actually genuinely was like, "Dude, that is actually really pretty."

[00:31:06]Um to the point to the point that you will see it in like pretty much every linear algebra book.

[00:31:16]Including this one. Including this bad boy right here.

[00:31:23]This one right here. This is the Fibonacci or this is the golden ratio.

[00:31:28]This number gets tiny and it disappears when the for like a large power of n.

[00:31:36]Um and you can prove you can prove that formula just like Gauss's formula. Gauss found like a speed up and it's just like an easy equation to calculate the triangle numbers.

[00:31:49]You can use linear algebra and this matrix to prove that uh the Fibonacci's um uh differ like the ratio of Fibonacci numbers uh is the golden ratio for large powers or for large Fibonacci numbers.

[00:32:12]Never cared about the fact that it's true, but the reason why was super beautiful when I saw that. Okay, so thank you. Uh let's do this proof.

[00:32:23]Here. So, um assume first A, B, and C are integers such that a b a plus b equals a plus c.

[00:32:49]Then um then b plus Oh, wait. Oh.

[00:33:01]Oops.

[00:33:03]You can add negative a to both sides of the equation.

[00:33:25]And by associativity we can regroup the parentheses.

[00:33:46]Um and you may be used to thinking negative a plus a equals zero. If we wanted to be really formal, we could do this. We could do one um commutativity.

[00:34:04]But clearly this is going to end up um equaling zero plus b zero plus c and so um I'm going to draw the double struck arrow to say implies like this line is true, then this line is true, then this equality is true, then this equality is true.

[00:34:35]B equals C.

[00:34:37]Q.E.D.

[00:34:41]Q.T. Square.

[00:34:44]So, that's it. That's what we wanted to achieve.

[00:34:49]Uh thanks everyone. Thanks for watching.

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