Let me explain why multiplication tables are often taught wrong, and show a bunch of underrated patterns within them: 0:00 - Part 1: Metamemorization 2:26 - Parts 2 and 3: Primes and Composites 5:58 - Parts 4, 8, and 9: Squares and Almost-Squares 13:30 - Parts 10 and 100 and etc: The Beauty of Zooming Out 19:40 - Part Infinity: Conclusionfinity Thanks for watching! Leave a comment about your favorite part. And consider showing this video to a teacher in your life (if teachers incorporated more of these patterns when teaching arithmetic, I don’t think as many people would hate multiplication tables!). If you want bonus content, I've been doing livestreams every few days on my @Domotro channel, which are saved like videos on that channel's "live" tab afterwards. There is also a Combo Class "Discord" where combo lords chat, and a "Patreon" for those able to help fund future episodes (see the video description for names of my supporters and links to those).

Love how dude just randomly feeds a squirrel in the middle of shooting.

one of the few people on the internet who can manage to scratch my itch for entertainment pertaining to mathematics

When I found out about the almost square numbers being very similar to square numbers, I tried doing my own research generalizing the pattern and extending it more down the line. The pattern would go: a²=(a-n)*(a+n)+n² which can be really useful if you needed to calculate something like 28*32, as 30*30 is easy and going 4 down leaves you with 896...

These videos are soothing to me. The chaos makes me feel safe and welcome, like it’s okay for me to be messy.

Every upload I watch is more chaotic than the last. Regardless, it is really cool to see the times table with some new perspective. Lots of cool patterns exist in it, as is the case for all mathematical structures!

One of my favorite examples of patterns being more useful than memorization is trig identities. I may forget that tan^2 + 1 = sec^2 but I do know the definitions/practical meanings of sin, cos, tan, and I know how they relate to csc, sec, and cot, so I can figure it out from there!

This is how Squirrel obesity starts.... - squirrel obesity makes squirrels spherical instead of squareical

There's actually a general case for the (n+1)(n-1)=n²-1 type of "almost square" that I actually find really useful. (n+x)(n-x)=n²-x². So 6×2=(4+2)(4-2)=4²-2²=16-4=12. And indeed, if you continue moving diagonally away from the squares on your multiplication table you'll see this pattern of subtracting swuares quite clearly. And what I mean by finding this really useful is that if I have to, say, multiply 40×46 in my head, I'll often find it easier to find the square of 43 and then subtract 9 from it. Not that I have 43² memoried, but I can work my way up from 40² (which is 4²×10²=16×100=1600). (n+1)²=n²+2n+1 after all, so I just add 81+83+85=(80×3)+(1+3+5)=249 to 1600 for 43²=1849, and thus 40×46=1840. Yeah, sure, I could also do 40²+(40×6), but often I personally find the "almost square" method reduces the number of things I have to keep in memory all at once (and also the fact that they use different operators, plus the ability to add on my fingers rather than doing even more mental multiplication, helps me avoid mixing things up). It's not always the right tool for the job, but it's a tool I've found to be worth knowing (it's also one of the first bits of math I derived from first principles without any help as a kid (without even using algebra), so I have an admitted soft spot for it).

Not only hes talking about a good topic but he is one of the few channels I saw that *actually has a personality.* Please, dont stop, we need more people like you.

An interesting property of square numbers that I always forget but find cool is a thing that I remembered this video is that if you pick a number, let's start simple with 1 and 2, then square it, so we get 1 and 4, then subtract the bigger one with the smaller one, we get 3, then we repeat the process with the bigger becoming the smaller number and its successor, so 2 and 3, then squaring it, we get 4 and 9, then we do 9-4 and we get 5, which is 3+2, and we can repeat that process a lot of times (for example 16-9=7, 25-16=9, 36-25=11, 49-36=13, 64-49=15 and so on)

This general idea of teaching by way of showing what patterns exist, indicating where they overlap in different representations, and then extrapolating that pattern outwards.. it's fire. It's very broadly applicable as so many patterns are overlapping and relatable across so many systems and compositions. One commenter mentioned that the only problem with teaching kids this way is that it's too much at once. I disagree to the extent that I believe that most humans, and especially some, are geared towards recognizing patterns and then seeing how they fit together and finally extrapolating what the broader system looks like.. we *want* to see how it fits together and our biology backs up that drive. There could be no foul in exposing young curious minds to some of the fundamental overarching patterns in many arenas, and I'm in agreement that it's probably the thing to do in fact.

I tend to have a few sparse multiplication results memorized to act as waypoints, then move to the desired multiplication with repeated addition.

It’s also very interesting how these “multiplication tables” generalize to other operations. Tables like these are incredibly useful for representing the abstract notion of function composition with regards to groups and their actions (and many other places, but I found it very helpful in that case.) Noticing patters akin to the square numbers or primes etc can give you a much better idea of the group you’re dealing with.

Stopping me forgetting my entirely memorised education and teaching me more interesting and useful ways to apply more solid concepts... I wish you were my maths teacher sooo bad

This dude gives me very strong Radagast the Brown vibes 🤣👍

Thank you for making brilliant free educational content. The atmosphere and manner of teaching is so comforting and wholesome. I am hooked.❤️‍🔥

Absolutely , less memorising times tables and more patterns and intuition. Another great video ! 😊❤️👍

I never properly memorised my times-tables. I fill in the gaps with addition or subtraction, depending on whether the closest value that comes to mind is above or below what I need. One thing nobody taught me about times-tables as a kid, but I realised when studying computer science at uni, and that I wish more people knew about, is that they're an example of the space-time tradeoff in action. It's a lookup table, trading increased memory for reduced compute time. You're spending a hundred slots in your memory to save the time you'd otherwise spend calculating those commonly-used values. For an actual use in a real program of such a technique, transparency in Duke Nukem 3D uses a 256*256 2D lookup table to find which colour in the game's palette should be used for a 1/3 transparent pixel (or 2/3 by swapping the order) when overlaying transparent walls or sprites. The actual calculations of what colour you'd get with a 2:1 mix of any two colours in the game's palette, and which colour in the palette was the closest match to it, was done by the developers during development and saved to a file that the game can then load from disk on startup. It takes 64KB of memory at runtime, which is not insignificant for a game that has to run on machines with 8MB of memory, but it's much faster than calculating three linear interpolations and then searching through the 256 palette entries for the closest match. Definitely worthwhile if it means you can run the game on a 486, rather than a Pentium.

I absolutely love this channel. You do such a great job at teaching! Keep it up!

Domotro is a legend. He should make a second channel where he proves everything from his main videos 😎

I feel like the filming has leveled up a bit and the aesthetic is really coming together with all of it. Exciting seeing shit start to really gel! As always the math is high quality infotainment, thank you D!

The only problem with teaching kids exactly like this is that it presents way too many concepts at once. The best way to first ever present a times table to a kid is as an addition table, showing different patterns with sum and subtraction on the times table, then presenting the division one as the perfect opposite. After all that, it would definitely be benefical to slowly present some of those concepts you showed here and quite a lot of those would be useful for future concepts.

There are two things I’ve discovered in this video by looking at the times table: The difference of consecutive same-parity integers happens perpendicularly to the perfect squares diagonal and symmetrically from the perfect squares diagonal until the difference becomes 0, reaching the outer edges. This relates to the other thing: The sum of same-parity integers form diagonals parallel to the perfect squares diagonal, but they’re offset by the integer they start on and are the addition of the same parity of that integer.

Man- this is SUCH a good maths channel! Great job, brother!😊🎉

"Hello 911? I need to report a fire next door - oh wait...he's just doing that thing again...nevermind"

Great video. I suspect the going up to 12 thing is a tradition from a combination of Imperial measurements, old school British/Roman currency with 240 pence to the pound in 20*12 units and some time based calcuations.

i love how this gives a new perspective on things taken for granted. also, SQUIRREL!! 🌰🐿

And the pattern, as shown in the 3 * 5 example, continues. For all numbers their product is: the average, squared - the distance from the average, squared e.g 7 * 4 Average = (7 + 4)/2 = 5.5 Distance = (7 - 4)/2 = 1.5 (5.5)^2 = 30.25 (1.5)^2 = 2.25 30.25 - 2.25 = 28 = 7 * 4. It extends to all real numbers, no matter the signage. e.g -7 * 4 Average = (-7 + 4)/2 = 1.5 Distance = (-7 - 4)/2 = -5.5 (the same absolute values as for 7 * 4, but swapped) 2.25 - 30.25 = -28 = -7 * 4

That squirrel is my spirit animal

2:10 Our brains *love* patterns. That's like the whole reason we evolved intelligence, pattern recognition. Even a dumb butterfly can memorize a location across the world. But smarter critters can and enjoy making deductions and noticing patterns. Crows are fascinating for their ability to do so

Ive been coming up with fun tricks for multiplying things by 15 because every morning when I make coffee I use a 15:1 coffee:water ratio by weight in grams! a fun little exercise every morning :)

15:19 I absolutely adore this cozy way of doing math on paper. Sure, computers can make accurate pictures, but it feels almost soulless

You're honestly a fantastic educator

I have ADHD, so I stubbornly refused to memorise anything... I "cheated" by figuring out "tricks" to avoid actually remembering the tables, like times x * 2 is just x + x and x * 5 is just (x*10) / 2 and x * 10 is simply adding a trailing zero etc. Which means rather than memorising; I figured out what multiplication actually means.

I found that polygonal numbers form straight lines, like square numbers but different slope. Triangular numbers like 6, 10, and 15 are all in a line

The biggest real life benefit of this video goes to the… you guessed it… yeah… to the squirrel 👏👌😹

Here's s fun thing i never thought about before: if you make a rectangle from with a corner in upper left corner of the product cells, and the opposite corner at the lower right corner of some cell with number n, then n is the area of that rectangle, and the two coordinates are describe the side lengths of the rectangle. I don't know why ive never considered that before.

The squirrel bits were so good holy shit. The times when you were pointing to things on the times table desperately needed highlighting though.

I love this backyard scientist aesthetic

Domotro - Explainer of Maths, Patron of Squirrels

Sharing this with my parents, one of whom is currently teaching 5th grade and the other of whom is a teacher's aide specializing in math for grades 3-8 or so!

You Positively Exude Witchy Energy. From your disregard for current conventions, to your attunement with nature and your urge to breakdown and understand the world you reside within. People like you are who generate improvement around the world.

When your math teacher goes into hunger coma:

Love your lab or studio, It design seems timeless 😅

How does zero appear an infinite amount of times in the 0-9 table?

This genius is a dark mage of our time, resurrecting the ancient power of math and conjuring a future where it's no longer feared, but embraced by the masses. These videos are mildly terrifying in the best way - the kind that strikes awe, reverence, and a burning curiosity soaked in eager urgency.

I think by the end of Grade -4 the entire video will be a giant Rube Goldberg machine on fire.

Get you a guy who caresses you like Domotro caresses almost-square numbers.

As a secondary school maths teacher in the UK, I often get pupils who have struggled to memorise all the tables. I get them to memorise the squares and work from there to get 8x7 etc as its the exact method that got me thru to Post Grad studies.

4:50 ok that is just adorable

wow reversal of entropy in the first 20 seconds - must be a powerful episode

Let's say you are asked to multiply two numbers which are not far away from each other. I'll take the Example with 21x28 First step: find the "middle number" between 1 and 8 middle number is either 4.5 So you can select anyone you want but I'll select always de smallest one; 4 in this case. Then 21x28 = 24x24 + 24 - (28-24)*(28-23) Thus: 24x24 + 24 - 4*3 Now, I dont know how much 24x24 is but I do know that: (a+b)² = a² + 2ab + b² hence: 24x24 = 20² + 2*20*4 + 4² And finally, 21x28 = 20² + 2*20*4 + 16 +24 - 12 = 20² + 188 = 588 If the two numbers are too far away like: 21x51,then: 21x51 = 21x25*2+21 and calculate 21x25 as 23² - 4 = 20² + 2*3*20 + 9 - 4 = 400+120+5=525 So: 525*2+21=1071 With Practise you can do this mentally pretty fasy

Yep, ive got the 12x12 times table memorized from 3rd grade still Now ive got the times table (cayley table) for the sedenions which are defined by 15 imaginary axis which each square to negative one (for 15 different directions for rotation) and 1 real axis for scaling. Technically every single combination of elements of the 15x15 sized cayley table exists for some setup for sedenions, because you can change whether the element squares to negative one or positive one, and the fact that their names are entirely arbitrary. The setup i have memorized is equivalent to a matrix with the format (reals) (along the diagonal), -i, -j, -k, ... (i being the standard imaginary numbers, and you have all of them one after the other along the first row all negative) then all the imaginaries go in their respective antidiagonals and when they cross the regular diagonal they switch to positive versions of themselves. The rest of the items are actually super easy to figure out because the pattern you got from filling out the first half with the information ive given you

I hadn't considered the almost square factors before. Looking at those diagonals reveals difference of two squares. If you know your square numbers, fanning out from them orthogonally, you just subtract square numbers for the distance from the square diagonal. That fact alone helps you figure out half the numbers in the table. As for your curve, I think it might be more clear to say greater than 100 or greater than 10. This is because you aren't going to start counting at 0 for that band, 0 being a special case. Not only will it improve the appearance of the curve, but it makes more sense logically.

19:42 I legit thought he took a bite out of a tennis ball

Thank you! now instead of doing 7x13 i can do 10x10 -3^3 because I learned that squares are squares away in the other diagonal. Can I do the opposite? like 14x14 is 10x18+16 ? Nice!

That squirrel shot was cool

He keeps the squirrel away by throwing clocks at the ground.

Most things fallen in any video of yours I've watched :) Once I memorized the primes up to 199, I felt like I could intuitively calculate all the other numbers in the cells. I memorized them because I like to count primes when I do breathwork, or am in line waiting for something, just as some fun for my brain. I never would've done it at a teachers request because I'm that kind of stubborn. Primes are the gateway drug to numbers. Math could be so fun in school if they let it be.

will there ever be a combo class video where nothing falls, breaks or gets set on fire?

The difference of squares trick is my favorite way to make people think I'm smarter than I actually am. By my ability to "effortlessly calculate" that 58 x 62 is 3596. Or 29 x 31 is 899. I'm not literally crunching the numbers in my head, I'm just cheesing the rules lol

very nice - there are chineese combo classes where children are taught multiplying quickly (like a quick reading, tapping pencil) i am awaiting children that will be using Parseval theorem better than i :)

yo domotro so that curve you found in the times table i think is called an asymptote (but its offset by abt 1 in each direction). i could be wrong tho but yeah

Love the maths... My PTSD is allergic to the banging. Will try with volume off and CC on. ❤❤❤

In France, we only learn tables up to nine. By the way, I don't really see how those patterns with the inverse function are useful, especially since we don't learn about functions and graphs until long after memoizing the times tables.

19:43 Is he trying to grow a cubic apple on the tree, in the plastic case? Or is it a message in a bottle?

Fire, squirrels and math! What more do you need?

i cant even physically imagine how dirty that entire setup must be

This table is the reason why we should have all converted ourselves to being base 2 creatures. That way we would've only had to "memorize" the truth table of the AND gate!

I thought this was an Explosions & Fire video because of the thumbail with the fire, lab coat, and face

This turned out to be unexpectedly very cool!

Domotro: "What's 12 x 7" Me, proving his point: "84-FUCK"

this is how I learned, i never learned math the right way and learned patterns

It seems complicated to people, but imo using patterns is infinitely more easy than rote memorization of the entire table. In fact, it blows my mind that people actually do that. I actually didn't even learn with a table. I just gradually picked up on various patterns myself as I progressed through math. Idk why, but my school didn't even use times tables for the gifted class. Only the regular class was forced to memorize them. I guess they just assumed we'd pick it up naturally? They were right, but I think most people really don't even need the table to get good at it.

pattern i found and used as a dumb baby: for 9 times a single digit number n, the tens digit is (n - 1) and the ones digit is (10 - n)

Aren't the part numbers ordinal? Wouldn't that mean that the last part isn't ∞, but ω? 🤔

Thanks!

It's good to see that the mad hatter has found a hobby.

Well... they only told me to memorize up to the multiplications for 10 in school, so I guess this doesn't apply to me

product = (sum/2)²-(diff/2)² product = average²-(average-smallest)² consider sum = a+b, diff = a-b, product = a×b, average = sum/2, smallest = a, for numbers a

This channel is lit

Time stable? I bet you love this.

That intro sounds crazy

As a kid I never memorised the times table, I just remembered every 3rd entry. I don't need to remember 1×7, it's obviously 7. I don't need to memorise 2×7, it's obviously 7+7=14 3×7 I memorised 4×7 is just 3×7+1=21+7=28 5×7 is 6×7-7 = 42-7=35 6×7 I memorised 7×7 = 6×7+7=49 ... Memorising 144 different answers never made sense to me when I can remember about 30 and never be more than two simple addition/subtractions away from the answer. I also never liked history. So many dates to memorise and they were just meaningless to me.

Now that was unexpected! And pleasantly so. :)

I learned the multiplication table in chinese (up to twelve), and it sounds sort of like a chant since it has only one syllable for the numbers 0 to 10. i'm at the point of my life at which i rarely do mental arithmetics and old memory starts to mutate, one day I realised that my muscle memory of reciting the multiplication table betrayed me after double checking the calculation with a calculator, because it sounded right even though the numbers are wrong

there may be one other interesting way to actually teach multiplication tables better and that is by counting since the connexion is a bit clearer between the factors of the same number then again, for some children even the dice representations that you were showing might help since the current memorization is so dry

Hey howya doin combolords

I was taught all this aswell though.

If two large numbers are close to any easy square, then they can be multiplied using a quadratic. solve: 999,996 × 999,997 let m = 1,000,000 999,996 × 999,997 = (m-4)×(m-3) = m²-7m+21 = 10¹²-7×10⁶+21 = 1,000,000,000,000-7,000,000+21 = 999,993,000,021 easier than multiplying for real. * any numbers separated by two is the square of the middle minus 1. or (x-1)(x+1) = x²-1

13:53 - I'm squinting...

As a dart player I accidentally memorized the three times table up to 20

here in germany we only had to remember the quadratic numbers up to 20, so only the diagonal, not the rest, thankfully

enough about square numbers, what about squirrel numbers numbers?

i learned somethin'!

Why the fuck wasn't I taught maths like this at school.

i like you clock wizard guy

Damn just found you. You should get a lav mic cause when you turn away it can get harder to hear you.

I actually had a moment the other day when I wished I memorized the multiples of 13, knowing I could have solved something much faster.