What was the first times table?

0:47 Welcome to Maths Is Not Boring by kidopoly.com. I'm Mira! And I'm Finn. We have a really special episode today, and a big shout out goes to Amelia in the UAE who asked us a fantastic question. Who invented Times Tables?

1:04 If you haven't already, tap follow wherever you're listening so you never miss an episode.

1:10 Now, Amelia, to answer your brilliant question, I want you to start by picturing something very ordinary. Imagine you lay out some wrapped sweets on a kitchen table. You line them up very neatly. You make three rows and you put exactly four sweets in each row.

1:28 Okay, I am picturing that right now. It makes a perfect little rectangle made of sweets. Three rows going down and four columns going across.

1:38 Exactly! Now, I want you to imagine you are not allowed to touch the sweets. You cannot move a single one. Just walk around to the adjacent side of the table and look at the exact same sweets from that new angle.

1:52 Wait, if I look from the side, the lines go the other way. Now it looks like four rows going down, with three sweets in each row.

2:01 Yes! Not one single suite has moved. The total amount of sweets on the table hasn't changed at all. But just by changing where you stand, three fours magically turned into four threes.

2:14 That is actually kind of wild when you really picture it. The numbers just completely swapped places, but the physical reality of what is sitting on the table is exactly the same.

2:25 It is! And mathematicians have a real formal name for this quiet magic lying at the heart of multiplication. They call it commutativity.

2:36 Commutativity? That sounds like commuting? Like when people commute to work on a busy train in the morning?

2:46 Actually, the travel connection is a more modern invention. The word commutativity comes from an old Latin root that simply means to swap or to exchange. It means the numbers can completely swap places with each other, and the total outcome stays exactly the same.

3:06 So if you multiply three and four, the answer is twelve. And if you multiply four and three, it is also twelve. Automatically, you literally get two maths facts for the price of one just by walking around the table. Precisely!

3:22 It is a massive shortcut built right into the universe. And people have relied on this swapping magic for thousands of years. Which brings us back to Amelia's question about who invented Times Tables. We have to take a trip back in time to about 305 BCE in ancient China.

3:42 Wait, 305 BCE? That is over 2,000 years ago. Did they even have paper books to write their Times tables in back then?

3:52 Paper wasn't widely used for everyday writing yet, so instead, scholars and bureaucrats wrote their calculations on long, thin strips of bamboo. And one specific set of those bamboo strips became a massive, brain-bending mystery for modern historians. A mystery? A collection of thousands of these ancient bamboo strips had been dug up from an ancient tomb.

4:27 They were actually remarkably well preserved. Because they were waterlogged, they were protected from oxygen and decay for over 2,000 years. Though they were covered in thick mud and smelled absolutely awful.

4:42 Oh wow! They must have been incredibly fragile. Like holding a wet leaf that is about to fall apart in your hands.

4:50 Exactly like that. They were given to Qinghua University in Beijing, and a dedicated team of scholars had to carefully clean them with incredibly soft brushes. They were racing against time to stop the mould from destroying the ancient text forever.

5:07 That sounds terrifying. Imagine sneezing and accidentally destroying a 2,000-year-old message from an ancient mathematician.

5:15unknown Ha!

5:15 I know! It gets harder too. The original strings that once held the strips together into a scroll had rotted away centuries ago. So the strips were completely scrambled. The scholars spent years slowly reading the ancient ink and piecing the strips back together, like the world's most stressful jigsaw puzzle.

5:37 And what did they actually find when they put the puzzle together?

5:40 Most of the strips turned out to be history books and ancient poetry. But in the 2010s, they realised that 21 of the strips stood out from all the rest. When laid side by side and lined up perfectly, those 21 strips formed a complete grid. They had found the oldest known decimal times table in the world.

6:04 Wait, decimal? You mean using base 10, the exact same way we count today?

6:10 Yes, a professor of mathematics would insist on us using that exact phrase. The oldest known decimal times table. The ancient Babylonians had older tables, but they counted in base 60. The Qinghua strips are base 10, making them deeply connected to how we do math today.

6:30 So over 2000 years ago, in ancient China, someone was using commutativity to figure out their taxes or trade, reading it right off those bamboo strips?

6:41 Yes, and because of the magic of commutativity, they only needed half the space. Once you understand that 5 times 7 is the exact same amount as 7x5, you don't need to write both of them out. The grid does the work for you.

6:57 Hold on though, I need to push back on this a little bit. Does this swapping magic work for every single kind of maths? Like, if I take my suites and subtract them instead of multiplying, it doesn't seem like commutativity would work.

7:12 Ah, Finn, you have hit on something absolutely crucial. No, it does not always work. Commutativity holds perfectly true for ordinary numbers when you multiply them and when you add them together. But subtraction and division, they absolutely do not swap. Not even a little bit.

7:33 Right, because if I have four wrapped suites and I take away three, I have exactly one suite left for myself. But if I only have three suites to start with, I cannot possibly take away four.

7:47 Exactly. The order matters completely for subtraction. If you swap the numbers, the whole reality breaks down. But for multiplication, the order does not matter at all. That is why commutativity is such a beautiful, special property.

8:04 It is almost like this invisible rule built right into the fabric of the universe. It doesn't care if you are an ancient scholar in China holding a delicate bamboo strip, or a kid sitting in your kitchen today with a pile of sweets. The rule stays true forever. So if commutativity is this universal rule that is everywhere, where else is it hiding in normal everyday life?

8:28 Aside from wrap sweets sitting on a table.

8:31 Oh, it lives everywhere you see a grid. The next time you open a carton of eggs in the fridge, I want you to just look at it. Let's say it is a standard box of six eggs. It is usually sitting there as two long rows of three eggs.

8:46 Oh, I see where this is going. If I pick up the carton and turn it sideways in my hands, suddenly it is three rows of two eggs. The eggs themselves didn't change at all.

8:58 Exactly! You just witnessed commutativity in your own kitchen. It also lives inside your favourite snacks. Think about a big chocolate bar with breakable pieces. The bar might be four pieces across and five pieces down, but turn it 90 degrees and suddenly it is five pieces across and four pieces down.

9:19 I see it now. It is all just rectangles turning on their side. Like looking at the giant square window panes on the side of a tall office building. Or walking into a cinema and looking at the long rows of folding seats.

9:33 Yes, every single time you look at a grid of objects, whether it is the little square tiles on your bathroom floor, or the hundreds of seats in a stadium, commutativity is staring right back at you. The shape of the whole rectangle never changes its actual size, just the way you are looking at it. That is the quiet, beautiful magic of multiplication, hiding in plain sight.

9:58 Alright Finn, quiz time.

10:01 Okay, I'm ready.

10:03 First question. What is the real mathematical name for numbers swapping places in multiplication? Next one. What material was the ancient Chinese times table written on? Question three. How many special strips made up that ancient times table? Here's number four. What year roughly do those ancient strips come from?

10:42 Last question. Who asked us the brilliant question that started today's episode? Great job! Let's see how you did. The first answer is commutativity. For question two, it was written on bamboo. Number three, the answer is twenty-one strips. Question four, they date back to about 305 BCE. And the final answer is Amelia in the UAE.

11:19 You know, I'm never going to look at a chocolate bar the same way again. I'm just going to be spinning it around to see the commutativity in action.

11:29 I love that! It is amazing how one little change in perspective completely transforms how you see the world. If you enjoyed seeing the world differently today, please leave us a five-star review. Just scroll down and tap the stars. It helps other curious kids find the show.

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