What is clock arithmetic?

0:47 Welcome to MassIs Not Boring by kidopoly.com. I'm Mira! And I'm Finn! If you haven't already, tap follow wherever you're listening so you never miss an episode.

0:59 Hey Mira, have you heard the Invisible Gorilla episode on Philosophy's Not Boring yet?

1:05 Yes, it is so wild. Psychologists run a test on selective attention, where they have people watch a video of a basketball game and count the passes.

1:15 Right! And because everyone is concentrating so hard on counting, about half the viewers completely miss a person in a full gorilla suit walking right through the middle of the game.

1:27 Exactly! It is fascinating how the brain handles numbers and attention. If you want to discover how concentrating can make an entire gorilla vanish in plain sight, you might consider giving Philosophies Not Boring a listen and a follow.

1:42 Okay, so what is our mathematical puzzle for today?

1:46 Well Finn, today I want to take you to a world where 11 plus 3 is 2.

1:52 Wait, no it's not. 11 plus 3 is 14. It has always been fourteen.

1:58 On an ordinary number line, yes, but I promise you, there is an object you look at every single day where 11 plus 3 is exactly 2, and it makes perfect, obvious sense.

2:10 An object I look at every day. Where numbers behave differently. Wait, let me picture this. 11 plus 1, 2, 3. Is it a clock?

2:23 Yes! Think about it. On an ordinary number line, numbers go on forever in a straight line. But on a clock face they go round and round. After 12 comes one again.

2:35 Oh, so if it's 11 o'clock and you wait three hours, it's not 14 o'clock. You just count past 12 and land on two.

2:44 Exactly! 11 plus 3 is 2. You just did arithmetic on a circle instead of a straight line.

2:51 Haha, that feels like cheating, but it is totally true.

2:55 People have known about remainders and wrapping numbers for a very long time. But in the year 1801, in Germany, a brilliant mathematician named Carl Friedrich Gauss wrote a massive, incredible book about it. What was the book called? It was called Disquisiziones Arithmeticae. Gauss was only 24 years old when it was published.

3:19 Whoa, that sounds like a serious, heavy book, just to talk about what a clock does.

3:24 It was. He didn't invent the idea of remainders, but he introduced a systematic theory for them. He proved that numbers that wrap round follow their own tidy rules, just like ordinary arithmetic.

3:38 So he treated this spinning clock math like real, serious mathematics?

3:43 Yes, he gave it a special notation, a way of writing it down so mathematicians could explore its hidden patterns. It was completely revolutionary.

3:53 You know, I actually did a wrapping number thing yesterday without even knowing it.

3:57 Wait, you did? How?

4:00 Yeah, I was looking forward to the weekend. Today is Saturday, and I was counting days, and I realised in seven days it won't be day number twenty something. It just wraps right back around to Saturday.

4:12 Finn, that is a brilliant example! Days of the week work exactly the same way. They wrap round after seven.

4:20 And months wrap around after twelve too. January always comes back after December. It never goes to month 13.

4:28 Yes. The secret to all of this is what mathematicians call the remainder. What matters is the remainder. What is left over after the full turns? So the full turns just disappear? Kind of. Imagine spinning round in a circle. If you spin completely around and face exactly where you started, that full spin doesn't change where you are pointing.

4:53 That is so strange to think about. You can have these huge amounts of time, days and days spinning past, but all that really matters is the little bit left over at the end. Exactly.

5:05 Whether the clock goes round one time or a hundred times, the math only cares about where the hand stops. The remainder is the star of the show. Now, mathematicians have a real name for this idea. They call it modular arithmetic.

5:21 Modular arithmetic. That sounds like building a spaceship out of modules.

5:27 It does. When we use a 12-hour clock, we are doing arithmetic modulo 12. The number where everything resets is called the modulus.

5:37 Wait, what if someone uses a 24-hour clock? Like my dad's digital watch.

5:43 That is exactly the same idea, just modulo 24. The circle is just a little bigger before it wraps around to zero. Using 12 or 0 at the top is just a choice we make.

5:57 Oh, mirror! I just realised something else. This math connects to so many things we use. It does? Like what? Like things that spin. When you look at a circle, whether it is hours or days, it wraps right back around again.

6:13 Finn, yes! That is one of my favourite ideas! Wrapping the days of the week is modular arithmetic in action.

6:21 They go in a circle. So Gauss was explaining the math of spinning things, working for clocks, days, and even secret Caesar codes.

6:31 It is beautiful, isn't it? One simple idea hiding everywhere you look. And it is even more everywhere today. The exact same modular arithmetic Gauss wrote about in 1801 is what checks the barcode when you buy a snack.

6:47 Wait, barcodes use wrapping math?

6:49 Yes! Barcodes have a special check digit at the end. The computer uses modular arithmetic and looks at the remainder to help spot mistakes if the scanner read the number incorrectly.

7:02 No way! So if the scanner reads a long number, the remainder has to match perfectly or it beeps an error. You got it!

7:10 And it's not just barcodes. Card numbers use it, and the secret codes that protect messages on a phone all rely on checking what is left over after the full turns.

7:21 So this isn't just about telling time. It is literally holding our digital world together.

7:26 It really is. And here is a fun fact about Carl Friedrich Gauss. That exact same year, in 1801, astronomers lost track of a newly discovered dwarf planet called Ceres. Did Gauss find it using clock math? Not quite clockmath, but he used his incredible grasp of numbers to predict exactly where it would appear in the sky.

7:50 And he was right! They called him the Prince of Mathematicians.

7:54 A prince of numbers? That is the coolest title ever.

7:58 Alright Finn, it is quiz time. Are you ready? I am ready. Lay it on me. First question. What is the real mathematical name for this wrapping math? Next one. What year did Carl Friedrich Gauss publish his famous book? Question three. What is the secret that matters most in this kind of math? The part left over.

8:33 Here's number four. What common thing wraps around after seven instead of twelve? Last question. Name one modern technology that uses this math to check for mistakes. Great job! Let's see how you did. The first answer is modular arithmetic. For question two, the book was published in 1801. The answer to question three is the remainder.

9:09 For question four, days of the week wrap around after seven. And the last answer is barcodes, card numbers, or secret phone codes.

9:20 I am never going to look at a Saturday the same way again. It's just a remainder, hiding in plain sight. I know.

9:30 Once you see the world looping in circles, you can't unsee it.

9:35 If you're listening and you loved spinning in circles with us today, please leave us a five-star review. It helps other kids find the show.

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10:00 And if you want to shout out on the show, or just want to say hi, email us at hello at kidopoly.com. Thanks for exploring with us. See you next time on Maths is not boring. Bye.