The questions that kids ask about science aren’t always easy to answer. Sometimes, their little brains can lead to big places that adults forget to explore. That is what inspired our series Science Question From A Toddler, which uses kids’ curiosity as a jumping-off point to investigate the scientific wonders that adults don’t even think to ask about. The answers are for adults, but they wouldn’t be possible without the wonder that only a child can bring. I want the toddlers in your life to be a part of it! Send me their science questions, and they may serve as the inspiration for a column. And now, our toddler …
What if there were no number 6? — Isaac R., age 5 1/2
“There are things … where you just stand there and you’re like, ‘At this point, am I teaching math or am I teaching philosophy?’” said Jordan Ellenberg, whose titles at the University of Wisconsin suggest that he is meant to be teaching math. But he’s right about the line between the two being thin, and a question like Isaac’s is a perfect example of that. To even start to contemplate an answer, experts told me, you have to first define the nature of “6”: Is it something humans invented, or is it a universal fact that humans just made up some names for?
Most of the mathematicians I spoke to for this story decided to go with the assumption that Isaac is asking about a world where 6 ceased to exist as a fundamental concept. We aren’t talking about simply renaming the thing we now call “6” and calling it “5” or “7” or “splorfledinger” instead. We are talking about literally being unable to take six objects and put them into a pile — the universe looks at a half-dozen eggs and says, “Nah.” All the mathematicians agreed that would have some serious consequences — maybe to the point where life as we know it ceases to exist at all.
What would happen if 6 vanished overnight? At least some of the other numbers would also cease to exist. If there’s no 6, said Caroline Turnage-Butterbaugh, a math professor at Duke University, then there can’t be a 7, 8 or 9 — or, really, any number greater than 5. Say you have a pile of seven items. You could take one away, ending up with the-number-formerly-known-as-6. But because 6 is just gone — there can be no pile of six — you can’t have 7, either, Turnage-Butterbaugh told me. It, too, winks out of existence by virtue of creating a scenario that could lead to an already nonexistent number. “All the other integers are out,” she said. “It’s very detrimental.”
Lillian Pierce, another Duke math professor, thought you’d have to eliminate 2 and 3, as well, because, otherwise, what would 2 x 3 equal? Anything that could multiply or divide into 6 would have to go. If losing 6 meant losing 2, Pierce wrote to me in an email, then we would also lose the concept of “even” and “odd” numbers.
And losing 6 would affect far more than just numbers, since numbers pervade so many things. Renate Scheidler, a math professor at the University of Calgary, said there’d be no six-string guitars, so music would sound different. Hexagons couldn’t exist, so bees couldn’t build their hives the same way, since the cells in a beehive are made of hexagons.
And, oh, one more thing: It’s possible that losing 6 would radically change all life on Earth. That’s because life on this planet is based on basic, building-block molecules made up of carbon and hydrogen atoms. Some of those molecules, including benzene and cyclohexane, are six-sided.
So a loss of 6 could be devastating. But that doesn’t mean you have to use 6 if you don’t want to. As long as 6 (or any number, really) still exists as a fundamental concept, we can choose many different ways of naming, counting and thinking about that number. We know, after all, that humans in other times and places didn’t count the way we do today — in sets of 10. The ancient Babylonians, for instance, had a mathematical system based on the number 60, which was possibly an outgrowth of choosing to count using the 12 knuckles on four fingers of one hand, rather than counting each finger on both hands to get to 10. If we put it up for a vote tomorrow, and all agreed, we could just choose to skip “6” and call a grouping of x x x x x x objects “7” instead, said Anne Shepler, a math professor at the University of North Texas.
And choosing to count in different ways goes beyond cultural differences. The researchers I spoke to told me about a system called modular arithmetic, a way of counting where you really could create a world without 6 — a place where 5 + 1 effectively equals 0.
Before you call “fake news” on me, consider that modular arithmetic is already something you use every day. The entire way we tell time is based on this. Lurking inside a mundane, civilian clock is a world without 13; a world where 12 is effectively the same thing as 0; a world where 7 + 7 = 2, and not 14.1 You are familiar with this already, and it’s not as strange as it first appears.
That’s why clock faces are a useful way of thinking about modular arithmetic, Turnage-Butterbaugh told me. A normal clock is using 12 as its modulus — the number the modular arithmetic counting system wraps around. In the case of a clock, we typically still use the modulus itself. But you don’t have to. Midnight and noon could just as easily be 00:00 as 12:00. Now imagine a clock face with 0 at the top, 3 at the bottom, and tick marks for 1 and 2, 4 and 5 on either side. That’s a visualization of a modulo 6, or mod 6, counting system. Start at the 5, count one place over. You end up on the 0, right? 5 + 1 = 0. Congrats, you’ve just made a world where math still works, but the 6 is gone. On Isaac’s next birthday, he will be 0.
So we can use modular arithmetic to remove 6 — or any other number — from the world whenever we want to, at least in a limited sort of way. It’s kind of like a clubhouse where only certain numbers are welcome. You couldn’t remove those numbers from the world entirely. But in the club, math functions just fine without them. And this trick is more than just a curiosity that helps us answer esoteric children’s questions. Modular arithmetic has practical uses, as well.
For example, one of the most well-known (and oldest) codes in the world is the Caesar cipher. This has been used by everyone from Julius Caesar himself to my fourth-grade classmates,2 and it’s based on substituting one letter for another — all the A’s in a message become C’s, for example. How do you decide which letter to substitute for A, though? That’s where the modular arithmetic comes in. The Caesar cipher is a mod 26 system — one digit for each letter of the alphabet. If A = 1 to start, then you can shift everything by two places and end up with A = 3. Because C was originally the third letter, A becomes C. (Meanwhile, 26 + 2 = 2 in mod 26 counting, so Z becomes B.) In this case, the key that cracks the code is knowing that we’re working with mod 26 + 2.
The takeaway from all of this — particularly if you are a young person interested in math — is that asking weird, philosophical, have-you-ever-really-looked-at-your-hand sorts of questions can lead to stuff that’s actually useful. And, also, that mathematicians absolutely love it when you ask these kinds of questions. As Shepler wrote to me in an email, “To a mathematician, any kid could be the next Einstein potentially ruined by misguided adults who don’t take him/her seriously.”