The Magic of Number 9 (Part 2)
The number 9 is kind of amazing. How amazing? Keep on reading for the exciting conclusion to Math Dude’s story about the magical number 9.
Jason Marshall, PhD
Listen
The Magic of Number 9 (Part 2)
Pick a number, any number. Then multiply it by 9. Next, add its digits together and keep doing that with each successive number you get until you end up with a single-digit number. Now subtract 5.
Find the letter corresponding to your number—where 1 is A, 2 is B, and so on—and think of a European country that begins with that letter. Take the last letter of that country and think of an animal that begins with that letter. Finally, take the last letter of that animal and think of a color that begins with that letter.
I bet you’re thinking about orange kangaroos in Denmark.
Am I right? Indeed, the results are in and math fans around the world have responded almost unanimously that orange kangaroos in Denmark is exactly what they get.
As we began to talk about in Part 1 of this series about the number 9, it wasn’t magic that let me know this, it was the amazing properties of the incredible number 9—properties that are used over and over again in lots of seemingly magical math tricks. Want to know how it works? Keep on reading!.
A Pair of Perplexing Puzzles
In our first episode about the magic of number 9, we talked about the fact that the first step in the orange kangaroos puzzle—the one where you multiply your number by 9—ensures that you end up working with a number that’s a multiple of 9. Interestingly, we also discovered that the puzzle posed a few episodes ago by math fan Natalie includes a procedure that also guarantees you’ll end up working with a multiple of 9.
In both puzzles, we’re first coaxed into creating this multiple of 9 number, and then we’re told to find what’s called its digital root. The digital root of a number is just a fancy term that means repeatedly adding up its digits (and those of each successive number we come up with) until we end up with a single-digit number.
Why do both puzzles have us go through this same seemingly strange rigamarole? To understand why, let’s take a look at exactly what happens when we find the digital root of a multiple of 9.
Digital Roots & Multiples of 9
Before we do anything too fancy, let’s just see what happens when we find the digital root of various multiples of 9 such as 9×2=18, 9×3=27, and 9×4=36. The digital root of 18 is 1+8=9 (hmm, kind of interesting), the digital root of 27 is 2+7=9 (hmm, definitely interesting), and the digital root of 36 is 3+6=9 (hmm, now that’s really interesting). This kind of looks like a pattern, right? It looks like the digital root of a multiple of 9 is always 9.
The digital root of a multiple of 9 is always 9.
But is this true for every single multiple of 9? As it turns out, yes! How do we know it? Well, we could keep checking larger and larger multiples of 9 to see if it holds, but that’s kind of tedious and there’s actually a much more clever and elegant way to show that it must be true.
First, a word of warning: This way is a little tricky, so do your best to follow along but don’t feel bad if you get a little lost. This proof is cool, but you can certainly understand where all of those orange kangaroos come from without understanding every last detail. Okay, here’s how it works:
How to Prove It
Let’s say we have a 2-digit multiple of 9 number that we can write xy. The x here stands for the digit in the 10s place and the y stands for the digit in the 1s place. If you think about it, you’ll see that we can also write this number 10x + y (since that’s just what the decimal number system means). Now here’s the clever part: Let’s rewrite this number as 10x + y = (9x + x) + y = 9x + (x + y)—in other words, a multiple of 9 plus the sum of the original digits.
If you think about this, you’ll see that it says that the sum of the digits is equal to our original multiple of 9 number minus some other multiple of 9 number. The fact that both of these are multiples of 9 must mean that the sum of the digits of a multiple of 9 number must also be a multiple of 9. And it means that the digital root of a multiple of 9 number must be a multiple of 9, too!
Of course, we only showed this for a 2-digit number, but you can use the same technique to show that it’s also true for numbers with however many digits you like. It’s a bit more work to do, but it does work!
Orange Kangaroos in Denmark
So what does any of this have to do with those orange kangaroos in Denmark? It’s actually surprisingly simple!
Once you know that the math part of the puzzle was designed solely to make us come up with the number 9, and then to subtract 5 from this number to arrive at the number 4. In other words, no matter what number you start with, the amazing properties of the number 9 that we’ve discovered guarantee that you’ll always end up with the number 4…and therefore with the letter “D.”
No matter what number you start with…you’ll always end up with the number 4.
Then, since Denmark is the only country in Europe that starts with “D” in English, you’re pretty much guaranteed to choose it. And once you’ve arrived at Denmark, kangaroo and orange are the most common things that people come up with—so they make for a fairly safe bet.
The tricky part is that all of the math at the beginning seems sufficiently complicated to ensure that everybody who randomly picks a number must end up at a different letter of the alphabet. Which makes knowing you’d come up with Denmark seem quite remarkable! But the truth is that all of that math was there to actually guarantee that it would happen! Pretty clever, right?
Wrap Up
Okay, that’s all the math we have time for today.
opens in a new window
Please be sure to check out my book The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Number 9 image courtesy of Shutterstock.
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.