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,
Episode #178

Magic Number 9

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 x 2 = 18, 9 x 3 = 27, and 9 x 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:


About the Author

Jason Marshall, PhD
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