I understand how to do it. But how does it work? How does crossing out 9s help you check if a basic arithmetic problem is incorrect?
Something to do with balancing the equation?
Thanks!
In: Mathematics
So the concept you’re looking at is called a digital root, and for those who are looking here wondering what we’re talking about: a digital root is the number you get when you add up all of a number’s digits and keep doing it until you get only a single digit, its digital root. For example, **482** –> **4+8+2=14** –> **1+4=5**, making 5 the digital root of 482.
But the important questions are why does it work, and why do we cross out any 9s and any pairs that make 9s? Well turns out, 9 is a special number, because in our counting system, its the highest single digit there is. If we go to any whole numbers higher than 9, we tick it over back to 0 and add a 1 to the next digit over, like the tens place. This has a funny little effect…
If we add **9** (a number 1 away from rolling over to the next tens place) to **9** (a number 1 away from rolling over to the next tens place, again), we end up with **18** (a number 2 away from rolling over to the next tens place). Add another 9 and you get **27** (a number 3 away from rolling over to the next tens place). Aaaaall the way up to **90**, where we can just fit a 9 onto it without rolling over, but don’t fret because now we’re just back to 9 + 9, back to where we started.
If you’re keeping “digital root” in mind, you might notice something: because of what I just explained, every time we add 9, the ones place goes down by 1, and the tens place goes up by 1. And hey, the digit root of that **18** is **1+8=9**… And that **27** is **2+7=9**! Well turns out what you’re doing to its digit root every time you add another 9 is adding 1 to it and then subtracting 1, which you can see means the digital root doesn’t budge, no matter how many 9’s you add.
Okay. So what? Well, that “tens go up 1, ones go down 1” works for any of the positive whole numbers, not just multiples of 9. **24**? Digital root is **2+4=6**… Add a 9! Now you get **33** (digital root **3+3= …6** again!). What that digital root of a number is telling us is how many numbers away from being a multiple of 9 it is! Look at 24 and subtract its digital root: **24-6=18**, a multiple of 9. How about that 33? **33 – 6 = 27**, another multiple of 9. This means you can describe whole numbers as its digital root + some amount of 9’s! **482** back at the start? Well 482 minus its digital root of 5 is 477, which is **53** 9’s!
But wait: digital roots are adding up all of their digits, and adding 9 to a digital root doesn’t change it. In that case, why even bother with the 9’s *already in* a number? Take **439**. **4+3+9=7+9**. We could add that 9 in as well, but then we’ll just end up with **16** –> **1 + 6** which is back to **7**, so just don’t bother with that 9, cross it out. And since anything that adds up to 9 will also be just adding another pointless 9, cross them out to. Digital root of **182**? **1+8+2=11** –> **1+1=2**… Or just drop the “9” in it (the 1 and 8) and you’re already at **2**, which is must faster.
Where does that leave us? Well if positive whole numbers are their digital roots plus bunch of 9’s, we can do some math tricks with arithmetic. Feel free to try them out for yourself, but basic arithmetic can be turned into “arithmetic between two single-digit numbers and a bunch of 9’s”. So if you multiply two numbers together, you’re multiplying two single-digit numbers and also there’s a bunch of 9’s there, which means the answer will be those two single-digit numbers multiplied together and a whole bunch of 9’s.
So yeah, you can check basic arithmetic with this method!
**2356/19= …125**? Let’s check if the digital roots… **2+3+5+6=16** –> **1+6=7**, and cross out the 9 from **19** to get its digital root of **1**. Great, so **7/1=7**, so our answer should have a digital root of 7! Now let’s check the digital root of the answer I had… **1+2+5= …8**. Uh oh. This means I’ve done something wrong! And look at that, punching it into a calculator shows that the answer is **124**, whose digital root is **1+2+4=7**, our missing answer!
What’s important to remember is that this is just a check to see if you got the answer *wrong*; it doesn’t check if you got it *right*. This is because it only checks if you have the correct *digital root*, but you can still have the wrong number of 9’s, and getting fractions involved makes things much trickier. I could go on about ways to just do arithmetic like this, but I think I’ve gone on enough, huh.
The process of “crossing 9s” is a simple hack to calculate remainder after division by 9. If two numbers are equal, their remainder after division by 9 is also equal (ANY mathematical operation, performed on two equal numbers, must give equal results). 9 is chosen just because it’s easy to calculate using that hack.
As to why the hack works – it’s because 9 is 1 less than 10, or 9 + 1, if you will. 20 is 2 * (9 + 1), or 18 + 2. 30 is 27 + 3. The first number is always a multiple of 9, the second number is the remainder. We only care about the remainder, which is, as you might notice, simply the digit itself. And since 100 = 99 + 1 and 1000 = 999 + 1 etc., adding up all the digits gives us the remainder for the whole number, no matter how many digits it has.
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