9s are interesting. As an accountant, early on I learned about the rule of 9s. When reconciling accounts, if something was off by 9, or a multiple of 9, it was likely because you transposed two digits somewhere along the way. For example if you added a 71 instead of a 17, that's a difference of 54, which is a multiple of 9.
As I went to school "in the old days before calculators", I was taught this method, and I think it teaches more about the nature of numbers in mathematics than just punching them up on a calculator. Cheers, Russ
A better way to explain the peculiar behaviour of 9 would be: If we add 9 to any number, we are increasing it's 10s place by 1 because 9 is the largest number in Base10. Also, we are reducing the units place by 1. This -1 cancels out the +1 thus rendering the addition of 9 in the digital route moot.
In Italy we call this "the test of nine", they taught me in primary school and I still use it after 10+ years when I can't use a calculator. So useful!
I've been randomly doing this with numbers in my head for no reason. I had no idea that it actually had an application! You made me really excited to learn about this; thank you!
Microsoft had a good reason for that, some programs might've seen the 9 at the begining of the version number and then run in 95/98 compatibility mode. I don't know why Apple's doing it, though.
My Father taught me this in the early '60's to check my grade school math homework and I quickly figured out that none of my teachers knew the method so I kept it a secret and enjoyed their amazement at my error free math answers on tests.
So this should work in any base, right? Except that you'll have to replace 9 with the base - 1. Or at least, whatever the representation of that number is in that base.
Josh LZK It should indeed work in any base, since adding b-1 to any number in base b can only have one of two possible outcomes: if the original number ends in a zero, the new number ends in b-1, or if not the penultimate digit increases by 1 (carrying over if that digit is b-1) and the unit digit decreases by 1.
I thought the point of this was to check if you did addition correctly. I mean if you add two numbers incorrectly and the root is always 1 (apart from 0 which you mentioned), it doesn't really help you much. One thing that does confuse me is that if you do this the long way (i.e NOT casting out the base - 1 number), you always get a digital root of 1 (again apart from a starting number of 0). BUT if you cast out the 1's (as binary is base 2) you always get 0. I feel like this is wrong, if anyone has a correction it would be appreciated. Cheers
Josh LZK Correct. This method would work really well for the ancient Sumerians and Babylonians, who used a base-60 number system and thus would have been casting out 59s, and would only have a 1 in 60 chance of a false negative.
His explanation of why it works isn't great because you could substitute any number for 9 in the formulas of the form 9n + a and get a similar result. For example (7n + a) + (7m + b) = 7(n+m) + (a + b). But if you tried to "cast out 7s" it won't work because, for example, 16 isn't a multiple of 7 even though its digits add up to 7. The reason casting out 9s actually works as opposed to other numbers is that all multiples of 9 have digits that add up to a multiple of 9 (eg 72 is a multiple of 9 and its digits add up to a multiple of 9). So when you take out the digits of a number which add up to 9 you are in effect subtracting a multiple of 9 from the number which doesn't change the digital root relative to multiples of 9.
He did specify that the "a" in the formula is the digital root of the number, which explains why using 7 wouldn't work, because you can't write every number as 7n+a where a is the digital root. You can only do that with 9.
Kashyap Nadig Variables are arbitrary numbers, thus numbers can be split up and manipulated "like variables" because those manipulations *are* happening to numbers, we just don't know/care what they are.
Actually you can do that with any number as long as you take the correct digital root, Kn+a, where K = Base - 1 and a = digital root in that base. so 7n + a would work in octagonal base.
Origami Swami I think I didn't explain my issue with his proof very well. If you watch the video he never actually proves that adding 9 doesn't change the digital root. (It's true but he never proves it, he just says it's true.) He then says, under that assumption, that every number can be written as 9n+a for some n where a is the number's digital root (again true but not proven in the video). Next he says if you add two numbers of the form (9n+a) + (9m + b) = 9(m+n) + (a+b) where (a+b) has the same digital root as the sum which is true but the only reason (a+b) is because adding multiples of 9 don't change the digital root (again, not proven in the video). The same equation would be true if you replaced 9 with 7, for example, but in that case a+b might not still be the digital root since adding a multiple of 7 can change it. He then explains that deleting digits which add up to 9 is the same as subtracting a multiple of 9 (which is true since all multiples of 9 have digits which add to a multiple of 9, although that again wasn't proven in the video) so "casting out 9s" leaves the digital root unchanged. So his statements aren't wrong but his proof is very incomplete. The key elements he left out are 1) proving that adding a multiple of 9 to a number does not change the digital root; 2) that multiples of 9 and only multiples of 9 have a digital root of 9; and 3) that "casting out 9s" is equivalent to subtracting a multiple of 9 from the number. None of these are horribly hard to prove but they're important enough not to hand wave them away since the method only works because of those facts.
This video is going to be very helpful if somebody ever finds themselves and 8 other people on a sinking ship with a numbered bracelet on their hand, and a mysterious voice telling them that they have nine hours to escape through the ninth door.
Thank you. My son in 5th grade is supposed to check his math homework this way, and I never learned. The written instructions gave me heart palpitations and shortness of breath. This, I understand! :-)
Some may not realize, in a long list of numbers to add, you can also cast out nines between the rows. For example, adding 647 + 3521 = will quickly give you a 1 as the root. 6+3=9, 4+5=9, 7+2=9. I learned this back in the early 80's when the cashiers at Radio Shack were writing prices on paper pads and adding them by calculator to tell me how much to pay for my latest project. I could easily verify their numbers while viewing their receipt pad upside down across the counter.
I don't know if it's been mentioned before already, but this is taught in every primary school in Italy. The expression "fare la prova del 9" (casting out nines) is also a colloquialism, meaning "to double check / to verify / to cross check".
I was waiting for him to mention it was a digital root, and just as he does I see this comment. Highly recommend 999 to everyone, one of my favorite DS games
Also, don't forget casting out elevens! Add the 1s, 100s, 10000s, digits, and subtract the 10's, 1000's, 100000s ... doing all arithmetic mod 11. Casting out elevens detects transposed digits, and combined with casting out nines means you'll fail to detect only 1 in 99 sums with mistakes, instead of 1 in 9. ETA: Oh, yeah, and it's hypothetically possible to cast out 7's and 13's by working with triples of digits, because 1001=7*11*13. But I've never learnt those algorithms well enough to do them on command.
At about 4:05 you can tell that he is truly spectacular at numbers by not following any particular instructions but rather knowing what happens. He has an intuition about them like people do with driving or playing games. He can jump straight to the answer but can also do how thought process. He downplays it, but it is a sign of mastery far beyond normal
I was thinking about this for years, since I was a kid. Only recently I figured out (and proved) that the digital root is mod 9 if we work with base 10.
I refer to these things as "cycles". I always try to use cycles in my arithmetic whenever I can either to do it or check it because position in cycles is much easier to manage than multiple digits. Well, we all already do this; digits are just cycles of length 10. This is a cycle of length 9. It's always kind of fun to find a cycle in the numbers.
I knew this in middle school. I didn't pay much attention in class, mainly because the teacher kept explaining stuff that I already knew. I would spend my time trying to find patterns in math and what kind of connections numbers had with each other, while all the other kids were busy trying to figure out that X was being used as a hypothetical number. I know that I knew about this, but I don't know if I discovered it by myself or if my teacher mentioned it once and I played around with it. I don't think I ever learned how to use the trick with multiplication, and certainly not division. This video brings me back to middle school and I love it.
In the 60's and 70's, our school's principal would often punish us by assigning us a math problem such as 37 to the 12th power. When you were done, he'd check it with this method. Today, I finally found it here. Thanks!
Thanks James. My maths is basic, to say the least, but you always manage to transmit your enthusiasm and make it interesting. Along with the Mathologer, you're my favourite maths teacher.
In binary you can cast out ones. All answers are correct! XDin base X, you can modulo by (x-1) easily taking the sum of the digits, and then check your arithmetic using that. Larger bases are more accurate but are more prone to you making mistakes in the first place.
Would love to see more videos like this showing practical, everyday math tips. I have an MS in engineering with plenty of courses on ODEs and PDEs, etc, but never learned this gem till now! Thanks!
Funny enough, I discovered this trick as a teenager while listening to a Coast to Coast AM episode about numerology (paranormal stuff on the show seemed fun). The guest said if you wanted to get your "numerological number" or something like that you add all the letters in you name as numbers and keep adding those digits until you get a single digit. The host asked if it mattered if you did it separately for first name and last name or if you had to do all the letters at the same time and the numerologist said it didn't matter. I started playing around with that and noticed that no mater how you do a sum, this reduction always gives you the same digit. Say, if you wanted to do 15+245+33 you can reduce them by doing all the digits at the same time: 1+5+2+4+5+3+3 -> 23 -> 5 or by reducing each number separately: 15->6, 245->11->2, 33->6 and then 6+2+6 = 14 -> 5 Either way, that reduction is the same as the reduction of the initial sum, 293 -> 14 -> 5. I guess I learned something useful from a pseudoscience :D
Mr IY if you are a true Indian then you should have known that this method is also there in Vedic mathematics. In fact it was found in Vedic maths first
Casting out nines was taught in schools all over Europe several decades ago, until pocket calculators became commonplace. It is well known (indeed, it was the subject one of the first Numberphile videos) that similar techniques based on modular arithmetic are routinely used in computer science for detecting and fixing failures in hardware and communications. It is less known, though, that by dropping some of the nice features of modular arithmetic a generalized framework for approximate computation called "abstract interpretation" is obtained. This is a core technique for software analysis tools.
My brother and I did this as a game where he'd ask what's 4 + 11, I'd say 15, he'd say no it's 6, until I caught on. Then I did this in school when I was bored and quickly realized that 9's became redundant. I never realized this was something people used to do to check their arithmetic. Really cool seeing this neat trick being shared online regardless.
I gasped and paused the video at 2:49. "What?! Adding the digits is the same as mod 9??!" I was so fascinated and surprised that I sat here and tried to outline the steps to a proof in my mind. Then for a general base n case. It took me quite a while but I see it now. Thank you for this lesson and the inspiration! I had not done something like this in a long time.
Instead of summing all the digits, you can take their difference to "cast out 11s." Take the first digit, subtract the next digit from it, then add the next digit, then subtract the next digit from them and so on (alternating between adding and subtracting.) As with casting out nines, any number that reduces down to zero is divisible by 11. 5049 => 5-0+4-9 = 0 More generally, just like in any base-n system you can add the digits to cast out (n-1), you can take their difference to cast out (n+1). In base 6, 49 is represented as "121." 1-2+1 = 0, so 49 must be divisible by 7.
To be fair it's not intended to be a perfect check, it's just a very fast way to spot check if you made possibly an error. It's handy when you want to do an instant "sanity check" that you did your calculations right without actually having to go through the whole process a second time.
I always used this to find out if a number is a multiple of 3 by checking to see if the sum of the digits was divisible by 3. This helps a lot for simplifying fractions and finding prime factorizations along with checking if a number is divisible by 5 or 2.
Yes! Finally touching on this. For those wondering, by the way, yes, it's a property of whatever base you're in. So if you're in a number system with base X, this rule will be true for the number X - 1. I find it interesting just as a quick way of getting the digit sum of various numbers quickly, which can be used for other fun things! Would be interesting to know how many functions you can use this trick for (subtraction, powers, roots, etc.).
It should hold true for subtraction as long as you interpret a negative number as 9 minus that number (for instance, -7 is 9(-1)+2, so its digit is 9-7=2). For powers you'll have to use algebra to figure out a formula but it might be doable. Powers step a bit off the realm of classic arithmetic in the integer ring though.
You can reduce the base of powers modulo n and have the result still be congruent modulo n. Additionally you could reduce the powers to modulo phi(n) where phi is Euler's totient function or Carmichael's reduced totient function. Both of which give 6 for n = 9. There's actually a lot of theory behind calculating exponents modulo n. It's really important for cryptography. See also "Exponentiation by squaring" and "Chinese Remainder Theorem".
Another similar fact that I like and actually use every once in a while is that you can add all the digits in a number together, and if the end result is divisible by three, then the starting number is divisible by three. Ex. 215 is 8 expressed this way and is not divisible by three, but 216 is 9 when expressed as such and equals 72 when divided by three (which itself also happens to be divisible by three, but that's neither here nor there as this was just an example I pulled from the air while sleep deprived.)
Gread you covered this topic. These are the tricks they should teach in school. Not the thing about calculating the digit sum, but the ofstriking of nines. (Which I btw had to discover on my own in 6th grade)
In short, this holds because of the Homomorphism Z -> Z/9 and the fact that any power of 10 is congruent 1 mod 9. Since 10 is congruent -1 mod 11, one can do the same trick modulo 11 with alternating sums of to digits. Doing both checks tells you that you're correct modulo 99, which reduces the chance of being wrong to 1:99.
I learned this in school! You can also use the digital root to check whether a number is multiple of 9 - its digital root will be 9 as well! And since 9 is a multiple of 3, you can also check whether a number is multiple of 3 by checking if its digital root is a multiple of 3!
I was never taught this, but happened upon this property some years ago and always wondered what it was called and what use it might have. Thanks for finally answering my uncertainty.
I predate affordable calculators and I still do long multiplication, addition etc on paper if I don't have Excel in front of me. I can't be bothered with calculators, you get too many typos. No-one ever taught me this when I was at school, and I went to primary school in the UK and elementary school in the USA. This is great. Thanks!
It's also very easy to check that everything is correct modulo 2. The sum between several numbers will be even if and only if an even number of inputs are odd. Same goes for differences because subtraction is just adding the negative. And the product between multiple numbers will be even if and only if at least one input is even. With this easy extra precaution, you can be sure if you messed up your arithmetic, you have to be off by a multiple of 18.
I totally used to do this in my head when I was younger, whenever I saw big numbers. But I never thought it was actually useful. i just noticed it as a cool thing you could do.
Now I understand why multiplying by 9 always gives a result with a digital root of 9 (except 0), which is how I remember the multiples of 9. I've used that trick for as long as I've needed to multiply. It's cool that I can finally prove why it works.
It's quite obvious but I still wanted to point out that it's nice about this is that every multitude of 9 consists of two digits that make a total of 9, so 1+8=9, 2+7=9, 3+6=9. This remains consistent until you get past 10*9. Since 11*9=99->9+9=18 but then again 18->1+8=9. What's beautiful as well as is that after that point every total becomes 9 (simply put, 1+a+b where a+b=8) again until 21 of which the total is 18 again, just as 22*9 which is 18 as well, then the total becomes 9 again where every number becomes 2+a+b where a+b=7. There's this really nice pattern of numbers here. From 71 to 77 for example as well the total becomes 18 but with 78 to 80 the total becomes 9 again. This pattern actually continues when it comes to even higher numbers, albeit a bit differently. from 101 to 111 the total is always more than 9 after that the total becomes 9 again until 121 to 122 and 131 to 133 and so on. Same with 201-222 and even 1000 to 1111. I don't think I can make an exact formula for this but the pattern is certainly there. Basically the totals are always 9 except when the number lies between (10*k+1)*9 and (11*k)*9, but it's the same with (100*k+1)*9 to (111*k)*9 and (1000*k+1)*9 to (1111*k)*9 and (1000o*k+1)*9 to (11111*k)*9 and so forth. If you all these specific multitudes, and look at how many of these apply, you can also calculate the total of a multitude of 9 of which the answer consists of. if you were to look at 43421 for example. It is between 40001 and 44444 so that's 1 "check", it's not between 43001 and 43333, so that's a "fail", it is between 43401 and 4344 so that's another "check" and lasty it's also between 43421 and 43422. That's 3 checks so the total number of digits of the final answer will have a value of (1+3 checks)*9= 36. Put it in a calculator and you get 43421*9=390789->3+9+0+7+8+9=36. I'm pretty sure this always works but I can't really come up with a formula for it.
Fun Coincidence: two days ago, while tutoring abstract algebra, we proved 9 divides a number iff 9 divides the sum of its digits; exactly why canceling 9s works! Now, two days later, I find this video. I love math!
The way you did the manual multiplication is different to what kids are being thought here. It's cool to see that there's another way to do manual multiplication like you did. It seems harder for me but i guess i can adapt it.
In Italy this is (or was) taught at every elementary school. It's called 'the test of the nine' ("la prova del nove"), which is also an idiom to mean the final confirmation of something. Does the expression exist also elsewhere?
9s are interesting. As an accountant, early on I learned about the rule of 9s. When reconciling accounts, if something was off by 9, or a multiple of 9, it was likely because you transposed two digits somewhere along the way. For example if you added a 71 instead of a 17, that's a difference of 54, which is a multiple of 9.
He should have said that this works because he is using base 10. For any base n you can cast out (n-1) and its multiples.
I find it odd that this even works in binary.
Your sarcasm level is high indeed!
eglerian yeah
eglerian so this would work in base 6 (hex-decimal?) too right?
tommygunsegs the method would then become casting out 5
As I went to school "in the old days before calculators", I was taught this method, and I think it teaches more about the nature of numbers in mathematics than just punching them up on a calculator. Cheers, Russ
Mathematics is way more than waste time with sums
I doubt you know more about number theory or algebra just because of that haha
??.
A better way to explain the peculiar behaviour of 9 would be:
If we add 9 to any number, we are increasing it's 10s place by 1 because 9 is the largest number in Base10. Also, we are reducing the units place by 1. This -1 cancels out the +1 thus rendering the addition of 9 in the digital route moot.
In Italy we call this "the test of nine", they taught me in primary school and I still use it after 10+ years when I can't use a calculator.
So useful!
I've been randomly doing this with numbers in my head for no reason. I had no idea that it actually had an application! You made me really excited to learn about this; thank you!
Casting out nines? Clearly Apple and Microsoft are fans of this method.
XD
However, the iPhone X is for the tenth anniversary of the iPhone, so the next iPhone should be the iPhone nine.
Maybe they release the iPhone X2 next instead of the iPhone 9
powww apple owned!
Microsoft had a good reason for that, some programs might've seen the 9 at the begining of the version number and then run in 95/98 compatibility mode.
I don't know why Apple's doing it, though.
i know right
My Father taught me this in the early '60's to check my grade school math homework and I quickly figured out that none of my teachers knew the method so I kept it a secret and enjoyed their amazement at my error free math answers on tests.
I studied that in elementary school. Thank you Italian Education System.
Here in Portugal, all the kids from my generation learned this in the 3rd grade. I'm really thankful for that!
A form of casting out nines was described by Hippolytus (170-235) in The Refutation of all Heresies. Wikipedia.
So this should work in any base, right? Except that you'll have to replace 9 with the base - 1. Or at least, whatever the representation of that number is in that base.
Josh LZK It should indeed work in any base, since adding b-1 to any number in base b can only have one of two possible outcomes: if the original number ends in a zero, the new number ends in b-1, or if not the penultimate digit increases by 1 (carrying over if that digit is b-1) and the unit digit decreases by 1.
I guess it kind of falls apart with binary. Also I think the accuracy of the check increases as the base number increases.
Why would it fall apart in binary? The digital root is just always 1, with the exception of 0.
I thought the point of this was to check if you did addition correctly. I mean if you add two numbers incorrectly and the root is always 1 (apart from 0 which you mentioned), it doesn't really help you much.
One thing that does confuse me is that if you do this the long way (i.e NOT casting out the base - 1 number), you always get a digital root of 1 (again apart from a starting number of 0). BUT if you cast out the 1's (as binary is base 2) you always get 0.
I feel like this is wrong, if anyone has a correction it would be appreciated. Cheers
Josh LZK Correct. This method would work really well for the ancient Sumerians and Babylonians, who used a base-60 number system and thus would have been casting out 59s, and would only have a 1 in 60 chance of a false negative.
His explanation of why it works isn't great because you could substitute any number for 9 in the formulas of the form 9n + a and get a similar result. For example (7n + a) + (7m + b) = 7(n+m) + (a + b). But if you tried to "cast out 7s" it won't work because, for example, 16 isn't a multiple of 7 even though its digits add up to 7.
The reason casting out 9s actually works as opposed to other numbers is that all multiples of 9 have digits that add up to a multiple of 9 (eg 72 is a multiple of 9 and its digits add up to a multiple of 9). So when you take out the digits of a number which add up to 9 you are in effect subtracting a multiple of 9 from the number which doesn't change the digital root relative to multiples of 9.
He did specify that the "a" in the formula is the digital root of the number, which explains why using 7 wouldn't work, because you can't write every number as 7n+a where a is the digital root. You can only do that with 9.
In a proof/explanation, why would you "substitute" a number in the first place? Numbers are not variables!
Kashyap Nadig Variables are arbitrary numbers, thus numbers can be split up and manipulated "like variables" because those manipulations *are* happening to numbers, we just don't know/care what they are.
Actually you can do that with any number as long as you take the correct digital root, Kn+a, where K = Base - 1 and a = digital root in that base. so 7n + a would work in octagonal base.
Origami Swami I think I didn't explain my issue with his proof very well. If you watch the video he never actually proves that adding 9 doesn't change the digital root. (It's true but he never proves it, he just says it's true.) He then says, under that assumption, that every number can be written as 9n+a for some n where a is the number's digital root (again true but not proven in the video). Next he says if you add two numbers of the form (9n+a) + (9m + b) = 9(m+n) + (a+b) where (a+b) has the same digital root as the sum which is true but the only reason (a+b) is because adding multiples of 9 don't change the digital root (again, not proven in the video). The same equation would be true if you replaced 9 with 7, for example, but in that case a+b might not still be the digital root since adding a multiple of 7 can change it. He then explains that deleting digits which add up to 9 is the same as subtracting a multiple of 9 (which is true since all multiples of 9 have digits which add to a multiple of 9, although that again wasn't proven in the video) so "casting out 9s" leaves the digital root unchanged.
So his statements aren't wrong but his proof is very incomplete. The key elements he left out are 1) proving that adding a multiple of 9 to a number does not change the digital root; 2) that multiples of 9 and only multiples of 9 have a digital root of 9; and 3) that "casting out 9s" is equivalent to subtracting a multiple of 9 from the number. None of these are horribly hard to prove but they're important enough not to hand wave them away since the method only works because of those facts.
This video is going to be very helpful if somebody ever finds themselves and 8 other people on a sinking ship with a numbered bracelet on their hand, and a mysterious voice telling them that they have nine hours to escape through the ninth door.
I haven't watched the video. So the question is is there a difference between the ninth and the 0th door?
wait thats why it is zero escape. It all makes sense now.
But is it the ninth door or is it the qth door?
en.wikipedia.org/wiki/Nine_Hours,_Nine_Persons,_Nine_Doors
I was looking for a comment with a reference like this xD
Why didn't I know this?
Fastex, I guess they stopped including it in mathematics textbooks at some point.
Cuz you a have calculator and you dont check if the calculator is right!
ma pure qui sei? ahaha
"pure" oltre a...?
marcus kron
Thank you. My son in 5th grade is supposed to check his math homework this way, and I never learned. The written instructions gave me heart palpitations and shortness of breath. This, I understand! :-)
i like the videos with/by james so much
Some may not realize, in a long list of numbers to add, you can also cast out nines between the rows.
For example, adding
647 +
3521 = will quickly give you a 1 as the root. 6+3=9, 4+5=9, 7+2=9.
I learned this back in the early 80's when the cashiers at Radio Shack were writing prices on paper pads and adding them by calculator to tell me how much to pay for my latest project. I could easily verify their numbers while viewing their receipt pad upside down across the counter.
I spy a nail and gear. Long live the nail and gear!
;)
I really want to have a Flaggy Flag planted on the Moon...
rebmcr definitely
oh...
I didn't get it :/
If you're getting this, i saw another thing in the unfocused background that had "hellointernet.fm" written on it, among other intriguing things
I don't know if it's been mentioned before already, but this is taught in every primary school in Italy. The expression "fare la prova del 9" (casting out nines) is also a colloquialism, meaning "to double check / to verify / to cross check".
The Zero Escape fan in me is smiling.
Dr. Grime's videos are my favorite. I barely need to do math for my job but I still love learning these tricks.
Checksum for BrownPaper OS.
His passion for mathematics is so infectious and captivating.
Gaming recommendation: Zero Escape: 9 Hours 9 persons 9 doors.
Uses this as a major part in the plot
Oh boy, I envy you.
Came here to see if anyone was going to mention 999, and I'm glad I found this. Hearing "Digital Root" is giving me PTSD, already...
I was waiting for him to mention it was a digital root, and just as he does I see this comment. Highly recommend 999 to everyone, one of my favorite DS games
I also came here to see if other viewers had played 999. I cannot recommend that game enough.
Also, don't forget casting out elevens! Add the 1s, 100s, 10000s, digits, and subtract the 10's, 1000's, 100000s ... doing all arithmetic mod 11. Casting out elevens detects transposed digits, and combined with casting out nines means you'll fail to detect only 1 in 99 sums with mistakes, instead of 1 in 9.
ETA: Oh, yeah, and it's hypothetically possible to cast out 7's and 13's by working with triples of digits, because 1001=7*11*13. But I've never learnt those algorithms well enough to do them on command.
2:25 "digital root"
Finally, my 9 Hours, 9 Persons, 9 Doors knowledge coming into play
At about 4:05 you can tell that he is truly spectacular at numbers by not following any particular instructions but rather knowing what happens. He has an intuition about them like people do with driving or playing games. He can jump straight to the answer but can also do how thought process. He downplays it, but it is a sign of mastery far beyond normal
Andrew Morris it was just multiplication relax
Cheesy Bread I know it was just multiplication, but there was something with how he did it that sent my genius detector off the charts
I was thinking about this for years, since I was a kid. Only recently I figured out (and proved) that the digital root is mod 9 if we work with base 10.
I refer to these things as "cycles". I always try to use cycles in my arithmetic whenever I can either to do it or check it because position in cycles is much easier to manage than multiple digits. Well, we all already do this; digits are just cycles of length 10. This is a cycle of length 9. It's always kind of fun to find a cycle in the numbers.
Hey, it's an other James Grime video!
Hi James Grime!!
I knew this in middle school. I didn't pay much attention in class, mainly because the teacher kept explaining stuff that I already knew. I would spend my time trying to find patterns in math and what kind of connections numbers had with each other, while all the other kids were busy trying to figure out that X was being used as a hypothetical number. I know that I knew about this, but I don't know if I discovered it by myself or if my teacher mentioned it once and I played around with it. I don't think I ever learned how to use the trick with multiplication, and certainly not division. This video brings me back to middle school and I love it.
Calculation *[GONE WRONG]*
*[GONE HEXUAL]*
Yes James!
James is probably he most interesting person on numberphile
Now I need Numberphile to play 999
Marion Beliveau Exactly what I was thinking.
My 4th grade teacher taught us this back in the 1970s. Had completely forgotten about it until I saw the video title. AWESOMESAUCE!
Is the title a reference to apple skipping the iPhone 9? Subtle brady
they copied mcirosoft
I wonder why Apple did that? At least Microsoft had a logical reason.
The iPhone X is the 10th anniversary iPhone, so they ignored the names of their other iPhones, and went with X.
Adnan Abbas
Skipping Windows 9
In the 60's and 70's, our school's principal would often punish us by assigning us a math problem such as 37 to the 12th power. When you were done, he'd check it with this method. Today, I finally found it here. Thanks!
"This man knows". Brady 2K17
Thanks James. My maths is basic, to say the least, but you always manage to transmit your enthusiasm and make it interesting. Along with the Mathologer, you're my favourite maths teacher.
All superbly interesting, but I think what we all need to know is when is next calculator unboxing video coming?
In binary you can cast out ones. All answers are correct! XDin base X, you can modulo by (x-1) easily taking the sum of the digits, and then check your arithmetic using that. Larger bases are more accurate but are more prone to you making mistakes in the first place.
Anyone else knew about this already from
zero escape 999?
No, I learnt it at primary school, like everyone should have done.
K is best Seven
Potassium What do you mean by "K is best Seven"?
K, the guy in the robot suit in VLR is better than Seven, the one with the 7 on his bracelet.
They're both large, amnesiac characters in the Zero Escape series.
Would love to see more videos like this showing practical, everyday math tips. I have an MS in engineering with plenty of courses on ODEs and PDEs, etc, but never learned this gem till now! Thanks!
Why'd u upload this early on a school day D:
Now i have to watch this in class
At least when you're called out you can give the excuse that it's actually a video about maths so it's totally school related.
Ask your teacher to watch it in class 😂
Hope it helps.
+Dannyx51 Math is great, but geography is important too. When would it not have been early on a school day?
Math class? That'd be nice. You can doodle with vihart while you're at it
Funny enough, I discovered this trick as a teenager while listening to a Coast to Coast AM episode about numerology (paranormal stuff on the show seemed fun).
The guest said if you wanted to get your "numerological number" or something like that you add all the letters in you name as numbers and keep adding those digits until you get a single digit. The host asked if it mattered if you did it separately for first name and last name or if you had to do all the letters at the same time and the numerologist said it didn't matter.
I started playing around with that and noticed that no mater how you do a sum, this reduction always gives you the same digit.
Say, if you wanted to do 15+245+33 you can reduce them by doing all the digits at the same time:
1+5+2+4+5+3+3 -> 23 -> 5
or by reducing each number separately:
15->6, 245->11->2, 33->6 and then 6+2+6 = 14 -> 5
Either way, that reduction is the same as the reduction of the initial sum, 293 -> 14 -> 5.
I guess I learned something useful from a pseudoscience :D
Crazy trick!
Cheers from India👏
Jeffrey you know the answer..
poo in loo prajeet
SUPERPOWER BY 2020
Mr IY if you are a true Indian then you should have known that this method is also there in Vedic mathematics. In fact it was found in Vedic maths first
cringe..
I love Grimeys videos, they are the best
Oh! It's the singing banana ♥♥♥♥
Casting out nines was taught in schools all over Europe several decades ago, until pocket calculators became commonplace. It is well known (indeed, it was the subject one of the first Numberphile videos) that similar techniques based on modular arithmetic are routinely used in computer science for detecting and fixing failures in hardware and communications. It is less known, though, that by dropping some of the nice features of modular arithmetic a generalized framework for approximate computation called "abstract interpretation" is obtained. This is a core technique for software analysis tools.
I tried casting out Ones to check my binary arithmetic. The results were... Not helpful.
Turns out I have spent my life finding digital roots of numbers I see everywhere, now I know what doing that is called, thanks Dr Grime!
I was born on 16th April 1991
16 = 4x4
April = 4
1+6+4+1+9+9+1 = 4
Tattoed this on my arm 4 years ago :)
Modified :P
But 1+6+4+1+9+9+1=31=3+1=4
At first, I thought it seemed needlessly complicated and time-consuming, but when you got to the actual "Casting out nines" part... Wow!
cgp grey flag in the background
It's the Hello Internet flag aka the mighty nail and gear
not a CGP Grey flag
My brother and I did this as a game where he'd ask what's 4 + 11, I'd say 15, he'd say no it's 6, until I caught on. Then I did this in school when I was bored and quickly realized that 9's became redundant. I never realized this was something people used to do to check their arithmetic. Really cool seeing this neat trick being shared online regardless.
*Zero Escape: 999* anyone?
I gasped and paused the video at 2:49. "What?! Adding the digits is the same as mod 9??!" I was so fascinated and surprised that I sat here and tried to outline the steps to a proof in my mind. Then for a general base n case. It took me quite a while but I see it now. Thank you for this lesson and the inspiration! I had not done something like this in a long time.
Bah, amateur, Singingbanana could explain it better, in less time and, AND be more entertaining.
i was having a bad morning due to no sleep and this popped up on my feed :)
IIIIIIIII
I
IIIIIIIII
Heard about this many times, but didn't know what it was about. Thanks James for another smart, fun and enlightening video.
Cool video .but can you cast out the nine of 9/11
just call it 11th of September and you're set
I've seen adding digits in numerology. Cool that there's a practical use of the same process.
Chuck Norris counted to infinity - twice!!
Chuck Norris jokes stopped being funny last decade.
Chuck Norris visited the Virgin Islands once. They are now just called the Islands.
oh look, 8 year olds do still exist!
never thought i would hear such words in 2017
you can't count to infinity. This joke isn't funny; it's just irritating.
Instead of summing all the digits, you can take their difference to "cast out 11s." Take the first digit, subtract the next digit from it, then add the next digit, then subtract the next digit from them and so on (alternating between adding and subtracting.) As with casting out nines, any number that reduces down to zero is divisible by 11.
5049 => 5-0+4-9 = 0
More generally, just like in any base-n system you can add the digits to cast out (n-1), you can take their difference to cast out (n+1). In base 6, 49 is represented as "121." 1-2+1 = 0, so 49 must be divisible by 7.
I think this is a dumb check, if you got two numbers the wrong way round it adds to the same number
To be fair it's not intended to be a perfect check, it's just a very fast way to spot check if you made possibly an error. It's handy when you want to do an instant "sanity check" that you did your calculations right without actually having to go through the whole process a second time.
It works 88.9% of the time, and once you find a mistake you could try to be more cautious going forward.
Once again James Grimes manages to blow my mind. Or should I say my nine.
I always used this to find out if a number is a multiple of 3 by checking to see if the sum of the digits was divisible by 3. This helps a lot for simplifying fractions and finding prime factorizations along with checking if a number is divisible by 5 or 2.
Yes! Finally touching on this. For those wondering, by the way, yes, it's a property of whatever base you're in. So if you're in a number system with base X, this rule will be true for the number X - 1. I find it interesting just as a quick way of getting the digit sum of various numbers quickly, which can be used for other fun things! Would be interesting to know how many functions you can use this trick for (subtraction, powers, roots, etc.).
It should hold true for subtraction as long as you interpret a negative number as 9 minus that number (for instance, -7 is 9(-1)+2, so its digit is 9-7=2). For powers you'll have to use algebra to figure out a formula but it might be doable. Powers step a bit off the realm of classic arithmetic in the integer ring though.
You can reduce the base of powers modulo n and have the result still be congruent modulo n. Additionally you could reduce the powers to modulo phi(n) where phi is Euler's totient function or Carmichael's reduced totient function. Both of which give 6 for n = 9. There's actually a lot of theory behind calculating exponents modulo n. It's really important for cryptography. See also "Exponentiation by squaring" and "Chinese Remainder Theorem".
Love the Nail & Gear framed picture in the background!
He is so smart
PLEASE SHARE THIS! Educate the world.
Thank you! My son's math called for us to do this and I had no clue. Looking at it didn't help me - your video sure did!
Another similar fact that I like and actually use every once in a while is that you can add all the digits in a number together, and if the end result is divisible by three, then the starting number is divisible by three. Ex. 215 is 8 expressed this way and is not divisible by three, but 216 is 9 when expressed as such and equals 72 when divided by three (which itself also happens to be divisible by three, but that's neither here nor there as this was just an example I pulled from the air while sleep deprived.)
ayy cant start a day properly without Dr Grime :D
so glad im this early
You just changed forever my high-school student life
I learned this 65 or so years ago. Thanks for the refresher!
Gread you covered this topic. These are the tricks they should teach in school.
Not the thing about calculating the digit sum, but the ofstriking of nines. (Which I btw had to discover on my own in 6th grade)
This was a cool video. I always liked how the number 9 has so many unique traits.
In short, this holds because of the Homomorphism Z -> Z/9 and the fact that any power of 10 is congruent 1 mod 9. Since 10 is congruent -1 mod 11, one can do the same trick modulo 11 with alternating sums of to digits. Doing both checks tells you that you're correct modulo 99, which reduces the chance of being wrong to 1:99.
I learned this in school! You can also use the digital root to check whether a number is multiple of 9 - its digital root will be 9 as well! And since 9 is a multiple of 3, you can also check whether a number is multiple of 3 by checking if its digital root is a multiple of 3!
This is a blast from the past!
I was never taught this, but happened upon this property some years ago and always wondered what it was called and what use it might have. Thanks for finally answering my uncertainty.
I predate affordable calculators and I still do long multiplication, addition etc on paper if I don't have Excel in front of me. I can't be bothered with calculators, you get too many typos.
No-one ever taught me this when I was at school, and I went to primary school in the UK and elementary school in the USA.
This is great. Thanks!
It's also very easy to check that everything is correct modulo 2. The sum between several numbers will be even if and only if an even number of inputs are odd. Same goes for differences because subtraction is just adding the negative. And the product between multiple numbers will be even if and only if at least one input is even.
With this easy extra precaution, you can be sure if you messed up your arithmetic, you have to be off by a multiple of 18.
I totally used to do this in my head when I was younger, whenever I saw big numbers. But I never thought it was actually useful. i just noticed it as a cool thing you could do.
Now I understand why multiplying by 9 always gives a result with a digital root of 9 (except 0), which is how I remember the multiples of 9. I've used that trick for as long as I've needed to multiply. It's cool that I can finally prove why it works.
I learned this in elementary school because my teacher found it interesting. Really interesting trivia that's usable for finding modulo 9 for example.
I first learned about this from John Conway's Book of Numbers. It's a great way to introduce people to modular arithmetic.
It's quite obvious but I still wanted to point out that it's nice about this is that every multitude of 9 consists of two digits that make a total of 9, so 1+8=9, 2+7=9, 3+6=9. This remains consistent until you get past 10*9. Since 11*9=99->9+9=18 but then again 18->1+8=9. What's beautiful as well as is that after that point every total becomes 9 (simply put, 1+a+b where a+b=8) again until 21 of which the total is 18 again, just as 22*9 which is 18 as well, then the total becomes 9 again where every number becomes 2+a+b where a+b=7.
There's this really nice pattern of numbers here. From 71 to 77 for example as well the total becomes 18 but with 78 to 80 the total becomes 9 again. This pattern actually continues when it comes to even higher numbers, albeit a bit differently. from 101 to 111 the total is always more than 9 after that the total becomes 9 again until 121 to 122 and 131 to 133 and so on. Same with 201-222 and even 1000 to 1111.
I don't think I can make an exact formula for this but the pattern is certainly there. Basically the totals are always 9 except when the number lies between (10*k+1)*9 and (11*k)*9, but it's the same with (100*k+1)*9 to (111*k)*9 and (1000*k+1)*9 to (1111*k)*9 and (1000o*k+1)*9 to (11111*k)*9 and so forth.
If you all these specific multitudes, and look at how many of these apply, you can also calculate the total of a multitude of 9 of which the answer consists of.
if you were to look at 43421 for example. It is between 40001 and 44444 so that's 1 "check", it's not between 43001 and 43333, so that's a "fail", it is between 43401 and 4344 so that's another "check" and lasty it's also between 43421 and 43422. That's 3 checks so the total number of digits of the final answer will have a value of (1+3 checks)*9= 36. Put it in a calculator and you get 43421*9=390789->3+9+0+7+8+9=36.
I'm pretty sure this always works but I can't really come up with a formula for it.
This is really impressive, it does seem to make the checking easier, ty for sharing
Fun Coincidence: two days ago, while tutoring abstract algebra, we proved 9 divides a number iff 9 divides the sum of its digits; exactly why canceling 9s works!
Now, two days later, I find this video.
I love math!
The way you did the manual multiplication is different to what kids are being thought here. It's cool to see that there's another way to do manual multiplication like you did. It seems harder for me but i guess i can adapt it.
I LOVE this trick, and I show it to everyone I can. You should talk about casting out 11's sometime!
0:56 - THAT NAIL AND GEAR FLAG YEAH
I figured out a version of this when I was a kid and it's basically why 9 is my favorite number.
Thumbs up for the NAIL AND GEAR!
In Italy this is (or was) taught at every elementary school. It's called 'the test of the nine' ("la prova del nove"), which is also an idiom to mean the final confirmation of something. Does the expression exist also elsewhere?
This is incredible useful and fun. Awesomely useful.