[Music] welcome to episode n of physics girl where n is an integer I'm Diana look sometimes I talk about things other than physics and today is one of those days because I love math and math is the language of physics so I want to share with you some of my favorite math tricks some are pretty simple but fun and some are less relevant here goes this is for my American or American curious friends so I've always calculated tips the same way and it's never failed me until recently I've always gone with the same method that 20% tip which is about average for good service is just two dollars on every 10 that you spend if you spend $30 that's 2 for $6 to $25 is 2 4 and an extra dollar because I'm a halfway up the next 10 easy but the other day I spent 170 dollars on something and I just froze because my typical method of 2 4 6 every 10 was gonna look ridiculous as I stood there telling all my fingers so I came up with a different method which is just moving the decimal place over one and multiplying by two and you get 34 this is a little bit trickier for fast math but it's faster than counting by twos 34 so ignoring the debates on whether tipping is a useful social construct there's a trick for you Elevens times tables are really easy until you get past 10 but there's a super simple trick for numbers after that 43 times 11 just take the 4 in the 3 and separate them add them and then put that 7 in the middle and that is the answer so easy and you can see why it works from regular long multiplication this works well if the sum of the digits is less than 10 but it still works if you carry when you add so 78 times 11 separate the 7 & 8 add 7 plus 8 equals 15 and then I just make sure I add the 1 to that 7 voila I wish I had known this trick earlier in life we've all done the trick we're multiplying by nines you use your fingers for example nine times seven put your seventh finger down and you get six tens on this side and three ones on this side 63 it's a really nice trick but it doesn't explain the coolest part which is why it works so if we were multiplying by ten so that'd be easy to see so 4 times 10 is 4 times fingers or 10 10 10 10 that's 10 20 30 40 if you multiply 4 times 9 it's almost like four tens but it's a single one less for each tens finger because nine is one less than 10 so as I count up my 10 fingers 10 10 10 10 I have to keep track of 4 ones that I take away we'll keep those four negative ones over here I'll set aside three of the tens fingers and I'm gonna take the last 10 and bring it over here as 10 ones fingers now I take away my 4 ones from those fingers and I'm left with six let's bring those back together with the tens and I've got three tens and six ones 36 so this works because I'm doing in base 10 and I have 10 fingers very convenient but if I wanted to do my 8 times tables I'd have to use 9 fingers and calculate in base 9 for example 8 times 5 is 40 4 in base 9 but that's not as useful side note when I was thinking about this problem I noticed this pattern pause the video to check it out if that's your thing but we're gonna move on what's the square root of 32 ah put down that calculator you could do it by trial and error knowing it's got to be less than six but more than five but that's no fun let's use geometry measure for inches on your pipe cleaner and bend it there to 90 degrees bend the pipe cleaner again at 4 inches and form a right triangle double checking that the angle is still 90 degrees now the length of this third side squared according to the Pythagorean theorem it should be a squared plus B squared or 32 because it's the hypotenuse so the length of this third side should measure the square root of 32 I measured five point six two five which is pretty close to the real answer which is pretty awesome so if you want to calculate how close you were without using a calculator just use the hypotenuse of that triangle to form another right triangle with sides length square root 32 diegans theorem says that the hypotenuse of that triangle would have to be 4 32 plus 32 or square root 64 or 8 try that and see how close you get to length eight inches using this method you can construct the square root of any number as long as you have enough pipe cleaner and a long enough ruler I have Alex from technicality here helping me with the last trick you ready for some math magic I'm excited I'm excited for my mind to move on ok without telling me the number think of a number less than a thousand and write it down okay now divide the number by 7 11 and 13 and then give me the remainders for each of those ok I did 7 I got remainder 1 my remainder for 11 is 3 for 13 was fine awesome okay now you gotta give me a second um Alex is this your number oh my gosh can you hold up your number that you originally wrote down yes right here whew yes ah are you blown away yeah magic do you have any guesses I guess something to do between the relationships with your being and seven that's very good guess because I made you do that I took your remainder numbers and I multiplied them by 715 and then 364 and then 927 yeah and then I and then I added those all together and then I subtracted the highest multiple of one zero zero one that was less than the number that those added up to seems pretty random right yeah what did you think it was really cool it was really cool it's my legal trick that you can play on your friends if like yeah the time to do it the math trick Diana just showed me always works if you do the math right try it on your friends the trick works because the Chinese remainder theorem now the Chinese remainder theorem isn't just a game it's actually used in the encryption algorithms and you can see it all throughout history the rule goes like this if you know the remainder of your chosen number after you've I need it by the three smaller numbers that don't have any common factors other than one that is these numbers are co-prime you can find the chosen number as long as it's less than the product of these three smaller numbers for example seven times 11 times 13 is 1001 once you know those remainders they only match a single number from zero to a thousand and one the remainders are like that numbers fingerprint pretty cool thank you for being a good sport and playing along yeah this video was your idea so I of course I'm going to make you do some math oh yes of course yeah thank you so much Alex for helping out with this last trick and happy physics things I'd like to thank Alex who you just saw from technicality he actually came up with the idea for this video and I thought it was really cool so I wanted to involve him and he's got his own YouTube channel which you should check out it's called technicality and I'll put a link in the description we've actually got another video with some fun science on his channel so I'll link to that as well and you should check it out toodles [Music]