Welcome to episode n of "Physics Girl," where
n is an integer.
I'm Dianna.
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 $2 on every $10 that you spend.
So if you spend $30, that's 2, 4, $6 tip. $25
is 2, 4, and an extra dollar because I'm
halfway up the next 10
Easy.
But the other day, I spent $170 on something.
And I just froze because my typical method of 2, 4, 6, 80--
For every 10 was going to look ridiculous
as I stood there counting on my fingers.
So I came up with a different method,
which is just moving the decimal place over one
and multiplying by 2.
And you get 34.
This is a little bit trickier for fast math,
but it's faster than counting by two's.
34.
So ignoring the debates on whether tipping
is a useful social construct, there's a trick for you.
11's 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 and the 8, add 7 plus 8
equals 15.
And then I just make sure I add the 1 to that seven.
Voila.
I wish I had known this trick earlier in life.
We've all done the trick where multiplying by nines,
you use your fingers.
For example, 9 times 7-- 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 10s, that would be easy to see.
So 4 times 10 is four 10's fingers, or 10, 10, 10, 10.
That's 10, 20, 30, 40.
If you multiply 4 times 9, it's almost like four 10's.
But it's a single one less for each 10's finger
because 9 is 1 less than 10.
So as I count up my 10's fingers-- 10, 10, 10 10--
I have to keep track of four 1's that I take away.
We'll keep those four negative 1's over here.
I'll set aside three of the 10's fingers.
And I'm going to take the last 10
and bring it over here as 10 1's fingers.
Now I take away my four ones from those fingers,
and I'm left with 6.
Let's bring this back together with the 10's.
And I've got three 10's and six 1's-- 36.
So this works because I'm doing it in base 10,
and I have 10 fingers.
Very convenient.
But if I wanted to do my 8's times tables,
I'd have to use nine fingers and calculate in base 9.
For example, 8 times 5 is 44 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 6 but more than 5.
But that's no fun.
Let's use geometry.
Measure 4 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 5.625, 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.
Pythagorean theorem says that the hypotenuse of that triangle
would have to be square root 32 plus 32 or square root 64,
or 8.
Try that and see how close you get to length 8 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 be blown.
OK.
Without telling me the number, think
of a number less than 1,000 and write it down.
OK.
Now divide that 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.
DIANNA: OK
The remainder I got for 13 was 5.
Awesome.
OK.
Now you've got to give me a second.
Alex--
Yes.
Is this your number?
It is indeed!
DIANNA: Oh, my gosh!
Can you hold up your number that you originally wrote down?
Yes.
Right here.
DIANNA: Whoo!
Yes!
Are you blown away?
[INAUDIBLE].
Yeah.
How'd you do it?
Do you have any guesses?
I guess it's something to do between the relationships
with the remainders in 7, 1, and 13.
That's a 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.
927?
Yeah.
And then I added those all together.
And then I subtracted the highest multiple of 10001
that was less than the number that those added up to.
Huh.
Seems pretty random, right?
Yeah.
What did you think?
It was really cool!
It was really cool.
It's a neat little trick that you can play on your friends
if you have the time to do it.
The math trick Dianna just showed me always
works if you do the math right.
Try it out on your friends.
The trick works because of the Chinese remainder theorem. now,
the Chinese remainder theorem isn't just a game.
It's actually used in 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 dividing it by the three smaller numbers that
don't have any common factors other than 1-- that
is, these numbers are coprime-- you can find
the chosen number as long as it's
less than the product of these three smaller numbers.
For example, 7 times 11 times 13 is 1,001.
Once you know those remainders, they only match a single number
from 0 to 1,001.
The remainders are like that number's fingerprint.
Pretty cool.
Thank you for being a good sport and playing along.
Thank you.
This video was your idea.
So I course am 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 physicsing.
Happy physics-ing.
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!
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