Hello and welcome to Maths with Jay.
In this video we're going to be looking at constructing
the first few lines of Pascal's triangle.
It gets its name from the French mathematician
Blaise Pascal who lived in the 17th century,
but the triangle actually had been around for hundreds of years
before even Pascal was alive. Anyway,
let's have a look at the triangle. If you want to make one of these
yourself, what you could do is get hold of some
isometric paper and combine
2 of the triangles into a diamond shape, in the sort of way I've done here, although I've
stretched it out,
so it doesn't quite look like equilateral triangles being added
together here.
Or you could maybe find some triangle dotty paper would have the same effect.
Okay, so let's have a look at the
way in which this triangle is constructed. We start off
by simply writing the number 1 in the first cell at the top of the triangle,
and then from now on,
every other cell that we fill in, is going to be done according to the same
rule, and that is
that each cell will contain the sum of the two numbers
that are in the row above it, to the immediate left
and right, so when we come down to looking at the second row,
looking at the first element to fill in,
we've got a number 1 to the right of it,
just above it, and nothing just to the left of it, so
nothing plus one is one. Similarly for the
next element in the second row
we're going to be adding 1 to nothing so we get 1; and then when we get onto
the third row, we will be adding
1 to nothing to get 1, and then,
when we get to the second element of this row
we're adding 1 and 1 from the row above
to get 2, and then the last element in this row we'll be adding 1
and nothing to get 1. On to the next row and, as you can see,
the first element is going to be nothing plus 1
will be 1; then we'll be adding 1 and 2
to get 3, then 2
and 1 to get 3 again, and then 1
nothing to get 1; so on to the next row;
and I'm going to put in the 1s at both ends to start with. I think that
you can see by now that
every row is going to start and end with a 1,
and then we'll add 1 and 3 together
to get 4, 3 and 3 to get 6,
and 3 and 1 to get 4, and
you can probably see that not only are we getting 1s at both ends,
but each row is symmetrical, isn't it?
So we can put the 1s at the ends of the next row
and then we know that
1 and 4 will be the same as the sum of 4 and 1.
So 1 plus 4 and 4 plus 1 are both 5.
And 4 plus 6
and 6 plus 4 is 10.
So we get 10s in the middle of this row. And because we can see
that we've got 1s at the start of each row and at the end of each row, let's just
put all of them in
to start with, and then if we look down the next diagonal on both sides,
we can see that we are just getting the counting numbers,
so let's put it in the rest of those diagonals,
from 6 down to 15, so now we've got less work to do -
we've completed the easy bits.
Right, so next we're going to add
together 5 plus 10 and then
10 plus 5, giving us 15
at both ends, and then adding together 10
and 10 we get 20 and then,
on the next row, so what we going to do now, is do a whole row at a time.
I think you can see that 6 plus 15 is 21,
15 and 20 is 35,
and then by symmetry at the other end, we will have 35 and 21
again. Then we'll be adding
7 and 21
to give us 28, 21 and 35 will add up to 56
and double 35 in the middle to give us 70.
And then, of course, repeating on the right hand side as well.
The next row, well, we'll get 8 plus 28
is 36, 28 plus 56 is 84,
56 plus 70 is 126,
and then we'll repeat on the right hand side.
And then the next row: 9 plus 36 is 45,
36 plus 84 is 120,
84 plus 126 is 210
and doubling 126 will give us 252.
On to the next row:
10 plus 45 is 55,
45 plus 120 (is) 165,
120 plus 210 is 330,
and 210 plus 252 is 462.
On to the next line:
11 plus 55 is 66,
55 plus 165 is 220,
165 plus 330 is 495,
330 plus 462 is
792, and double 462
is 924.
And the next line: 12 plus 66 is 78,
66 plus 220 is 286,
220 plus 495,well the numbers are getting a bit bigger now,
BUT: we can simplify these by thinking of this:...
What I would do is say 495 is just
5 away from 500, so let's think of that as being 500 and take 5 away from
220, so that makes that 215,
so it's easier to add together 215
and 500 to get 715.
And do the same sort of thing for 495 plus 792:
so 495 is five away from 500
so if we take five away from 92 we'll get
87, so we're adding 500 and 787
which will be 1287 or 1,287.
And then 792 plus 924 - well 792
is 8 away from 800, so that's going to be
800 plus... now we need to take the
8 away from 924 so that's 916,
so we're adding 800 to 916 to get 1,716.
OK, so that's that row sorted. Next row: 13 plus 78 is 91.
78 plus 286 is 364.
286 plus 715:
well that's a nice one isn't it, because 86 plus 14 is
100, and 200 add 700 is 900, so that's 1001.
And then 715 plus 1,287 -
well that's 2002. And 1,287 plus 1,716 is 3003.
And then - the middle one - well, doubling 1716
gives us 3432.
And the last line that we're going to look at: adding 14 and 91 is 105,
91 plus 364 is 455,
364 add 1001 is 1,365,
And these are fairly easy to do now, aren't they? You certainly shouldn't need a calculator
for these:1001 plus 2002 will be 3003,
2002 plus 3003
is 5005, and 3,003 plus 3432
is
6435
and we've reached the middle. So we're done. The triangle,
in fact, goes on forever but we're just looking
at the first few lines of it.
Now, you know I'm always going on about
the fact that you should check your answers. Something like this, although it's
really
simple, in that you're only adding together two numbers,
it would be really easy to make an error quite early on
and not realise you've done that, to be going on doing the whole thing
with one small error creating bigger errors
later on. So let's think about how we could check our answers here.
Well, have you thought of adding together
the total of each line, seeing if there's maybe some kind of pattern
that we could see? So going right up to the top:
Well, the first line has obviously just got
1 in it, the second line, the second row has got 1 and 1, 1 plus 1
is 2, the next one, 1 plus 2 plus 1
is 4, the next one, when we add
1, 3, 3 and 1, we get 8
so the numbers we're getting are 1, 2, 4, 8,...
Can you see
a pattern developing here? The next number will be 16,
the one after that is 32.
So in fact, hopefully you've noticed that the sum of each row
is double the previous row.
And if you think about it, that's happened because
each number is contributing
twice to the next row. It's being added in twice, once it's
the left-hand number and once it's the right-hand number.
So that's logical why that should be happening.
But that really gives us a way of checking our answers.
If we add together the numbers in each row as we go along
and make sure that they are following that pattern
then it's quite likely that we haven't made a mistake.
So having got our Pascal's triangle, and deciding we're happy that we haven't made
any mistakes,
the next thing to do is to start playing with it: What we could do
is look for number patterns within the triangle itself:
So one pattern that you might have noticed is
the triangle numbers or the triangular numbers.
Then you might like to look for the tetrahedral numbers.
Once you've done that,.....hidden away
are the Fibonacci numbers, so those are more difficult to spot.
And what I thought you might like to do next, is do some colouring in:
How about colouring in all the even numbers
in the triangle?