WEBVTT

e9353e9f-c35c-4e73-8791-1b4152bd90ff-0
00:00:05.440 --> 00:00:06.480
Hi, my name's Richard.

63d886bd-76ca-4263-9342-7c6732310934-0
00:00:06.480 --> 00:00:10.160
We're going to be looking here at Tamura
2021 paper one question 9.

bdd9091c-490c-4b86-958e-249fbd39c9db-0
00:00:10.160 --> 00:00:10.760
Here it is.

870b1068-67ce-4aba-8e88-3d9911b377a4-0
00:00:10.760 --> 00:00:16.453
It says find the area enclosed by the
graph of the modulus of X plus modulus of

870b1068-67ce-4aba-8e88-3d9911b377a4-1
00:00:16.453 --> 00:00:16.880
y = 1.

a2091c22-369f-44ba-a3a1-401394b9d72b-0
00:00:17.160 --> 00:00:20.742
So the key thing for this question is
having a good way of dealing with this

a2091c22-369f-44ba-a3a1-401394b9d72b-1
00:00:20.742 --> 00:00:21.440
modulus symbol.

8af173ee-cf0e-4889-9e0d-b86200828374-0
00:00:21.760 --> 00:00:24.000
And we're going to think about that
separately to begin with.

264934df-ac9d-4461-9fba-b46a6c77d559-0
00:00:24.320 --> 00:00:28.024
So when we see the modulus of X,
we know that the value of this will

264934df-ac9d-4461-9fba-b46a6c77d559-1
00:00:28.024 --> 00:00:30.440
depend on whether X is positive or
negative.

30a70cd9-dd16-48e5-b72b-9ca46f7775a5-0
00:00:30.880 --> 00:00:36.400
So the modulus of X is equal to X itself
if X is greater than or equal to 0.

3b9e457c-bbd2-40c7-8ed4-f0b789ef1bc4-0
00:00:36.640 --> 00:00:41.365
So think the modulus of three is 3/3 is
greater than or equal to 0,

3b9e457c-bbd2-40c7-8ed4-f0b789ef1bc4-1
00:00:41.365 --> 00:00:44.840
but it's equal to negative X if X is less
than 0.

bcb53cab-eea6-4f40-bdbe-2fa8d54cafc5-0
00:00:45.440 --> 00:00:48.400
So you might think of the case of X being
say -4.

7f2ded13-f4bf-4d59-b5d4-500f0ef1f656-0
00:00:48.600 --> 00:00:52.360
If you want the modulus of -4,
we have to take the negative of -4.

fc15704c-82fa-4b04-a08c-d487f2a337c6-0
00:00:52.360 --> 00:00:54.520
So we take the negative of that value to
get it.

bda7f662-0cca-4a4b-add4-7bf3beef65df-0
00:00:55.080 --> 00:00:58.240
So similarly,
Y will break down into two cases as well.

acbf0fed-dfb1-419b-90da-5abc40fb52ce-0
00:00:58.520 --> 00:01:03.000
The value that this takes will depend
upon whether Y is negative or greater

acbf0fed-dfb1-419b-90da-5abc40fb52ce-1
00:01:03.000 --> 00:01:04.120
than or equal to 0.

352daa98-6c13-4f9a-9d4a-89ac23d9850d-0
00:01:04.720 --> 00:01:07.880
So we have two cases for X and we also
have two cases for Y.

33796852-c6cb-4fc3-a6d0-c7b2ce4e064f-0
00:01:07.920 --> 00:01:11.840
And because we're they're together,
that will give us 4 cases overall.

182159ef-559c-44aa-a810-19534c5f78ba-0
00:01:12.400 --> 00:01:19.520
So a good way to think about this will be
to set up our axis to draw this graph.

afb88118-fb07-41d0-aecf-461845286766-0
00:01:20.200 --> 00:01:22.880
And we're going to have to think about
four different cases.

93bb80ab-c61d-42bf-99e7-4dd94066de6e-0
00:01:22.880 --> 00:01:28.029
So we've got the case when X is greater
than or equal to 0 and Y is greater than

93bb80ab-c61d-42bf-99e7-4dd94066de6e-1
00:01:28.029 --> 00:01:28.920
or equal to 0.

ca9e42d2-00b1-484c-a8e0-23aa6dc19a0a-0
00:01:29.200 --> 00:01:33.309
So that will correspond to any part of
the graph which is in this quadrant here,

ca9e42d2-00b1-484c-a8e0-23aa6dc19a0a-1
00:01:33.309 --> 00:01:36.760
X greater than or equal to 0 and Y
greater than equal to 0 in here.

71226527-30e0-427a-983f-532901fdc03d-0
00:01:37.440 --> 00:01:42.036
Then we've got another case over here,
which is when X is less than 0 and Y is

71226527-30e0-427a-983f-532901fdc03d-1
00:01:42.036 --> 00:01:46.050
greater than or equal to 0,
which will give us any part of the graph

71226527-30e0-427a-983f-532901fdc03d-2
00:01:46.050 --> 00:01:47.040
in this quadrant.

5d6e5524-f1d7-4bf6-8b11-d9e560c98bad-0
00:01:47.040 --> 00:01:53.280
And we have two other cases down here,
X less than 0 and Y less than 0.

3b736ecf-f95d-4d34-a83a-11f3c6c64d65-0
00:01:53.720 --> 00:01:58.880
And over here we have X greater than or
equal to 0 and Y less than 0.

af28602b-ce17-4be6-be8a-2d7b2e23d01c-0
00:01:58.880 --> 00:02:01.640
So we're going to think about these four
separate cases.

1b62fb32-8f78-4b1d-8c92-cd4c463adb9e-0
00:02:02.160 --> 00:02:05.581
So let's start with probably the one
that's the most straightforward to think

1b62fb32-8f78-4b1d-8c92-cd4c463adb9e-1
00:02:05.581 --> 00:02:08.520
about, which is when X&
Y are both greater than or equal to 0.

356c54c5-01e1-4926-b2ee-f6f7c90a5160-0
00:02:09.120 --> 00:02:13.581
So in this case,
the modulus of X is X and the modulus of

356c54c5-01e1-4926-b2ee-f6f7c90a5160-1
00:02:13.581 --> 00:02:14.120
Y is Y.

8b17f4de-7b39-4f2d-bb5f-c38a14b921a0-0
00:02:14.520 --> 00:02:20.560
So the the equation we're looking at
becomes X + y = 1 up here.

a1038fb6-76a0-4c56-bdad-09bae4f0cf2c-0
00:02:21.000 --> 00:02:23.360
So this is a very familiar shape to us.

b43cccab-cc27-4f95-8cf2-3ed33ce035c7-0
00:02:23.360 --> 00:02:24.360
It's a straight line.

c4691c68-8d22-49d6-a4ae-e23420cc7dde-0
00:02:24.720 --> 00:02:28.292
We can see that the gradient would
actually be -1 if we imagine the X coming

c4691c68-8d22-49d6-a4ae-e23420cc7dde-1
00:02:28.292 --> 00:02:29.360
over to the other side.

c7c7768a-9f5f-44ac-8d16-dd0de4db54d6-0
00:02:29.840 --> 00:02:34.120
And we can see that this graph will look
like this.

f99e1b77-52b3-4bdf-b3db-3ac5036a2e62-0
00:02:34.520 --> 00:02:38.040
This is the .10 and this is the .01.

ab4223fc-5369-4955-aaf3-266f25d42a27-0
00:02:38.560 --> 00:02:41.080
And these are all the points for which X
+ y = 1.

072cafe4-3c32-43f0-ac83-06869eaf49bb-0
00:02:41.080 --> 00:02:44.400
So in this quadrant we will have this
part of the graph.

c265f48d-3be8-4802-8127-15cbc6aae4ec-0
00:02:45.520 --> 00:02:47.280
So now perhaps we'll look at this
quadrant.

a120f3f9-cdea-4620-a011-95924b2f4d08-0
00:02:47.840 --> 00:02:51.440
Now we have X less than 0 and Y is
greater than or equal to 0.

c5a200cf-4a00-48e2-a845-89d99fc65410-0
00:02:51.440 --> 00:02:55.200
So when X is less than 0,
this term is actually negative X.

624f97c0-8b4b-42ce-bf24-fc1e27caf5e6-0
00:02:55.640 --> 00:03:02.160
So our equation over here actually looks
like this minus X + y = 1.

6bcbdb91-ed61-4ee4-9e3f-f0c7e1411d0e-0
00:03:02.760 --> 00:03:05.440
Think about that rearrangement y = X + 1.

c42924fa-1b2e-4a8b-89fb-e59f2e5ad632-0
00:03:05.720 --> 00:03:09.738
We can see that this is going to be a
straight line whose gradient is 1 and

c42924fa-1b2e-4a8b-89fb-e59f2e5ad632-1
00:03:09.738 --> 00:03:11.960
whose intercept on the Y axis is also one.

2e9e73c8-1187-4403-829d-a4ceb29f8895-0
00:03:12.520 --> 00:03:15.200
And so we can imagine that that will look
like this.

b45ecc09-ff14-4ba8-9f8c-1119bf0cbb46-0
00:03:15.560 --> 00:03:23.040
This point here will be -1 So this is a
part of y = X + 1 in that quadrant only.

4f1b1790-7bfa-480e-ada6-a07e175df2a9-0
00:03:23.880 --> 00:03:26.416
OK,
moving down into the bottom left quadrant

4f1b1790-7bfa-480e-ada6-a07e175df2a9-1
00:03:26.416 --> 00:03:27.960
where they're both negative.

94f2e9fc-38db-4ed6-8291-adb2c3cc4199-0
00:03:28.440 --> 00:03:32.760
So here this will become minus X and this
will become minus Y.

8f997d5b-a3f7-4dc0-9b0f-378fb8d94ee4-0
00:03:33.240 --> 00:03:38.400
So we have -X -, y = 1 as our equation.

6ff4722c-847a-4a99-92b1-264382f3cdde-0
00:03:38.680 --> 00:03:44.077
We can imagine rearranging this again,
perhaps imagine the Y coming over,

6ff4722c-847a-4a99-92b1-264382f3cdde-1
00:03:44.077 --> 00:03:46.120
we would have y = -, X -, 1.

4c54bb3a-842a-47ae-bcf2-1a0a3ec6d0ec-0
00:03:46.160 --> 00:03:49.560
So that's got a negative gradient and an
intercept of -1.

2ea6cfc0-d7e4-4e34-9709-63391b412a6a-0
00:03:50.160 --> 00:03:51.960
So it will look something like this.

a973ba60-9a3e-4cdf-87df-dd88d15bd69d-0
00:03:52.080 --> 00:03:53.960
And this value here will be -1.

df7136b8-4643-42f7-ac82-8615aa01f895-0
00:03:54.520 --> 00:03:58.900
And finally, over in this quadrant,
X is now greater than or equal to 0,

df7136b8-4643-42f7-ac82-8615aa01f895-1
00:03:58.900 --> 00:03:59.920
Y is less than 0.

21d0afa2-32ed-4a4b-b2c0-c3484145bc15-0
00:04:00.360 --> 00:04:07.534
So our equation now looks like X -,
y = 1 because Y is negative and the

21d0afa2-32ed-4a4b-b2c0-c3484145bc15-1
00:04:07.534 --> 00:04:14.210
modulus of Y will be minus Y and this
will rearrange to y = X - 1,

21d0afa2-32ed-4a4b-b2c0-c3484145bc15-2
00:04:14.210 --> 00:04:16.999
which is this straight line.

c8a4e51e-4397-4acf-b314-48868bc5b3a0-0
00:04:17.680 --> 00:04:22.100
So having looked at all four quadrants,
we can see that the graph of mod X plus

c8a4e51e-4397-4acf-b314-48868bc5b3a0-1
00:04:22.100 --> 00:04:24.200
mod y = 1 is actually this shape here.

70213924-6425-4d6c-b777-8dbf1b6525b5-0
00:04:24.840 --> 00:04:28.280
And we are asked to find the area
enclosed within this shape.

3922cb62-6561-4d5e-b62e-49803668a492-0
00:04:31.040 --> 00:04:32.600
Several ways you might do this.

29c51b21-4a04-4caf-8052-4d3ed72be59d-0
00:04:33.000 --> 00:04:36.120
One way you can imagine each of these
triangles.

4c07a20d-76af-463f-b0c3-84e292adda97-0
00:04:36.800 --> 00:04:39.680
Each of these triangles is half of a unit
square.

8d2124e7-d49b-48a9-82a4-f102398b1f4b-0
00:04:40.240 --> 00:04:42.960
So this will be a one by one square and
we have half of it.

26d6b7e9-098e-4e5c-976a-78faa939407e-0
00:04:43.280 --> 00:04:48.154
So we might imagine that this is 1/2 and
this is 1/2 and this is 1/2 and this is

26d6b7e9-098e-4e5c-976a-78faa939407e-1
00:04:48.154 --> 00:04:50.080
1/2 and the area is there for 2.

fc2edaa4-5338-4d1f-9e12-9aca31f63a80-0
00:04:50.880 --> 00:04:54.486
Another way you might think about this is
to think about how long this line is the

fc2edaa4-5338-4d1f-9e12-9aca31f63a80-1
00:04:54.486 --> 00:04:55.920
length of the side of the square.

1dd36814-1e2b-4c6b-b921-ae98ffa08fb3-0
00:04:56.200 --> 00:04:58.800
And by Pythagoras theorem that is root 2.

6fda3387-3a1e-4adb-a29e-e78372d0dab4-0
00:04:58.800 --> 00:05:01.040
So we have root 2 times root 2 giving 2.

0c8bba14-24eb-4d35-83bd-6a84e3f71706-0
00:05:01.360 --> 00:05:04.844
Whichever way you do it,
you can see that the answer here would be

0c8bba14-24eb-4d35-83bd-6a84e3f71706-1
00:05:04.844 --> 00:05:06.040
this value C to up two.

6852bfa6-f4e5-472c-a6d8-abb5c56304df-0
00:05:07.280 --> 00:05:09.440
OK,
so let's just reflect on this question.

5b4de9ea-9fd5-4dae-9d3e-396cbc2967c2-0
00:05:09.640 --> 00:05:13.091
I think the key thing that came out of
this question was having a way of dealing

5b4de9ea-9fd5-4dae-9d3e-396cbc2967c2-1
00:05:13.091 --> 00:05:14.200
with the modulus function.

0d237a56-c3f3-42e6-b55a-33581d6fbdd3-0
00:05:14.520 --> 00:05:18.560
Realising that the modulus of X is
actually a function defined in two cases.

2c69693e-f097-438a-b6d4-23cbc3067bde-0
00:05:18.560 --> 00:05:22.880
You've got the case of X greater than or
equal to 0 and the case of X less than 0.

d4aabe7b-a88c-4eeb-a7f1-9920411909f8-0
00:05:23.120 --> 00:05:25.920
And so this then naturally divided up
into four cases.

37eb0e10-eae7-4b49-adab-18917b5cccdc-0
00:05:25.920 --> 00:05:28.080
Given that we've got both mod X and mod Y.

67aceaa8-a731-4429-b079-64b7dce1d0e9-0
00:05:28.520 --> 00:05:32.691
Another way into this problem may have
been to think about what's happening when

67aceaa8-a731-4429-b079-64b7dce1d0e9-1
00:05:32.691 --> 00:05:33.000
X = 0.

05e1040b-faab-40d8-89f7-5a9e395409a8-0
00:05:33.360 --> 00:05:37.125
So if you imagine X is 0 for example,
we would have mod Y being one,

05e1040b-faab-40d8-89f7-5a9e395409a8-1
00:05:37.125 --> 00:05:41.600
which means that Y is either plus one or
-1, which will give us these two points.

76ef519c-4e25-44ba-aa17-530990e67ed4-0
00:05:41.920 --> 00:05:45.927
And if we set Y equal to 0,
it gets something similar with ** is +1

76ef519c-4e25-44ba-aa17-530990e67ed4-1
00:05:45.927 --> 00:05:48.520
or -1,
which will give us these two points.

617e33c8-8431-41b2-92a8-9c9a5095f662-0
00:05:48.520 --> 00:05:51.040
Perhaps we can spot some other points on
this graph.

71c18dbb-0d12-49a3-a9ad-e1133afe6e82-0
00:05:51.280 --> 00:05:54.198
For example,
it's quite easy to see that X is 1/2 and

71c18dbb-0d12-49a3-a9ad-e1133afe6e82-1
00:05:54.198 --> 00:05:57.280
Y is 1/2 would be on the graph which is
this point here.

7346d6d8-3a83-44cb-bbbb-fdbbd20529f0-0
00:05:57.600 --> 00:05:59.480
But you can also start to flip the signs.

53708ce3-f2b6-4562-aa75-8753b3cb7fd5-0
00:05:59.480 --> 00:06:03.480
There you get X is minus half and Y is
plus 1/2 would work too.

7b5fd29d-9e0d-4052-81bf-a24cf81bb098-0
00:06:03.960 --> 00:06:06.888
So perhaps you can pick up some points
and start to build up a picture of what

7b5fd29d-9e0d-4052-81bf-a24cf81bb098-1
00:06:06.888 --> 00:06:08.000
the graph looks like that way.