| Transcript
for the Math
Jam "AoPS Classes Math Jam"
on May 20. |
| Math Jam hosted by rrusczyk
(Richard Rusczyk ). |
rrusczyk19:34:41
Hello, and welcome to an Art of Problem Solving Math Jam. Today we'll be discussing the MATHCOUNTS/AMC 8 Basics, Introduction to Number Theory, and Introduction to Geometry courses that start in the first week of June.
rrusczyk19:34:58
My name is Richard Rusczyk. I founded Art of Problem Solving and have written several Art of Problem Solving textbooks.
rrusczyk19:35:06
Before we get started I would like to take a moment to explain our Virtual Classroom to those who have not previously participated in a Math Jam or one of our online classes.
rrusczyk19:35:39
The classroom is moderated: students can type into the classroom, but only the moderators can choose a comment to drop into the classroom. So, when you send a message, it will not appear immediately, and may not appear at all. This helps keep the class organized and on track. This also means that only well-written comments will be dropped into the classroom, so please take time writing responses that are complete and easy to read. Also, only moderators can enter into private chats with other people in the classroom.
rrusczyk19:36:07
Note that it is not possible for the instructor to personally respond to every comment that you submit during the Math Jam -- please do not take it personally if your comment is not posted or responded to! I will try to respond to all questions to the extent that I can. I will let you know when to start asking questions about the classes.
rrusczyk19:36:30
(Our usual classes have assistants to field a lot of the extra questions -- so you will get all your questions answered in classes.)
Augustine19:36:58
I can not hear anything
rrusczyk19:37:06
There is no audio in the classroom.
rrusczyk19:37:14
This link explains why we do this:
OakGroveVistor19:37:43
should we click on it?
rrusczyk19:38:08
Only if you want to read about how the classroom works now. You can look at it later if you like.
rrusczyk19:38:40
There will be a full transcript of this Math Jam in the Math Jam section of the site about 30 minutes after we finish.
rrusczyk19:38:57
In this Math Jam, I will briefly describe the course, then go through an example problem or two. Then, I will hold a question-and-answer session about the class.
rrusczyk19:39:44
MATHCOUNTS/AMC 8 Basics
rrusczyk19:39:49
This summer, we are offering two different MATHCOUNTS/AMC 8 classes: MATHCOUNTS/AMC 8 Basics and Advanced MATHCOUNTS/AMC 8.
rrusczyk19:40:05
The MATHCOUNTS/AMC 8 Basics course will meet for 12 weeks on Mondays, starting June 1, at 7:30 PM Eastern / 4:30 PM Pacific. Each class is 90 minutes, and each is 7:30 - 9 PM ET (4:30 - 6 PM PT). This class is for students just getting started with the type of problem solving required for success in MATHCOUNTS and the AMC 8.
rrusczyk19:40:30
Our Advanced MATHCOUNTS/AMC 8 is for more experienced students, such as those who are training for State MATHCOUNTS, with hopes of qualifying for National MATHCOUNTS. The Advanced MATHCOUNTS/AMC 8 course will meet for 12 weeks on Fridays, starting June 5, at 7:30 PM Eastern / 4:30 PM Pacific. Each class is 90 minutes, and each is 7:30 - 9 PM ET (4:30 - 6 PM PT).
rrusczyk19:41:14
Both classes include some material from our previous MATHCOUNTS Problem Series class. The Advanced MATHCOUNTS/AMC 8 class will be a little more challenging than the old class, and the MATHCOUNTS/AMC 8 Basics course will be a little easier.
rrusczyk19:41:28
Each class will be taught by Joshua Zucker. Joshua has been a Math Olympiad Summer Program invitee, a member of the first US Physics Olympiad team, and a top-10 scorer on the Putnam. He holds a BS in physics and an MS in mathematics from Stanford, as well as an MS in astrophysics from UC Berkeley. He has taught at levels ranging from summer camps for gifted elementary school students through remedial arithmetic at community college. He was a middle and high school teacher in Palo Alto, CA and was formerly a problem writer for MATHCOUNTS.
rrusczyk19:41:47
(I will take questions after we go through a few problems. Please hold your questions about how classes work until then.)
rrusczyk19:42:16
While there is overlap in topics between the two classes, there will be almost no overlap in problems. Topics covered include methods of counting, probability, algebraic techniques, geometry, word problems, number theory, and more.
rrusczyk19:42:44
This class is a Problem Series class, meaning that the major focus of the class will be working through various contest problems. Although there will be weekly problem sets for each class posted on the message board, students do not submit their homeworks to be graded, and there is no personalized instructor feedback on the solutions. (However, the instructors will be monitoring the message board to answer questions and comment occasionally on the students' discussions there.)
Mewto5555519:43:00
The Mathcounts class helped me get as far as I did in Mathcounts (2nd written, 1st masters)
Watermelon9919:43:45
AT Nationals?
O M G19:43:45
woah was that this year's nationals?
rrusczyk19:43:49
Thanks for the plug. For those of you keeping score at home, he's referring to the National contest. I witnessed his mastery of the Masters round. Great fun.
rrusczyk19:43:59
Each course will also include a whole class that is a single giant Countdown Round contest!
rrusczyk19:44:23
We'll now look at a few sample problems from the MATHCOUNTS/AMC 8 courses. Hopefully these problems will give you an idea of which course is right for you. I will take questions about the course after the problems.
rrusczyk19:44:38
We'll start with a couple problems that are examples of easier problems. These are problems that would be included in the Basics class, but would only be used as a warm-up (or not at all) for the Advanced MATHCOUNTS/AMC 8 course.
rrusczyk19:45:01
connaissance19:46:14
38
aopsaccount213119:46:15
38 digits
remember19:46:15
38
rrusczyk19:46:45
Bold claim! Why is this the case?
rrusczyk19:47:10
What should we look for when we are counting digits?
Watermelon9919:47:44
multiples of 10
MarkMike19:47:44
multiples of ten
rrusczyk19:48:03
Powers of 10 help us count digits. How can we get powers of 10 into this problem?
limac19:48:37
4^20 = 2^40, and 2^40 = 2^36*2^4, thus 5^36 and 2^36 = 10^36 which is 1 with 36 zeroes. and 2^4 = 16, so it is 10^36*16 = 16 with 36 zeroes.
remember19:48:50
you convert the 4^20 to 2^40 and combine with the 5^36 to get 10^26*2^4 so 38 digits (phew)
connaissance19:48:50
=16 x 10^36 which has 36+2 digits
rrusczyk19:49:01
rrusczyk19:49:22
Nothing fancy -- no high-powered math needed!
rrusczyk19:49:28
Let's try another.
rrusczyk19:49:41
d10a10v1019:50:04
whats ! mean?
rrusczyk19:50:11
The exclamation point is a 'factorial'. 4! = 4 x 3 x 2 x 1, and factorials for all other positive integers are defined similarly - we multiply all the numbers from the given positive integer down to 1.
rrusczyk19:50:24
Do we have to multiply all that out to evaluate the given expression?
InsDel19:56:27
No, we can factor out some of the factorials
amanjaria19:56:27
Cancel 5! from all terms: 6! / 6 + 1 + 1 = 6!/8 = 6 x 5 x 4 x 3 x 2 x 1 / 8 = 90
professordad19:56:27
No, we don't have to multiply this out--all we need to do is use the distributive property.
remember19:56:27
no factor out the 5! in the denominator and simplify to 6!/8 = 90
TXMath19:56:31
no. because you can cancel out factors first
changle19:56:31
distribute 5! and then divide out 5!. THen you get 6! divde 8
rrusczyk19:56:40
We don't have to multiply. We can factor instead.
yankeesfan19:56:49
5!6!/6!+5!+5!=5!*6!/6(5!)+5!+5!=5!6!/8(5!)=6!/8 I think.
augnov19:56:49
6!5!/5!(6+1+1)=720/8=90
rrusczyk19:56:56
rrusczyk19:57:07
Also, clever factoring is extremely useful in math problems, and often is helpful in MATHCOUNTS.
rrusczyk19:57:18
The next two problems are examples of problems that would be on the harder end of the Basics class and the easier end of the Advanced MATHCOUNTS/AMC 8 class.
rrusczyk19:57:38
rrusczyk19:58:18
How do we tackle this? 99! is a very scary number: 99*98*97*96. . . *2*1.
rrusczyk19:58:22
That's huge!!!
rrusczyk19:58:41
Are we afraid of those big numbers?
connaissance19:58:57
all the terms above 5! are divisible by ten
professordad19:58:57
99! 's unit digit is 0, because there is a 5 and even number in it
connaissance19:58:57
so their units digit is zero
rrusczyk19:59:03
We don't get intimidated by the 99!. We only want the last digit. Most of these terms have last digit of 0.
rrusczyk19:59:08
Which ones don't?
MathTwo19:59:26
1, 2, 3, 4
qwertythecucumber19:59:26
1! ==>4!
frogbandit19:59:27
the ones that dont are 1! 2! 3! and 4!
bbhattacharya19:59:27
1,2,3,4
InsDel19:59:27
1!,... 4!
skylord581619:59:27
The ones under 5!
rrusczyk19:59:41
All the ones from 5! and on end in 0, because 5! ends in 0.
rrusczyk19:59:56
So, our problem is just the last digit of 1! - 2! + 3! -4!
HiDN42820:00:26
which can be easily calculated by saying 1-4+6-24
rrusczyk20:00:36
Indeed, and what does that equal?
HiDN42820:00:45
sorry, 1-2+6-24
rrusczyk20:00:52
Ah, I didn't catch your typo :)
rrusczyk20:00:55
Glad you did.
rrusczyk20:01:04
And when we compute that, what do we get?
modularmarc10120:01:13
-19
chewndi20:01:13
-19
k00lperson20:01:13
-19
Watermelon9920:01:13
-19
rrusczyk20:01:19
1 - 2 + 6 - 24 = -19.
rrusczyk20:01:25
So, is the answer 9?
sxiaohu20:01:46
no it's 1
Jobo20:01:46
no
modularmarc10120:01:46
no its 1
InsDel20:01:46
No, it is 1
rrusczyk20:01:52
Why?
qwertythecucumber20:02:14
no, subtract 9 from 0 you get one.
frogbandit20:02:14
-19 + 120 = 1
dajoe20:02:14
no, because you have to subtract 9
connaissance20:02:14
you must add a multiple of 10
connaissance20:02:14
our answer is POSITIVE
llamafoo20:02:14
10-9 is one
limac20:02:14
no we do ...0-...9 which is 1.
rrusczyk20:02:22
The answer isn't 9 because our sum is clearly positive. All those terms with a zero units digit clearly have a positive sum, so our expression equals -19 + (something big that ends in zero). Thus we have a final digit of 1, not 9.
rrusczyk20:02:30
Any questions about that?
rrusczyk20:02:55
Ask specific questions -- I can't help with "I don't get it".
Plutonium20:03:01
how do we know that the really big thing is 10...
rrusczyk20:03:12
All of the factorials from 5! on end in 0.
rrusczyk20:03:20
So, 99! is some huge number that ends in 0.
kathyisjjmom20:03:23
What's a unit digit? SOrry i asked this so late into the problem
rrusczyk20:03:27
The last digit.
Tribefan20:03:29
If we stopped at 98! would the answer by 9?
rrusczyk20:03:39
Right -- the last digit would be 9 then.
alexgnow20:03:50
Why is everything after 5! end in 0?
rrusczyk20:03:57
How do you make 6!
InsDel20:04:15
5! is 5*...2*1
k00lperson20:04:15
6 times 5 times 4 times 3 times 2 times 1
amanjaria20:04:15
6x5x4x3x2x1
skylord581620:04:15
6x5!
rrusczyk20:04:24
Exactly: 6! = 6 x 5!.
rrusczyk20:04:32
We know that 5! ends in 0, so 6! must, too!
RoFlLoLcOpT20:04:40
to get a 0 at the end you have to have a 10, which is 2*5. The first factorial to have a 5 in it is 5!
rrusczyk20:04:44
We can continue this forever.
inoit20:04:47
why if you stopped at 98! the answer will be 9 can u please explain
Hoopologygirl2420:05:05
the answer would be negative.
connaissance20:05:05
the answer would be negative
AncientSC220:05:05
because 98 is negative
rrusczyk20:05:10
If we stopped at 98!, then our answer would be *negative*.
AncientSC220:05:28
it alternates between postigive and negative
rrusczyk20:05:31
Right.
rrusczyk20:05:43
1! -2! +3! -4! + . . . +97! - 98!
rrusczyk20:06:06
One more problem for MATHCOUNTS, and then we'll discuss the class.
rrusczyk20:06:24
rrusczyk20:06:51
Where do we start here?
MathTwo20:07:16
find p, p+1
AncientSC220:07:16
find prime numbers right next to each other
rrusczyk20:07:20
And what are they?
qwertythecucumber20:07:32
the only prime numbers one away from each other is 2 and 3
sxiaohu20:07:32
one is even and one is odd and the only even prime number is 2 so its 2 and 3
remember20:07:32
find the least values of p and p+1(?)
splatyango20:07:38
Only odd numbers are prime except 2. meaning it has to be 2 and 3
rrusczyk20:07:54
Since either p or p + 1 is even, one of them must be even, so one of them must be 2, the only even prime. Since 1 isn't prime, the other one must be 3.
Now, how do we find the smallest composite number that is neither a multiple of 2 nor 3?
rrusczyk20:08:02
Is the answe r5?
lxu120:08:17
5 isn't composit
sxiaohu20:08:17
5 is prime
batteredbutnotdefeated20:08:17
not composite
istos20:08:17
no
professordad20:08:17
No, because it isn't composite
rrusczyk20:08:39
Exactly. 5 is a prime, so it cannot be the answer to the problem. We need a composite number, not a prime number.
rrusczyk20:08:53
(A composite number is a number that has a factor besides 1 and itself.)
RoFlLoLcOpT20:09:14
but multiplying by 5 gives a composite answer, which is 25
llamafoo20:09:14
it is 5^2 becuase it is 5x5
bbhattacharya20:09:14
25
professordad20:09:14
You multiply 5 by itself.
Tribefan20:09:14
it's 25
O M G20:09:14
so would it be 25 (I think I did something wrong I don't know)
bpgbcg20:09:14
no, the lowest must be devisible by 5 and a number not divisible by 1,2,3,or 4. Thus it must be 5x5, or 25
rrusczyk20:09:34
Exactly; we need a number with two prime factors (at least), and we can use the same prime factor twice.
rrusczyk20:09:39
But we can't use 2 or 3.
rrusczyk20:09:46
So, the number we want is 5*5 = 25.
rrusczyk20:10:18
These problems are about average difficulty for the Basics class (maybe a touch on the easy side).
rrusczyk20:10:32
These problems are way, way easier than average for the Advanced class.
rrusczyk20:10:51
Before going on to the other two classes, I'll take questions now about the MATHCOUNTS class.
batteredbutnotdefeated20:11:01
let's do average for advanced class!
rrusczyk20:11:08
You'll see that in the Math Jam next week.
treeswitheyes20:11:15
How much do the MATHCOUNTS/AMC 8 classes overlap with the the intro series?
rrusczyk20:11:50
We don't have problems in common, but a lot of the concepts overlap. The Intro series classes are much more thorough, and go much deeper into the math than the Problem Series.
professordad20:11:52
You mentioned something about homework--how do you submit homework?
rrusczyk20:12:19
For this class, the homework is to discuss extra problems we put on the class message board. You do not turn in material to the instructor.
InsDel20:12:22
How will the classes be taught? Show problems and then discuss answers, like here?
rrusczyk20:12:34
The classes will be much like what you just experienced.
rrusczyk20:12:47
They'll go a touch faster because there are assistants to handle some of the questions.
k00lperson20:12:54
When in the next mathjam class?
rrusczyk20:13:02
One week. See the calendar on the site.
changle20:13:06
so advanced is for those who are preparing for state
rrusczyk20:13:19
For competitive states like CA and TX, etc., yes.
jiwhan20:13:23
what happens if we cant attend classes during certain days?
rrusczyk20:13:49
We make a full transcript of every class. This transcript is available to all students in the course within 24 hours of class-end, and you can review that whenever you want.
Tribefan20:13:52
Do you have to enroll in order to read the transcript?
rrusczyk20:14:07
The transcript for the course, yes. The transcript for this Math Jam, no.
rrusczyk20:14:37
Details for how to get the class transcript are in the Course Information handout you get when you enroll.
Hoopologygirl2420:14:50
how much does each course cost?
rrusczyk20:15:01
This is described on the Class Enroll page:
d10a10v1020:15:17
when we are in the real class will it be the same as how we type now and you read it before posting it?
rrusczyk20:15:19
Yes
Outlands20:15:26
how do i speak to you
rrusczyk20:15:32
You type as you just did.
Ari20:15:47
How do i enroll in a class?
rrusczyk20:15:52
Visit that link above.
inoit20:15:54
is tehre a textbook for teh mathcounts class, if so wat is it
rrusczyk20:16:02
There is no text for the MATHCOUNTS class.
Watermelon9920:16:05
what level does the class correspond to in regular school math
rrusczyk20:16:16
Our classes are all more challenging than regular school math.
batteredbutnotdefeated20:16:18
so the real class will look similar to this math jam in structure and view
rrusczyk20:16:20
Yes
treeswitheyes20:16:23
Will you ever post someone's wrong answer?
rrusczyk20:16:31
Rarely.
sxiaohu20:16:34
is there audio in other classes like the course?
rrusczyk20:16:37
No audio.
rrusczyk20:16:46
That explains why.
limac20:16:55
Do we get to take actual Mathcounts/AMC 8 tests in the classes?
rrusczyk20:17:09
We don't have actual tests, but we will have a class-wide Countdown round.
alexgnow20:17:38
So, homework is extra, for yourself right?
rrusczyk20:17:41
Yes
professordad20:17:43
How will the countdown round work out--like FTW?
rrusczyk20:17:50
Like the FTW with everyone in at once.
rshila20:17:53
what are the requirements to join the basic amc 8? is it too difficult for a person who is doing this for the 1st time?
rrusczyk20:17:59
We expect to have a lot of first timers.
kathyisjjmom20:18:01
oh yeah - do you go with the students' pace? or do the students have to keep up?
rrusczyk20:18:26
The students have to keep up -- they should ask questions in the course if they have questions. There will be assistants there to help (unlike tonight).
d10a10v1020:18:29
i am in algebra 1 now is this class too hard?
rrusczyk20:18:45
We will have many students in the MATHCOUNTS classes who are in algebra 1.
k00lperson20:18:49
where do we find the homework?
rrusczyk20:19:15
There is no homework for the Math Jam. For the courses, this is explained in the Course Information document.
Jobo20:19:20
where do we find the transcripts for AMC 8 after we enroll
rrusczyk20:19:36
On the Class homepage, which you access by clicking My Classes.
MathTwo20:19:50
You explained we take one session on the Countdown round. Do we take a session on the Sprint round, etc.?
rrusczyk20:19:59
No; all the classes will help with all the rounds.
rrusczyk20:20:24
Email me your math background at classes@artofproblemsolving.com if you are still unsure what class to take.
batteredbutnotdefeated20:20:44
is intro to geometry harder than this class
rrusczyk20:20:50
Yes, much, much harder.
kathyisjjmom20:20:53
how long of a wait will there be to answer a given question? because different people have different paces
rrusczyk20:21:08
When we have assistants in the class and are doing math, the answer will come quickly.
changle20:21:14
when is the deadline for enrolling?
rrusczyk20:21:36
10 days after the course starts (although the MATHCOUNTS class usually fills, so enroll earlier than that)
TXMath20:21:38
Will we eventually get solutions to all the problems including homewordk?
rrusczyk20:21:40
Yes
Bud20:21:43
will you take questions at the end of each problem?
rrusczyk20:22:19
You can ask questions at any time. The instructor won't deal with them all publicly; some will be answered in 1-1 chat with an assistant, or with whispers (an instructor can send a comment only to you).
cocoarules20:22:22
What if we don't have enough time to answer our questions for each problem?
rrusczyk20:22:40
If you have questions still after class, you can ask them on the class message board.
rrusczyk20:23:03
(We won't stop the entire class for one student -- the assistants will work with those students.)
Riley Wonderful Ace20:23:05
how does the whole thing work? is it like this? will you have different questions we wll need to answer?
rrusczyk20:23:15
Yes, this is how the class session works.
mathsmart34720:23:18
how many people will there be per class??
rrusczyk20:23:29
It varies a lot; 30-70
Bud20:23:34
what types of questions will there be?
rrusczyk20:23:50
All sorts; the type of questions that appear on MATHCOUNTS.
inoit20:23:56
is there a challenge set for the mathcounts class
rrusczyk20:23:58
No
sxiaohu20:24:18
Will it be more chaotic with so many replies popping up or will it be like a PM to a student?
rrusczyk20:24:30
The regular classes are not as chaotic as the Math Jams.
kathyisjjmom20:24:34
will we focus on different types of maths? such as geometry, algebra, trig, stuff like that?
rrusczyk20:24:52
We will cover a variety of areas of math in the MATHCOUNTS class. No trig, though.
llamafoo20:24:54
If I miss a class, can I read the chat transcript for that class?
rrusczyk20:24:56
Yes.
d10a10v1020:25:00
what is the difference between a class and a math jam?
rrusczyk20:25:23
This is a Math Jam. You enroll in classes, and all you do there is math (rather than talking about how the classes work).
kathyisjjmom20:25:29
for mathcounts, are we taking this class JUST for the competition, or can this help us in school work as well?
rrusczyk20:25:48
It will help with contests and with school (although it will be a good deal harder than school much of the time)
mathsmart34720:25:50
should we ever type in the answer or suggest how we solved the answer??
rrusczyk20:26:00
Both! You'll get a feel for when to do what.
dajoe20:26:07
are you really the true mr. rusczyk?
rrusczyk20:26:12
I look a lot like him.
rrusczyk20:26:31
Let's do a little more math now, and then I'll take more questions.
Watermelon9920:27:08
Can I have your autograph Mr.Rusczyk?
rrusczyk20:27:18
If you make it to National MATHCOUNTS :)
no.1nyyfan20:27:19
I would like to know about the number theory class
rrusczyk20:27:23
Your wish. . . .
rrusczyk20:27:24
Introduction to Number Theory
rrusczyk20:27:34
Many students who already know how to solve MATHCOUNTS level problems about divisibility, base numbers, and divisor counting might think they have little need for this class, but many of the students who have made the most of this class were participants at national MATHCOUNTS and the AIME and found many of the problems discussed in class very challenging.
rrusczyk20:28:09
(I will take more questions after we discuss this class; if I didn't get to yours, you can ask again, or write me at classes@artofproblemsolving.com)
frogbandit20:28:11
did Mewto get your autograph?
rrusczyk20:28:20
I think so, and I made sure to get his.
rrusczyk20:28:26
The Introduction to Number Theory class covers divisibility problems, clever uses of prime factorization, base numbers, linear Diophantine equations, the Euclidean Algorithm, and covers the mechanics of modular arithmetic in thorough detail. There are also many other topics covered in this course that are rarely mentioned in math books at all, but these topics become increasingly important as you move to higher mathematics.
rrusczyk20:28:56
For example, the topics covered in this course are crucial to an understanding of such areas as cryptography and computer science.
d10a10v1020:29:17
what is cryptography
rrusczyk20:29:21
Codebreaking.
rrusczyk20:29:26
And codemaking
rrusczyk20:29:35
The following mini-lesson is excerpted from one of the Number Theory classes.
rrusczyk20:29:41
COUNTING DIVISORS
rrusczyk20:30:01
Once we know how to find the prime factorization of numbers, we can begin to use this tool to solve other problems.
rrusczyk20:30:08
One such problem is answering the question, 'How many positive divisors does a particular integer have?' This kind of counting problem is common in number theory.
rrusczyk20:30:33
rrusczyk20:31:08
Let's start this discussion by considering the prime factorization of 200. What is it?
bbhattacharya20:31:37
2*2*2*5*5
AncientSC220:31:38
2*2*2*25
limac20:31:38
5^2*2^3
professordad20:31:38
2^3 x 5^2, I think
AncientSC220:31:38
2*2*2*5*5
treeswitheyes20:31:38
2^3x5^2
rrusczyk20:31:40
We have 200 = 2^3 * 5^2.
rrusczyk20:32:04
How can we use this to describe factors of 200? That is, tell me about the prime factorization of any factor of 200.
alexgnow20:32:27
What do you mean?
rrusczyk20:32:39
I mean, could a factor of 200 have 3 among its prime factors?
Watermelon9920:32:48
they contain 2 or 5
Amaryllis20:32:49
It will have either a 2 or 5 as a factor.
TXMath20:32:49
it has to be divisible by 2 or 5
rrusczyk20:32:53
Can it have anything else?
alexgnow20:32:59
No
frogbandit20:32:59
no SIR
Funkymunk20:32:59
No, it has to include numbers from the prime factorization of 200
rrusczyk20:33:10
Exactly; it has to have only prime factors that 200 has.
rrusczyk20:34:03
We can't have any other prime factors in our divisor.
rrusczyk20:34:16
Now, what can the power of 2 possibly be?
Outlands20:34:19
what does "^" mean again
Bud20:34:36
to the power of
limac20:34:37
"to the power of"
rrusczyk20:34:38
That's the symbol you can use for powers. 2^3 is 8, for example
rrusczyk20:34:44
Now, what can the power of 2 be?
treeswitheyes20:35:00
0,1,2,3
bbhattacharya20:35:00
a's max is 3 & b's max is 2
bbhattacharya20:35:00
0,1,2,3
Watermelon9920:35:00
0, 1, 2, or 3
treeswitheyes20:35:00
0,1,2,3
rrusczyk20:35:13
Right -- we can have 0,1,2, or 3 factors of 2 in a divisor of 200.
Watermelon9920:35:17
it cant be more than 3 though
rrusczyk20:35:20
Why not?
Watermelon9920:35:52
because 200 has only 3 twos
Hoopologygirl2420:35:52
200 doesnt include more than2^3
amanjaria20:35:53
because you only have three 2's available
professordad20:35:53
Because 200 divided by any power of 2 greater than 3 will not result in an integer
rrusczyk20:36:00
If the factor divides into 200 evenly, it can't have more than 3 twos in its prime factorization, because 200 only has 3 twos.
rrusczyk20:36:14
So, how many choices do we have for the power of 2?
treeswitheyes20:36:35
4
alexgnow20:36:35
4
limac20:36:35
4
bbhattacharya20:36:35
4
yankeesfan20:36:35
4 choices
tornado.adv420:36:35
4
MathTwo20:36:35
4 choices, 0, 1, 2, 3
rrusczyk20:36:45
How about the power of 5?
bpgbcg20:37:10
3
changle20:37:10
3
limac20:37:10
3
professordad20:37:10
You have three choices (0, 1, and 2)
changle20:37:10
3 choices 0,1,2
InsDel20:37:10
0,1,2
rrusczyk20:37:40
rrusczyk20:38:11
So, a divisor of 200 can have 0, 1, or 2 factors of 5. That means we have three choices for how many 5's the divisor can have.
rrusczyk20:38:19
So, how many divisors does 200 have?
bbhattacharya20:38:41
12
Riley Wonderful Ace20:38:41
12
O M G20:38:41
3x4=12
simi1520:38:41
12
treeswitheyes20:38:42
We have 4 choices for the powers of 2 and with every choice we can pair it with one of the possible powers of 5 so there are 12
sxiaohu20:38:42
12, 4 choices* 3 choices= 12
rrusczyk20:38:47
rrusczyk20:38:56
rrusczyk20:39:12
Nice explanation, treeswitheyes. Speaking of trees, we can see this more clearly with a tree:
rrusczyk20:39:20
rrusczyk20:39:50
Indeed we can see that this is true by listing the divisors:
1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, and 200 are all the positive divisors of 200.
amanjaria20:39:53
so these are the kinds of problems in Intro to Number Theory?
rrusczyk20:39:55
Yes
rrusczyk20:40:01
So, why is it that we multiply the number of possible values for the exponent of each prime together?
rrusczyk20:40:15
Why don't we add, instead?
limac20:40:36
because for EACH of the 4 we have 3
alexgnow20:40:36
for each possiblity, there is 3
llamafoo20:40:36
the cases are independent
rrusczyk20:40:45
The reason we simply multiply the numbers of values for the exponents together is because we can select the values for each exponent independently from the values of the other exponent(s).
Riley Wonderful Ace20:40:56
you first choose a value for a and then you choose a value for b
rrusczyk20:41:00
alexgnow20:41:07
for each possibility of a, there are 3, sorry
rrusczyk20:41:10
Exactly.
rrusczyk20:41:16
rrusczyk20:41:25
In general what can we say about the number of positive divisors of an integer n with a prime factorization
rrusczyk20:41:32
rrusczyk20:41:38
How would we find the number of divisors of that?
rrusczyk20:41:56
(That's the prime factorization written over there on the right.)
frogbandit20:41:59
what do the ps mean?
rrusczyk20:42:03
The p's are the primes
inoit20:42:07
what does e stand for
rrusczyk20:42:18
The e's are the exponents in the prime factorization.
rrusczyk20:42:27
How do we use them to count the number of divisors?
Riley Wonderful Ace20:42:29
what does the m stand for?
rrusczyk20:42:37
The number of primes in the prime factorization.
rrusczyk20:42:43
And the number of factors is . . .
remy114020:42:50
(e_1+1)(e_2+1)(e_3+1)...(e_m+1)
professordad20:42:50
You find (e1+1) x (e2+2) x ..... x (em+1).
sxiaohu20:42:50
(e1+1)*(e2+1)...*(Em +1)
bbhattacharya20:42:50
it has (e1+1)(e2+1)...(em+1).
rrusczyk20:43:00
rrusczyk20:43:11
rrusczyk20:43:26
rrusczyk20:43:34
To make this method more clear, we will now work through a couple exercises.
rrusczyk20:43:40
How many positive divisors does 60 have?
rrusczyk20:43:49
Where do we start?
professordad20:44:13
You find the prime factorization
llamafoo20:44:13
prime factorization
changle20:44:13
you find the prime factorization
tornado.adv420:44:13
prime factorization of 60
rrusczyk20:44:17
Which is?
MathTwo20:44:23
60=2^2*3*5
simi1520:44:23
prime factorization: 2x2x3x5
Hoopologygirl2420:44:23
2^2 *3*5
bpgbcg20:44:23
2^2*3*5
rrusczyk20:44:29
rrusczyk20:44:35
And how do we count the divisors?
sxiaohu20:44:52
12 choices, There are 3*2*2 choices
limac20:44:52
3*2*2
batteredbutnotdefeated20:44:52
3*2*2=12
ashmath20:44:52
3x2x2=12
Funkymunk20:44:52
3*2*2=12
rrusczyk20:45:33
rrusczyk20:45:47
We have 3 choices for x (0, 1, 2), 2 choices for y (0 or 1), and 2 choices for z (0 or 1).
HiDN42820:45:51
You add one to every exponent then multiply all together
rrusczyk20:45:54
Exactly.
rrusczyk20:46:13
skylord581620:46:16
HiDN42820:46:18
it would be (x+1)(y+1)(z+1)
rrusczyk20:46:21
Exactly.
rrusczyk20:46:39
We'll try one more number theory problem that's a little harder.
alexgnow20:46:51
Why do you add one to every exponent
InsDel20:47:35
To account for 0 being a choice
no.1nyyfan20:47:35
because of the possibility of 0
tornado.adv420:47:35
you have to include 0 as an exponent
limac20:47:35
because the exponents are 0-n inclusive
rrusczyk20:47:37
Suppose we have 2^n in the prime factorization. The choices we have for the power of 2 in a divisor then are 2^0, 2^1, 2^2, up to 2^n. There are n+1 choices there!
rrusczyk20:47:57
Exactly -- this is because you could choose 0 as your exponent. And:
Funkymunk20:47:59
Because of the amount of numbers in the list 0,1,2,3...e is e+1
rrusczyk20:48:04
Precisely.
rrusczyk20:48:09
Now, for the harder problem.
rrusczyk20:48:16
Find the number of positive integral divisors of 792 that are even.
rrusczyk20:48:28
First, we make the problem a little easier.
InsDel20:48:32
What is an integral divisor?
Augustine20:48:32
What are integral divisors?
rrusczyk20:48:42
divisors. Nothing fancy about "integral"
rrusczyk20:48:53
How many divisors does 792 have?
rrusczyk20:49:02
What do we do?
mathguy99920:49:11
prime factorize first
mathguy99920:49:11
prime fctorize
Hoopologygirl2420:49:11
prime factorization
rrusczyk20:49:16
And what is it
frogbandit20:49:23
the prime factorization is 2^3 * 3^2 *11
skylord581620:49:23
MathTwo20:49:23
2^3*3^2*11
rrusczyk20:49:29
rrusczyk20:49:36
So, what is the number of divisors of 792?
remember20:50:07
4*3*2 divisors
simi1520:50:07
4x3x2 = 24
changle20:50:07
There are 2x4x3=24
InsDel20:50:07
4*3*2
inoit20:50:07
24
ashmath20:50:07
4x3x2=24
alexgnow20:50:07
4*3*2=24
yankeesfan20:50:07
4*3*2=24 divisors
RoFlLoLcOpT20:50:07
rrusczyk20:50:11
InsDel20:50:16
4*3*12
professordad20:50:19
And then you eliminate all the odd divisors.
rrusczyk20:50:41
That's one way to do it--find the odd divisors and eliminate them. Is there another way?
alexgnow20:50:47
How do you eliminate the odd divisors?
rrusczyk20:50:55
Good question; how could we do that?
rrusczyk20:51:20
(We'll look at two solutions here.)
rrusczyk20:51:23
Here's one:
bpgbcg20:51:26
to find even divisors, you use only the ones that have at least one even factor
tornado.adv420:51:26
restrict the exponent of 2 (a) so that it is not 0
rrusczyk20:51:38
This is how we can find the divisors that must be even.
rrusczyk20:51:50
We make sure the exponent of 2 in our divisor is not 0.
rrusczyk20:52:02
rrusczyk20:52:11
tornado.adv420:52:33
3
simi1520:52:33
3
HiDN42820:52:33
There would 3.
Bud20:52:33
3
treeswitheyes20:52:33
3
rrusczyk20:52:40
professordad20:53:06
So you would calculate a x (b+1) x (c+1)
TXMath20:53:06
is the answer 18
yankeesfan20:53:06
3*3*2=18 even divisors
bpgbcg20:53:06
3*3*2=18
professordad20:53:06
3 x 3 x 2 = 18
rrusczyk20:53:09
rrusczyk20:53:33
rrusczyk20:53:43
Now, how could we have counted the odd divisors?
tornado.adv420:54:17
all values where a = 0
qwertythecucumber20:54:17
make 2^0
bbhattacharya20:54:17
1*3*2=6
bpgbcg20:54:17
eliminate the 2^a term and solve for the number of devisors
TXMath20:54:17
restrict a to 0
professordad20:54:17
We could count (2+1) x (1+1)
RoFlLoLcOpT20:54:21
restrict a to 0
llamafoo20:54:21
restrict a to 0
rrusczyk20:54:23
We can count the odd divisors by making the exponent of 2 equal 0.
rrusczyk20:54:51
Then, we still have 3 choices for b and 2 for c, which means we have 3*2 = 6 odd divisors.
rrusczyk20:55:05
Then, we could subtract this from 24 to get 18, as before.
rrusczyk20:55:14
Here's a question from earlier:
changle20:55:17
why do you make sure the exponent of two is not 0
rrusczyk20:55:26
This was when we were counting even divisors.
qwertythecucumber20:55:45
every even number has a 2 in its factorization
tornado.adv420:55:45
if there is a 2 in the prime factorization of a number, it will be even
rrusczyk20:55:48
We made the exponent of two nonzero to force the divisor to be even.
batteredbutnotdefeated20:56:12
is the math jam over in 5 min?
rrusczyk20:56:19
We'll go a little longer, for sure.
Riley Wonderful Ace20:56:30
would the problems in the number therory class be cover in public school? I am in 6th grade.
rrusczyk20:56:46
No; much of what we teach in the number theory class is not taught in middle or high school.
rrusczyk20:56:53
As the course continues we begin to discuss more difficult questions such as 'How many of the divisors of 360 are even?' and 'How many of the divisors of 9800 are perfect squares?'
rrusczyk20:57:00
Here are a few more kinds of problems that we will be tackling in class:
rrusczyk20:57:08
rrusczyk20:57:16
rrusczyk20:57:21
rrusczyk20:57:25
rrusczyk20:57:30
rrusczyk20:57:34
rrusczyk20:57:40
The Introduction to Number Theory class will be taught by Naoki Sato. Naoki joined AoPS in 2005 after a successful career in investment banking. He won first place in the 1993 Canadian Mathematical Olympiad, and represented Canada at the 1992 and 1993 International Mathematical Olympiads, winning a bronze and silver medal, respectively. He has also served as deputy leader for the Canadian IMO team in 1997, 2002, and 2006. A native of Toronto, Canada, Naoki earned a Bachelor's in mathematics from the University of Toronto, and a Master's in mathematics from Yale University.
rrusczyk20:57:48
One more word about this number theory class. If you do not know modular arithmetic well enough to use it on most any problem up to AMC-12 level, this is a class that you would benefit from taking.
rrusczyk20:57:52
The course will meet for 12 weeks on Thursdays, starting June 4. Each class starts at 7:30 PM Eastern / 4:30 PM Pacific, and is 90 minutes long.
kathyisjjmom20:57:57
will this help us in our studies? because i'm not planning to compete in anything
rrusczyk20:58:28
Yes; the approach to math we teach is the approach that I used to make all of my college classes easy, and not just math classes.
treeswitheyes20:58:31
I don't know modular arithmatic at all, is that ok?
rrusczyk20:58:38
Then this is a good course for you.
Riley Wonderful Ace20:58:41
what is a diophantine equation?
rrusczyk20:58:51
An equation for which we only want integer solutions.
yankeesfan20:58:55
I would like to know more about the Intro to Geometry class
rrusczyk20:59:01
We'll get to that in a few minutes
skylord581620:59:11
I'm not familiar with the term, "modular arithmetic." What does it mean?
rrusczyk20:59:35
That's a sign that this would be a good course for you! It's one of the big ideas of number theory, and will take a good long time to explain.
limac20:59:38
The Intro to NT will cover everything in the Intro to NT book?
rrusczyk20:59:41
Pretty much.
Plutonium20:59:51
if I think some of the number theory is comfusing, what course should I take?
rrusczyk21:00:00
It sounds like the number theory class would be good for you.
MathTwo21:00:02
When will the book intermediate number theory come out?
rrusczyk21:00:08
Not for years.
Outlands21:00:10
i have to go now. can i be excused for the night.
rrusczyk21:00:17
Sure. You can leave whenever you like.
Watermelon9921:00:20
How long are the classes and math jams?
rrusczyk21:00:25
Classes are 90 minutes.
rrusczyk21:00:32
Math Jams go until we're finished.
vcez21:00:36
Is intermediate geometry out yet?
rrusczyk21:00:38
No
kathyisjjmom21:00:57
WHo writes these books?
rrusczyk21:01:06
Me :) And other people who love math and love teaching.
MuffinMan0921:01:13
if we have the NT book do you suggest taking the class too?
rrusczyk21:01:46
The class offers structure, other students to work with, a very experienced instructor, and detailed feedback on your work.
rrusczyk21:01:49
Speaking of which:
rrusczyk21:01:54
The homework for the class consists of weekly problems that will be posted to the class message board -- for these problems, you do not turn your solutions in, however you may post them to the message board if you like. The class also has 4 longer problem sets for which you should write up your full solutions and submit them. These solutions will be evaluated by AoPS graders, and you will receive detailed feedback.
kathyisjjmom21:01:56
WE have mathcounts at my school, does it teach the same material?
rrusczyk21:02:05
Only if you are at a very, very, very, very good school.
batteredbutnotdefeated21:02:08
What is NT?
rrusczyk21:02:13
Number Theory.
remy114021:02:20
Why is the Intermediate Number Theory seminar called a "seminar" instead of a regular class?
rrusczyk21:02:37
No Challenge Sets in the Intermediate class (homework to be turned in)
alexgnow21:02:45
How would math competitions look like on a college resume?
rrusczyk21:02:54
Good. MIT and CalTech explicitly ask for them.
ashmath21:03:00
I teach some of it in my middle school! I'm a 6th-8th grade teacher and I'm going to have a bunch of kids in my class next Wednesday for your next Math Jam! We had 7th in state math counts this year and I think this will help the kids as well as improve their problem solving skills. Thanks for offering the classes!
rrusczyk21:03:21
Sounds like your kids have a very good teacher!
relagha21:03:25
do we have challenge sets in this class
rrusczyk21:03:36
Yes. There are 4, one every three weeks.
kathyisjjmom21:03:40
If i think this Math Jam was already hard, i should be taking something really easy right?
rrusczyk21:04:17
You should take either Algebra 1 or the Intro Number Theory class - - those classes have books to support them, and are not at the breakneck speed of the problem series class.
Bud21:04:19
if i thought this was hard should i take algebra 1?
rrusczyk21:04:23
Probably.
kathyisjjmom21:04:26
How do you have all this time to teach these classes?
rrusczyk21:04:29
Time machine.
Superman62621:04:34
Number Theory is after Algebra right?
rrusczyk21:04:48
Yes; we recommend that students have some exposure to algebra before the number theory class.
augnov21:04:56
which course do i take first? Algebra1 or Nmber theory
rrusczyk21:05:09
If you have never taken an algebra class, we'd recommend Algebra 1 first.
rrusczyk21:05:18
After that, either Intro Number Theory or Intro Counting.
treeswitheyes21:05:20
Is pre algebra enough?
rrusczyk21:05:35
For many students, prealgebra is enough for our Intro Number Theory.
frogbandit21:05:49
is this math jam finished? its been 90 minutes
rrusczyk21:05:55
We'll be going over quite a bit.
alexgnow21:06:11
What is the order of classes recommended?
rrusczyk21:06:25
After NT/Counting comes Intro Geometry (the hardest of the Intro courses).
rrusczyk21:06:36
Then Algebra 3 and then the rest of the Intermediate courses in any order.
MathTwo21:06:39
whens intermediate geometry out
rrusczyk21:06:42
Not for years.
Funkymunk21:06:51
Would it be non-wise to take Geometry before NT?
rrusczyk21:07:00
Depends. We have plenty of students do it.
augnov21:07:06
i will take algebra1 in this Sep. at school. If I take Algebra1 in here, isn't it boring at school?
rrusczyk21:07:10
That could well happen.
rrusczyk21:07:24
Of course, Algebra 1 in school could be boring even if you don't take our class :)
li0854021:07:52
Do you have any classes in the morning or weekend?
rrusczyk21:07:56
Not currently.
TXMath21:07:58
Does the intro Geometry cover material that is not typical of a class at school?
rrusczyk21:08:10
Yes; it covers much more challenging material and is more rigorous.
O M G21:08:15
When will this end I have a state test tomarrow?
rrusczyk21:08:21
You can leave whenever you like.
changle21:08:24
what are number bases
rrusczyk21:08:30
Take the class and see ;)
limac21:08:32
Is proof first introduced in the geometry courses?
rrusczyk21:08:43
No; we do proof in all our classes!
dajoe21:08:57
im in algebra one, is the intro to geometry too hard?
rrusczyk21:09:16
You can look at the diagnostic test on our site to get an idea of that.
Lilac21:09:32
Can we get on to Geo?
rrusczyk21:09:57
Introduction to Geometry
rrusczyk21:10:03
In the Introduction to Geometry class we cover all the fundamentals of geometry. We will start with a few days covering the basic tools such as triangle congruence, similarity, power of a point, relationships between angles and circles, etc., then dive into using those tools and more to solve increasingly difficult problems.
rrusczyk21:10:10
Most of the problems in the course will be at the MATHCOUNTS and AMC-10 level of difficulty, but we will be throwing in a few harder problems occasionally to show how to use very basic ideas to solve very challenging problems.
amanjaria21:10:12
just one last question: what would you say is the daily time commitment required by students
rrusczyk21:10:29
5-8 hours/week in the subject classes, 3-4 hours a week for MATHCOUNTS/AMC classes.
rrusczyk21:10:35
Here's a sample problem:
rrusczyk21:10:45
In the diagram shown, DEOC is a square. The radius of circle O is 6 in. What is the number of inches in AC? Express your answer in simplest radical form.
rrusczyk21:10:53
rrusczyk21:11:06
rrusczyk21:11:20
In order to find AC, what will we need?
MathTwo21:12:12
measure of OC
treeswitheyes21:12:12
OC
bbhattacharya21:12:12
AO & OC
yankeesfan21:12:12
The length of OC.
bpgbcg21:12:12
the pythagorian therum, AO, and OC
rrusczyk21:12:16
We need AO, which we already have, and OC. How can we find OC?
rrusczyk21:12:29
Here is a link for the figure:
professordad21:13:07
And to find CO, you would find the diagonal of square DEOC and divide that by sqrt2
changle21:13:07
you can find oc by using the pythagorean theorem because you know the diagonal of the square
rrusczyk21:13:28
OC is a side length of a square. If we could find the diagonal, we could find the side length. How can we find the diagonal?
rrusczyk21:13:34
Ari21:13:55
That is the radius
batteredbutnotdefeated21:13:55
it's the radius of the circle
treeswitheyes21:13:55
its the radius
InsDel21:13:55
OD is 6in, being a radius
alexgnow21:13:55
radius, 6
rrusczyk21:14:07
We know the 6 inch radius of the circle is also the diagonal of the square. Now what is the side length of the square?
batteredbutnotdefeated21:14:35
6/sqrt2
Watermelon9921:14:35
6/sqrt(2)
Ari21:14:35
3 times the square root of 2
O M G21:14:35
3 root 2 because 6 divided by root 2
rrusczyk21:14:41
Because the diagonal has length 6 and OC is a side, we have:
rrusczyk21:14:45
rrusczyk21:14:51
rrusczyk21:14:59
And how do we find AC?
Funkymunk21:15:36
Phythagoream Theorem (However you spell it -.-)
MathTwo21:15:36
6^2+3sqrt2^2
professordad21:15:36
You find the square of 3sqrt2, add that to 36, and take the square root of that sum
Superman62621:15:36
then we use the pythagorean theorem
rrusczyk21:15:49
We have a right triangle, so we use the Pythagorean Theorem. What do we get as our answer?
bpgbcg21:16:03
6^2+3squrt2^2=AC^2
Plutonium21:16:16
3sq root of 6
O M G21:16:16
3 root 6
changle21:16:16
6^2+3sqrt2^2 so you get sqrt54
topofmath21:16:16
6^2 + (3sqrt2)^2 = 54, sqrt54 = 3sqrt6
rrusczyk21:16:21
rrusczyk21:16:35
Here is one question I saw a couple times:
yankeesfan21:16:41
How do you get 3sqrt2?
rrusczyk21:17:03
How did we get 3sqrt(2) for CO?
rrusczyk21:17:08
topofmath21:18:16
COsqrt2 = DO, DO = 6
professordad21:18:17
We got 3sqrt2 for CO by finding DO / sqrt2
rrusczyk21:18:39
If you don't know the relationship between the side of a square and the hypotenuse
Funkymunk21:18:41
We can divide by sqrt(2) since the diagon of a square is sqrt(2) * s, where s is the length of a side
rrusczyk21:18:51
you can also use the Pythagorean Theorem
rrusczyk21:18:53
We know that OC = CD and in right triangle OCD, we have OC^2 + CD^2 = OD^2, so OC^2 + OC^2 = OD^2. We know that OD = 6 (it is a radius), so 2OC^2 = 36, which means OC^2 = 18.
rrusczyk21:19:06
We'll do one more problem.
rrusczyk21:19:13
Sector OAB is a quarter of a circle of radius 3 cm. A circle is drawn inside this sector, tangent at three points as shown. What is the number of centimeters in the radius of the inscribed circle? Express your answer in simplest radical form.
rrusczyk21:19:18
rrusczyk21:19:37
Our goal is to find the radius of the small circle.
rrusczyk21:19:41
img=http://www.artofproblemsolving.com/Classes/IntroGeom/Images/03624113.gif
rrusczyk21:19:48
That is a link to the picture
alexgnow21:19:51
What's a tanget, it's popped up on the SAT soo many times
topofmath21:20:23
Tangent is line touches circle once
rrusczyk21:20:26
A line is tangent to a circle if it touches the circle in one point. Two circles are tangent if they touch at only one point.
rrusczyk21:20:33
What are we likely to do in this diagram?
Mewto5555521:20:37
draw some radii to points of tangency
rrusczyk21:21:20
We want the radius of the small circle, so we will draw some radii. But why draw the ones Mewto says to (other than "because Mewto says to", of course)
topofmath21:21:44
radius is perpindicular to tangent?
rrusczyk21:22:33
We draw radii to the points of tangency because such a radius is perpendicular to the tangent. And why do we like that?
InsDel21:22:35
To make right triangles?
rrusczyk21:22:41
We really like right angles.
topofmath21:22:47
Yay right triangles
rrusczyk21:22:50
And right triangles.
rrusczyk21:22:58
Draw a radius to each point of tangency and label each with length r.
rrusczyk21:23:03
rrusczyk21:23:07
Now what?
sxiaohu21:23:24
you have a square
rrusczyk21:23:38
We notice that HOGP is a square. How will the square help us?
Funkymunk21:24:25
batteredbutnotdefeated21:24:25
its diagnol
professordad21:24:25
The diagonal, PO, is rsqrt2
rrusczyk21:24:45
And why does the diagonal of the square help?
batteredbutnotdefeated21:24:53
makes a triangle?
rrusczyk21:24:56
What else?
topofmath21:25:25
It intoduces radius of quarter circle
topofmath21:25:25
PO + r = 3
professordad21:25:25
And that makes FO, which is radius of circle O, so you know FO is 3
Funkymunk21:25:25
rrusczyk21:25:31
Drawing a diagonal in the square will give us a radius of the large circle (FO) in terms of r.
rrusczyk21:25:35
Because the side length of the square is r, the diagonal is r*sqrt2.
rrusczyk21:25:42
bbhattacharya21:25:56
completes big radius
rrusczyk21:25:58
Now we're close. How do we finish?
topofmath21:26:26
r + rsqrt2 =3
professordad21:26:26
r(sqrt2+1)=3
changle21:26:26
r+rsqrt2=3
rrusczyk21:26:40
We set up an equation and solve it. We know that OF is a radius of the quarter circle, so OF = 3. However, we can write OF = OP + FP = r*sqrt(2) + r.
skylord581621:26:55
rrusczyk21:26:58
Now we have an equation we can solve for r. What is r?
professordad21:27:01
Use distributive property and you get r(sqrt2+1)=3
rrusczyk21:27:06
Keep going.
yankeesfan21:27:17
r=3/(1+sqrt(2))
professordad21:27:17
r = 3/(sqrt2+1)
rrusczyk21:27:23
topofmath21:27:30
simplify
1330221:27:30
r = 3/(sqrt2+1)
rrusczyk21:27:35
How do we simplify?
Funkymunk21:28:00
Conjugates
topofmath21:28:01
3sqrt2 - 3
limac21:28:01
rationalize
professordad21:28:01
Multiply by (sqrt(2)-1)
bbhattacharya21:28:01
*sqrt2-1
changle21:28:01
you multiply it by 1-sqrt2
tornado.adv421:28:01
multiply by the conjugate to rationalize the denominator
rrusczyk21:28:05
We multiply both the top and bottom by 1-sqrt2.
rrusczyk21:28:10
rrusczyk21:28:29
(Don't worry too much about that last step -- we teach that in the algebra classes, not the geometry classes.)
rrusczyk21:28:36
Notice that we don't just sit and stare at the problem and wait for it to solve itself. We have to add lines and variables so we can build equations.
rrusczyk21:28:41
The first of those two problems is on the easy end of problems we will discuss. The second is a bit easier than average. All the geometric tools we use to solve problems, such as all the special relationships we used to solve these two problems today, will be taught in the class. We don't expect students to have any background knowledge in geometry.
rrusczyk21:28:49
You can find more questions like those we cover in the course by checking out the Post Test for the course here:
rrusczyk21:29:05
This course is a full geometry course. The equivalent (and a bit beyond) of a typical honors geometry course.
rrusczyk21:29:08
The course will meet for 24 weeks on Tuesdays, starting June 2, at 7:30 PM Eastern / 4:30 PM Pacific. Each class is 90 minutes.
rrusczyk21:29:13
The Introduction to Geometry class will be taught by Naoki Sato. Naoki joined AoPS in 2005 after a successful career in investment banking. He won first place in the 1993 Canadian Mathematical Olympiad, and represented Canada at the 1992 and 1993 International Mathematical Olympiads, winning a bronze and silver medal, respectively. He has also served as deputy leader for the Canadian IMO team in 1997, 2002, and 2006. A native of Toronto, Canada, Naoki earned a Bachelor's in mathematics from the University of Toronto, and a Master's in mathematics from Yale University.
rrusczyk21:29:24
This course will use a textbook in conjunction with the course: our own Introduction to Geometry book. The material covered in the textbook is roughly equivalent to the material covered in the course. You can see the table of contents and some excerpts from the book here:
rrusczyk21:29:39
The book is required for the course. Students will be able to read additional material that complements the lectures, and will have access to a large number of practice problems at varying levels of difficulty. We recommend that students read the corresponding chapter(s) in the book before each lecture, and attempt some of that chapter's Review and Challenge Problems after each lecture.
rrusczyk21:30:00
The homework for the class consists of weekly problems that will be posted to the class message board -- for these problems, you do not turn your solutions in, however you may post them to the message board if you like. The class also has 8 longer problem sets for which you should write up your full solutions and submit them. These solutions will be evaluated by AoPS graders, and you will receive detailed feedback.
rrusczyk21:30:15
We are finished doing math tonight. I will stay and answer questions about the courses.
topofmath21:30:25
when can we get transcript for t his mathjam
rrusczyk21:30:36
In the transcripts section of the Math Jam page on the website.
limac21:30:38
Is the geometry a summer program?
rrusczyk21:30:47
It starts in the summer and runs through the fall.
changle21:30:49
will there be the same classes later on in the year?
rrusczyk21:30:51
yes
shikido21:30:56
How are special symbols like radicals entered?
rrusczyk21:31:15
With LaTeX. You can read about it on the website (click LaTeX on the side of the website).
InsDel21:31:34
If you are away from a class, how would you be affected by turning in the solution sets late?
rrusczyk21:31:48
That's usually OK, just warn us ahead of time.
limac21:31:51
Another question, is doing the geometry from Vol 1 enough or should I also go through the Intro to geometry?
rrusczyk21:32:17
Depends on what kind of student you are. If you find the Volume 1 problems pretty easy, then you don't need the intro class.
inoit21:32:20
is a negative number (such as -3, -5,-7 or -(any prime number) ) composite or prime, since the number can be broken into -1x that number?
rrusczyk21:32:40
The rules for negatives (when we speak of them being prime or composite) are basically the same as their opposites.
rrusczyk21:32:42
-3 is prime
rrusczyk21:32:46
-6 is composite
rrusczyk21:32:50
-1 is neither.
MathTwo21:32:53
In the Intro Geometry book it said the Intermediate Geometry book should be coming out in 2009. Is there a misprint in the book?
rrusczyk21:33:01
The best laid plans of mice and men . . .
rrusczyk21:33:10
Things haven't gone according to plan ;)
rrusczyk21:33:16
We built a lot of other things instead.
treeswitheyes21:33:19
My computer won't download LaTeX, is there another program?
rrusczyk21:33:30
You can still use it on the site without it being on your coputer.
InsDel21:33:32
Will I be able to access this overseas (i.e. timezone problems?)
rrusczyk21:33:37
Yes, you should be able to
skylord581621:33:39
What if I miss the first two weeks of class? I am going to be in transit to Canada and probably won't recieve internet.
rrusczyk21:33:57
That's OK. Bring the book with you and try to keep up with that, and read the class transcripts when you get back.
Funkymunk21:33:59
Does the geometry class focus on proofs?
rrusczyk21:34:08
Yes; there are a lot of proofs in the geometry class.
rrusczyk21:34:17
(And not the annoying two-column proofs, either.
simi1521:34:20
will the intro to amc8 course go through the fall
rrusczyk21:34:25
No, it is just the summer.
Superman62621:34:27
whats LaTeX?
rrusczyk21:34:34
A typesetting language for math.
rrusczyk21:34:45
The way most professional mathematicians write math documents.
Superman62621:35:01
do we need to use it?
rrusczyk21:35:04
No
rrusczyk21:35:11
Any more questions about the courses?
InsDel21:35:18
I've also seen it for publishing other (nonmath) books, too.
rrusczyk21:35:23
Absolutely.
Superman62621:35:34
what's the age range for the Algebra class?
rrusczyk21:35:44
Pretty wide. 5th - 9th grade.
inoit21:35:47
can u use latex w/o downloading it in the classroom
rrusczyk21:35:49
Yes.
rrusczyk21:35:58
We will explain how at the start of the course.
Funkymunk21:36:05
Should we use Mathematica for typing up challenge set solutions if we have it, or would LaTeX be better?
rrusczyk21:36:22
I don't know of many people using Mathematica for typesetting.
rrusczyk21:36:49
If you have any more questions about the classes later, please write me at classes@artofproblemsolving.com.
skylord581621:36:55
Can I be in the AMC8? I homeschool.
rrusczyk21:37:16
Yes, but you'll have to ask the AMC how to take the contest. Their website is www.unl.edu/amc
MathTwo21:37:40
I'm homeschooled, and I took the AMC8.
inoit21:37:43
should u review the textbook before a class?
rrusczyk21:37:52
I very, very, very strongly recommend you do this.
rrusczyk21:38:05
I did this all the way through college, and it was part of the reason I found college very easy.
rrusczyk21:38:31
Since I had reviewed the material once before class, during class I could focus on the harder parts of the material rather than getting bogged down with the easy stuff.
rrusczyk21:38:43
Work ahead. It makes everything easier.
rrusczyk21:38:53
Thanks for coming tonight; I look forward to working with many of you in classes in the coming years!
Superman62621:39:14
What's your favorite math competition?
rrusczyk21:39:22
AMC, ARML, MATHCOUNTS, Mandelbrot. There are so many good ones it's hard to pick a favorite! See you all on the site later!
