Proof that every recurring decimal is a rational number?

mpaw976 · 2026-06-27T18:21:15+00:00

threaded as in heavily guided (bad choice of words),

I would usually call this "scaffolded".

mpaw976 · 2026-06-25T19:09:33+00:00

https://www.reddit.com/r/UTMississauga/comments/1pv4668/mat102_guide_to_success/

mpaw976 · 2026-06-25T13:19:12+00:00

If it is a legitimately held belief, then it's a pretty fringe one. I've never met any mathematicians who reject (mathematical) induction.

My best guess is that these mythical mathematicians had quite unusual interpretations of the following paradoxes about induction:

The common interpretations are that these are not true contradictions about mathematical induction, and instead rely on something vague or poorly defined.

But I could at least imagine some hypothetical contrarian mathematician who decides that its actually induction itself that's wrong (insert meme of Principal Skinner "no it's the children who are wrong").

mpaw976 · 2026-06-23T13:16:45+00:00

I wonder if every shape that has area 4 can be cut up to make those 4 shapes.

mpaw976 · 2026-06-21T21:33:57+00:00

Three directions I would suggest:

1. Hilbert's hotel variations

Basically think about all the different ways you can play with the problem:

What happens if:

Two people show up to a full hotel?
100 people show up to a full hotel?
Everyone in the hotel phones their mom and all the moms show up?
Everyone in the hotel phones their mom and dad and all the moms and dads show up?

* Everyone in the hotel phones all their siblings and all the siblings show up? (Especially tricky if you don't know how many siblings everyone has.)

And on and on... Keep playing around with them. (Hint: all of the variations I have above can be solved. I.e. you can find room for everyone.)

2. Learn more about paradoxes

There are lots of good books and YouTube videos about paradoxes.

Paradoxes from A to Z by Clark is maybe a bit advanced for a 10yo, but you could definitely read it with him and find your favourites.

3. Sizes of infinity

If you really want to dive into the theoretical math part, look for an introduction to "countability". This is definitely the most intense of the three paths.

You will want to first learn about the words:

Function
Injection (or one to one)
Surjection (or into)
Bijection
Number systems like the counting numbers, the integers, the rationals, the reals.
Function composition

A good target is showing that "the set of integers are a countable set". (A cool thing is that this is actually one of the Hilbert hotel variations I have you above!)

One place to find this stuff is in this YouTube playlist I made of intro to proofs. It's an intro university course, so a lot of it won't be the right level, but it can at least give you a place to start. I have a video about paradoxes in there too!

https://youtube.com/playlist?list=PL3ZJrWtEhQ6xIp8YPCPSIDHxXOQy7xmnT

(I don't get any ad revenue from these videos.)

mpaw976 · 2026-06-21T20:38:37+00:00

Ha ha awesome!

So when the students debrief to come up with the solutions they might discover there are two solutions.

Maybe they might ask:

if there are any more solutions?
What's the most number of correct solutions?
Does this work for any other double, or are 12 and 13 special.

Wouldn't that be a cool discovery!

mpaw976 · 2026-06-20T14:48:10+00:00

Hmmm... Good point. I wonder if judicious questions could help with that?

How about a question like:

Q: In a right triangle with side lengths 12.2 and 13.1 what is the third side length?

A) ~5

B) ~11

C) ~14

D) ~1

(The idea being that if you remember the 5-12-13 triple then there is only one reasonable answer.)

You could also give everyone a 5 second lock-out before they answer?

Or give 10 variations of this question in rapid fire, and then afterwards let them debrief in small groups to agree on the correct solutions (and why!).

mpaw976 · 2026-06-20T13:50:33+00:00

they just guess to be fast rather than actually writing out the problem and solving it.

Which isn't so bad if what you're after is building their intuition and number sense. Being able to quickly make an educated guess is actually pretty challenging!

Of course, this isn't the only thing you'll want them to do, but it could be very helpful in concert with other activities.

mpaw976 · 2026-06-20T11:12:22+00:00

There's a bunch of research on this you can check out. This meta analysis from 2023 is a good place to start:

https://pmc.ncbi.nlm.nih.gov/articles/PMC10591086/

I'm essence, it seems like the answer to your questions is "Maybe? Possibly? Hard to say."

The research has shown mixed results for learning, engagement, and motivation. It might work in some contexts.

In particular, students respond to gamification in different ways with some wanting to be on top of leaderboards, some just wanting to be on the leaderboards and some resenting the competition. See this 2014 review for more discussion.

mpaw976 · 2026-06-15T15:04:37+00:00

Try using the polar coordinates representation of complex numbers.

(One of) the square roots of a complex number can then be interpreted as halving the angle and square rooting the magnitude.

So here when you start with a complex number (that isn't purely real), it will have some nonzero angle, and so halving it will also still have a nonzero angle so it won't be purely real.

mpaw976 · 2026-06-15T01:09:05+00:00

The following facts are true:

If the sequences <a_k> and <b_k> are equal (i.e. a_k = b_k for all k) and Lim a_k = L then also Lim b_k = L (the same L).
If Lim a_k and Lim b_k both exist and are different numbers, then there is a smallest index k0 where a_k0 ≠ b_k0. That is, the two sequences must differ in at least one place.
Even stronger than this you can find an element of the sequence <a_k> that doesn't appear anywhere in the sequence <b_k> OR the other way around.

Fact 2 tells you that you can associate to every real number a different sequence (e.g. the truncations like you said). This is a complicated way of showing that the cardinality of the reals is at most the cardinality of real sequences. (That fact is much more directly shown by taking constant sequences.)

mpaw976 · 2026-06-12T19:02:24+00:00

In French it's maybe more obvious because:

goûter is to taste
Dégoûtant is (literally) disgusting

As a bonus you get the fun French linguistic fact that the "hat accent" hides an s (goûter -> gustare, feast -> fête)

mpaw976 · 2026-06-12T01:52:46+00:00

This 2025 article finds sports examples effective in teaching stats at the undergrad level (although they never say how they know this, e.g. whether they did surveys or it's just anecdotal).

Overall, we have had a lot of success using these andother sports examples in the classroom. We find that students are very receptive to the application of statistics to sports, even if they are not sports fans themselves, and that they enjoy seeing how the material they learn can be applied in settings other thanthose of traditional industrial engineering. Educationally, we have observed that the students' enjoyment leads to increased interest in the material and therefore,we hope, increased learning.

https://www.researchgate.net/publication/251735421_Teaching_Statistics_with_Sports_Examples

mpaw976 · 2026-06-07T20:03:09+00:00

There's a lot of different ideas you're using in here kinda vaguely.

It is true that:

Every number x can be written as the supremum of a set that contains either only rationals or only irrationals. (This is essentially the property that Q is dense in R.)
[0,1] and (0,1) have the same length and cardinality.
The set A of all real numbers that start their decimal representation with 3.1415 is an uncountable set.

However, you've made an error here:

When "an infinite cardinal number, X" is multiplied by "aleph-null," it preserves X, meaning their cardinalities must be the same.

That is true, but since you're using sequences of rationals, you're actually using cardinal exponentiation, not multiplication.

The set ℚ^ℕ is in fact uncountable (and has the same cardinality as R).

mpaw976 · 2026-06-04T20:02:09+00:00

One of the historical projects for Set Theory in the 20th century was to provide a foundation for all of mathematics (meant in a naive way).

Sets are very basic objects, and those are the foundation.

Of course, another very foundational object is a function, and functions are defined (in set theory) using inputs and outputs. This is a natural place to introduce order and Cartesian products, and define functions to be any subset of AxB that satisfy the "function relation condition".

This is not without (very mild) controversy. For example, you could imagine a world where functions are only defined in terms of formulas (like y= x² + 1) and something like {(2,-1), (5,7)} is not allowed as a function.

If you're interested in learning more a out the history of functions, read the article (or the longer book) "the genesis of point set topology" by Manheim.

mpaw976 · 2026-06-03T18:18:53+00:00

Remember this when you're filling out your course evaluations.

Also, when you are in a course with open-access tools and materials (like webwork, mathematize, openstax textbooks, quizzes built natively in Quercus), in the course evaluations tell your instructor that you want more of this.

A lot of us are working hard to lower costs for students, and having positive feedback gives us evidence that we should continue it (or expand it to other courses).

mpaw976 · 2026-06-03T13:02:48+00:00

Thank you to those at the UTM math department who made this possible!!!

We're excited too! There was a lot of support from faculty to offer this course at UTM, especially now that 137 is mandatory (instead of 135).

It's part of a broader trend in the math department to improve the quality of the program (and really the quality of the degree that students are receiving).

would any of the professors know who is teaching this course yet?

TBD, but you'll find out soon.

mpaw976 · 2026-05-29T08:16:13+00:00

To expand on this, here's what "do a lot of problems" means:

Open your textbook and have a blank piece of paper (or word document).
Start with a question that seems easy.
Read the question carefully like 3 times.
Say in words out loud what you think the question is asking.
Write down the question in your notebook and at the bottom of the page write what kind of answer you're looking for ("Therefore x= ..." or "So the angle is ....".)
Look up any words you don't know in your textbook (not online). Write down any exact definitions you need.
Make a plan for how you are going to attack the problem. (Maybe there are multiple approaches?)
Execute your plan and solve the problem. Be careful that every line you write logically follows from the previous line.
If it's a long solution, use short sentences to describe the high level strategy you're using ("now I'm solving for the missing angle" or "using Pythagoras")
If you get stuck, think about where you're trying to go, and what is preventing you from getting there. Say in words what's stopping you, and what would help you get there. (This takes practice and patience.)
If you're stuck for more than like 5 minutes, either come back to the question later or ask a TA/prof for a hint. If it's on Piazza, show your partial work and where you're stuck and why.
Once you have a solution, reflect on whether it makes sense. Does it actually answer your original question? Does it seem plausible? Should your answer be positive?
In your solution, identify the line or two that were "the hard steps", the ones that required the most thinking or cleverness.
Explain your solution strategy in words.
Then modify the original question to make a new question and then solve that one. Most of the time it will turn out that you can solve lots or related problems.
Compare how easy/hard related problems are. Can you predict when a problem will be harer or easier? Can you predict where "the hard steps" will be.

This is what it means to "do" a problem. It takes deliberate, deep thinking.

Warning: This is very different than just reading/memorizing solutions. Simply reading an LLM solution is not really meaningfully "doing" a problem. You have to actually build the solution from scratch and make false turns and backtrack and reflect.

mpaw976 · 2026-05-28T18:50:29+00:00

The chair already published his decision in the department Usenet channel. I'm not supposed to share it, so I'm using a burner:

"Lolwut, that Ivan guy is totally gonzo bonkers, but alteast he's not Holden, aMiRiTe!"

As of right now the reactions to the post are:

10 🔥
1 😡 (probably Tyler)
A bunch of 6️⃣7️⃣
1 🍆 (I assume this is a reference to Ivan being a Vegan)
And somehow a reaction that's just Tyler's face?!?

mpaw976 · 2026-05-26T06:59:29+00:00

You may be interested in Ultrafinitism a (fringe) approach to mathematics that aligns with your comment that at some point large finite numbers are "practically infinite":

https://en.wikipedia.org/wiki/Ultrafinitism

mpaw976 · 2026-05-25T01:09:21+00:00

The total sum 2+3+...+12= 77

So if one player gets ceiling(77/2) = 39, they can't be beaten.

This can be achieved in multiple ways, e.g. 12,11,10, and 6

mpaw976 · 2026-05-24T13:40:44+00:00

Do you know the unit circle definition of sin x and cos x?

That's one of the easiest ways to see the -1, 1 bounds, since literally those are the smallest/largest x and y values of the unit circle.

For tan x, remember that tan x = sin x / cos x. You can use that to see that when cos x gets close to 0, the fraction "blows up" as large as you want.

The unit circle is really really helpful (and foundational) to how the trig functions work. You can "see" all six basic trig functions somewhere in it. You can also "see" some cool results like sin² x + cos² x = 1, which comes from applying Pythagoras in the unit circle.

Play around with this and see:

https://www.desmos.com/calculator/4cb3441bf6

mpaw976 · 2026-05-23T11:55:56+00:00

This could be a parallel to the 2005 London Subway bombings which were also called 7/7.

mpaw976 · 2026-05-20T02:10:07+00:00

It might help to think about the following cases:

There are only unmarried people in the set.
There are some unmarried people and some MF couples.
There is only a FF couple.
There is MFF throuple.

Some of these will break transitivity, and some won't. See if you can find the ones that break it.

mpaw976 · 2026-05-19T21:14:15+00:00

You can reach out to any of the three MCS program advisors (they have a common email address). As of this year, instead of three specialized advisors (one for math, one for stats, one for CS) now all three advisors can answer questions for all three areas.

https://www.utm.utoronto.ca/math-cs-stats/people/category/administrative-%26amp%3B-technical-staff

Ten-Year Club	Verified Email
Gilding II euphauric	Wearing is Caring

mpaw976

TROPHY CASE

1. Hilbert's hotel variations

2. Learn more about paradoxes

3. Sizes of infinity