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I'm Josh. This is my blog about mathematics, computer science, and everything in between. 

[numb3rth30ry]

A divisibility lattice for the number 60. 

Mathematics is beautiful. <3

[functionaldesign]

I have been wondering what this is called for weeks.

[the-real-numbers-deactivated202]

I'm gonna go fucking bezerk

[the-real-numbers-deactivated202]

"what change did you make?"

"I software engineered"

"yes but what change did you make?"

"code changes"

"yes... But what code changes"

"uh,, adding... Stuff"

[kubleeka]

X

[numb3rth30ry]

A lovely animation depicting a few partial sums for the Fourier series of a square wave by stacking rectangles.

Mathematics is beautiful. <3

[the-real-numbers-deactivated202]

The Jews: pi is three

Other cultures: I’m pretty sure it’s a little bit mo-.

The Jews: fuck you, pi is three.

[dukeofriven]

That’s not how I interpret this text. This is saying that the evidence of G-d’s wonders can be seen in every circle, because a circle’s ratios forever match the proportional circumference that could be measurable by Judah ha-Nasi, relevant in this case not only through his compilation of the mishna but - and this is where divine wonder comes in - for having had three hands.  

[the-real-numbers-deactivated202]

Mathematician 1: this has a solution?

Mathematician 2: what is it

Mathematician 1:

Mathematician 2: wig okay let’s just call it a day then

[counter-example]

mathematician 3: we’ve proved no solution exists

mathematician 4: wow my life’s work has finally paid off with an answer

[vacuously-true]

mathematician 5: I finally have an existence proof!

mathematician 6: I have a constructive proof

mathematician 5:

mathematician 5: … fuck you

[matan-matika]

Have you ever wanted to end a proof not with an unfilled square or “Q.E.D.,” but rather with “#get rekt”?

[the-axiom-of-choice]

Sometimes we write “QEmfD” for “Q E motherfucking D”.

[the-real-numbers-deactivated202]

[vacuously-true]

Too soon?

[the-real-numbers-deactivated202]

Lee Gao

[hurricane-euler]

Any advice on how I can become better at math?

This is all classroom-based because I cannot learn without external motivation, but here are some things that help:

1. Read the textbook before class

2. Take notes but only as a means of remembering what happened in class, not as an end in themselves. The studyblr tag is not a model here

3. Ask questions

4. Go to office hours

5. Do small talk with your professors

6. If it’s been 3 hours and you haven’t made any progress, do something else

7. Getting 8 hours of sleep is more important than studying

8. Start early. Do the first problem the day after you get the assignment. The rest will come. Start studying a week in advance, even just 15 minutes

9. Your gut knows more math than you think

10. But don’t skip steps

11. It’s okay to feel like an idiot, it means you probably aren’t one

12. Math doesn’t have to be your whole identity

13. No one study/note-taking/homework method works all the time. My strategy changes for every class. 

14. Remember to stop and smell the roses/flirt with the sexy primes/make fun of Pythagoras/fangirl over the FTC/admire adorable proofs/micdrop your QED/etc

[hyrodium]

Inspired by this twocubes’ post and asked to make an animation of it, I made a gif.

[nixcraft]

Basic Linux Commands. Very useful for new Linux users or programmers

[model-theory]

you’re forgetting sl! Cant program without Steam Locomotive

[the-real-numbers-deactivated202]

oh

[the-real-numbers-deactivated202]

yes, you can in fact make a mixed base “numeral system” with unique representations by using factorial bases, like so:

3×5! + 4×4! + 1×3! + 0×2! + 1×1! + 0×0!  = 46310

These numbers are called the factoradics. Apparently they’re nice to enumerate permutations with.

[the-real-numbers-deactivated202]

(this post has an error, it’s 463 in base 10. My subscript got taken out by Tumblr :( )

[third-degree-math]

Cursed Image of the Week #2

[complex-variable]

If this doesn’t satisfy you, then (sqrt 2)^(log_2(9)) is a concrete example where irrational^irrational = rational.

[just-evo-now]

Almost all irrational numbers can *only* achieve rational powers using irrational exponents

[complex-variable]

The set of algebraic irrationals which could achieve rational powers using irrational exponents form a subset of the countable set of algebraic numbers, hence has (Lebesgue) measure zero.

Since transcendental^rational can never be rational, and transcendentals numbers are irrational but not algebraic, one can now say that almost everywhere irrational numbers can never achieve a rational power using rational exponents.

But since for any transcendental number T and any rational number R we have T^{log_T®} = R, and log_T® is irrational, it can indeed be concluded that almost everywhere, irrational numbers can only achieve rational powers via irrational exponents.

[complex-variable]

‘Almost everywhere’ being necessary because there are examples in each case for irrational/rational bases to an irrational/rational exponent giving either an irrational or rational answer.

An obvious irrational that can achieve a rational power by a rational exponent being (sqrt2)^2 = 2.

[functionaldesign]

Say what you will about Java, but javadocs is (are?) a great tool, and more tools like it should be supported by language developers and communities.

[shieldfoss]

As far as I can tell from the spec, `int main()` and `int main(int, char*[])` are both valid ways to declare a main function.

Indeed so it is (I went and looked it up because it started bothering me that I didn’t know)

[serinemisc]

I’m gonna come clean here and admit that when I said “from the spec”, I actually mean “from Stack Overflow answers citing the spec”, because I’ve never actually read the C++ spec in my life.

[the-real-numbers-deactivated202]

Hottest take: ℕ ≝ ℤ^+ ∖ {2} ∪ {0}

[the-real-numbers-deactivated202]

Fuck the number 2

[godelitonthemountain]

Primes are so much better in this set

[the-real-numbers-deactivated202]

THEY’RE. ALL ODD.

[functionaldesign]

Well now you have even numbers that are not composite.

[the-real-numbers-deactivated202]

The best part is “even” is typically defined as “a multiple of 2” and not “the stuff that comes between odd numbers” so “even” may not even exist here.

They’re just extra primes?

[functionaldesign]

The numbers of the form n+n (not 2n, that string is meaningless) dont necessarily generate a prime ideal, but they also might not be factorable. For instance, 6 is not composite as you cannot factor it (3 times what?) However 24 = 6*4, but 24 can be factored to 8*3, neither of which have 6 has a factor. You lose the "every number is either prime or composite" when you poke holes in the integers like this.

[the-real-numbers-deactivated202]

Hottest take: ℕ ≝ ℤ^+ ∖ {2} ∪ {0}

[the-real-numbers-deactivated202]

Fuck the number 2

[godelitonthemountain]

Primes are so much better in this set

[the-real-numbers-deactivated202]

THEY’RE. ALL ODD.

[functionaldesign]

Well now you have even numbers that are not composite.