Hello! My name is Colin and I am a mathematician.

Today is the birthday of Maria Gaetana Agnesi, who wrote the first book about integration *and* differentiation. If you need help making difficult things simpler, give me a shout -- I'm usually happy to help.

Hello! My name is Colin and I am a mathematician.

Today is the birthday of Roger Hargreaves. If things are Topsy-Turvy and Impossible, and you need some Helpful Magic, don't Worry -- I'll be Happy to see what I can do.

For example, for the 47 times table, write down the grid with 7 in the top right:

| 7 4 1
| 8 5 2
| 9 6 3

Then prepend the five times table, knocked down by one for each line you've crossed:

\( \begin{pmatrix} 47 & 94 & 141 \\ 188 & 235 & 282 \\ 329 & 376 & 423 \end{pmatrix} \)

Boom!

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I think these two grids are all you need to memorise. Suppose you're multiplying by \(10a+b\), [^0]. Rotate the grid so that \( b\) is in the top left. If \( b \lt 5\), write down the \(a\) times table in front, but bump it up by one every time you cross a vertical line. If \(b\gt5\), write down the \(a+1\) times table, but drop it by one each time you cross a vertical line.

[^0] Sigh, because mathematicians: \(a\) and \(b\) integers, \(0\le a\lt10\) and \(1\le b\le 9\), \(b\ne 5\). Honestly.

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Oo, I stumbled on this: buttondown.email/hillelwayne/a -- I think (with a little thought) that might be the killer insight into memorising times tables.

Today while the youngest swims, I'm playing with this: 507movements.com/mm_152.html

It took a while to convince myself it was really an ellipse, but Desmos to the rescue: desmos.com/calculator/9hsixaci

I've made one of those daily word game things! But it's not just Wordle in a hat, it's a totally different puzzle and I'm really pleased with how it came out.

andrewt.net/puzzles/cell-tower

OK ... this was a *lot* of fun !

1:

⬜⬜⬜⬜⬜⬜⬜⬜➡️⬇️⬜⬜⬜⬜⬜
⏹⏹➡️➡️➡️➡️➡️➡️⬆️➡️➡️➡️➡️➡️➡️

andrewt.net/puzzles/cell-tower

Another proof 

@icecolbeveridge
along the lines of the √2 proof:

Suppose p/q is a reduced rational solution. Then clearing denominators, 𝑎𝑝²+𝑏𝑝𝑞+𝑐𝑞²=0. If exactly one of 𝑝,𝑞 are even, then this is two even terms and an odd one. If neither of them is even, then it's three odd terms. So they're both even, contradicting "reduced" fraction.

cc @jimsimons @ColinTheMathmo

My proof 

@icecolbeveridge (hope my first tray at a content warning works). Here's mine

Any odd square is 1 mod 8, but \(b^2-4ac\) is 5 mod 8.

My proof 

If \( ax^2 + bx + c =0 \) has rational solutions, it can be written as \((px+q)(rx+s)\), with \(p\),\(q\),\(r\) and \(s\) integer.

\( a= pr\) and is odd, so \(p\) and \(r\) are both odd.

Similarly, \( c = qs\) and is odd, so \(q\) and \(s\) are both odd.

\( b \) is odd, but must equal \( ps+qr\), which is even. Contradiction.

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I just stumbled on a nice little proof I'd forgotten about:

Show that \( ax^2 + bx + c =0 \) has no rational solutions if \(a\), \(b\) and \(c\) are all odd.

The curb cut effect: for example, media descriptions right here on Mastodon help people who use screen readers, but they are also useful for explaining the joke for people who don’t get it, for translations of text, and for copy-pasting what would otherwise be just an image of text.

Something I've wondered more than once on birdsite:

\( 1729 = 12^3 + 1^3\), so it's a multiple of 13.

\(1729 = 10^3 + 9^3 \), so it's a multiple of 19.

It's also a multiple of 7. Where does that come from? (i.e., is there a way to find *all* of the factors of a taxicab number given the cubes that sum to it, or similar?)

I used to put my most common programming errors on stickies on my monitor. Now I have one saying "Don't start anything new in the last half-hour of the day".

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