Colin Beveridge @icecolbeveridge@mathstodon.xyz

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.

[Correction in first para: *7 in the top left.]

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!

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.

Oo, I stumbled on this: https://buttondown.email/hillelwayne/archive/how-to-memorize-a-larger-multiplication-table/ -- I think (with a little thought) that might be the killer insight into memorising times tables.

A bit nicer if it's parameterised: https://www.desmos.com/calculator/bwg1jtlxm6

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

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

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.

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.

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".