I read this and I find myself wanting to build a VM, install Jupyterlab, learn how to write in Latex and load up Sagemath. It’s a sickness. I may very well start playing with the algebra for each of the little tricks I know. Better off just practicing but there are so many shiney toys to play with. That said, it would be nice to have a little place to keep my toys.
I have a similar idea. I am working on a compendium of all the 2x2 mult & squaring tricks. Hope,like you, to be able to display it in a nice format, I’m hoping to do a website. A pretty presentation is very important, it makes it much easier to memorize the patterns.
I’ve mostly been working on techniques I’ve gathered here and there but there’s a lot to be harvested from previous discussions here, especially Kinma.
Anyway, we should compare notes, from time to time - that way we can fill in holes, catch each other’s mistakes etc. It would be good to develop this out of all the community contributions and make it available as an organized resource
Thank you for your kind words. You probably shouldn’t encourage me As I mentioned, I’m trying to compile list of these techniques.
I probably should have stopped while ahead but I wanted to include Art Benjamin’s method of squaring 2 digit numbers in the list of Anker techniques. I doubt he invented it but it often goes by his name.
Starting with the result we had before, the identity
(A+d)(A-d) = A^{2} - d^{2}
Shift terms back and forth across the =
A^{2}= (A+d)(A-d) + d^{2}
Choose A to be the number to be squared, Choose d to bring one of the terms to a multiple of 10
63^{2}=(63-3)(63+3) + 3^{2}
...=60*66+9 =3969 <-- do not know why the editor impose this styling on this line
27^{2}= (27-3)(27+3) + 3^{2} = 24*30+9 = 729
Again; great way of showing this technique!
Anker. We keep the name.
It probably does not come as a surprise that I would love to help!
Ok - this is great. I’ll start organizing my material and writing it up in MD so it can post here. I will use as much HTML as the MD allows because I want this material for a web page. See how that works out. When I have most of that, I’ll go through posts.
I’ll also try and introduce the Positional Notation - that name is agnostic about the actual symbol used. I think it would help people if they learned to think about numbers that way and it makes some of the calculation patterns much clearer. If it’s accepted we can see which symbol people prefer
, : |
The formulation:
(a,d)(a,c) = a(a+1), 0, dc
when d+c = 10Is succinct and shows you exactly what to calculate and where to put it.
I’ve been concentrating on 2x2 but those techniques can often be extended to larger numbers, so I might explore some of that.
I’m going down a similar path but I loaded up a Ubuntu VM and texmaker for pretty.
HTML is far more accessible but I should learn Latex. Similarly, I am likely to install JupiterLab and SageMath to the mix just because notebooks that compile themselves are cool and emacs was never my thing no matter how hard I tried. I tried Org-Mode for a bit but it was like trying to do things using tools from the '80s. Great idea but I could never get myself productive with all the overhead that came with it.
I’m going to have to do a couple of short courses on how to express myself appropriately in math. Happily, this is all just simple algebra for now so I mostly just need to learn how to structure a simple proof. (thank goodness for youtube).
Little projects should hopefully; tidy up my algebra, learn me to do simple proofs, learn me some tex, provide me with a little book of math tricks with associated notes and proofs. Load up a few other fun tools and warm up to some other math skills in a way that is organized enough to come back to when I put it down and am distracted by some other shiny toy for a year or two.
I think most people find math tricks very accessible BUT I don’t think many care to grok why they work and/or how they generalize. I also suspect that actually doing these simple kinds of proofs till they are natural is far better than reading them.
What I would find really useful are examples of how to express a couple of these simple algebraic proofs formally. (i.e. In proper math geek form rather than the hand-wavy stuff I do with some algebra in the middle) so that I had a nice pattern to reproduce when doing proofs. ( I am pretty sure the “Great Courses” has a course on proofs video and there was something called “Prove It” that I should go back and find - vague memory)
The algebra required is very simple and should remain so. We are dealing with polynomials, whole powers mostly no higher than 2. Mostly we are playing with the Quadratic polynomial. You can expect to improve rapidly with practice. What you are aiming for is well within your reach.
Yes, people are afraid of algebra. And strangely uncurious about how these 'recipe’s work. IMO understanding why they work makes easier to remember , to use and to adapt. I have more to say on that but another time.
We can certainly do that exercise here if you like. Present what you have in mind and we’ll do a formal proof.
A great deal of mathematics is taking things to pieces and rearranging them so that it’s easier to work with or suitable for some technique. In fact mathematicians are notorious for this. The algebra lets you do this, but you have to have a nose for what to look for and that comes with experience.
In your first post you seemed to be rambling around trying to find something useful. I’ve done a lot of that. Pages and pages of math that went nowhere. There’s a high chance of making an error because you aren’t sure what right looks like. That will improve rapidly after you’ve worked through a few.
Realize that it’s quite possible to speak ‘ugly algebra’ and many do. Many of the formulations I’ve seen out there, while mathematically correct do not give you a feel for what’s really going. So after you’ve worked something out, think about how you would have liked to have seen it explained and write that out for yourself. Try to make your algebra eloquent. I find that very helpful.
I once sat in a lecture while the Prof worked a homework problem for the whole hour- literally three boards full of algebraic expansions. It was an impressive tour de force. But the solution that I had handed in was one line! That’s luck. He was a very smart guy but I found it, he didn’t, so he had to brute force the proof. But this is what happens when several people look at a problem together you get luckier
Thanks, I appreciate your thoughts… I did a little digging and found reasonable templates from the AMS, American Mathematical Society // Latex AMSBook and some decent discussion of formatting Theorems, Proofs, Lemmas. Right now I am mostly thinking about tooling and typesetting. I suspect after a couple of dozen I should be pretty comfortable as you are correct the majority of the common efficient “tricks” are usuallyq either difference of squares or rebasing of some sort. I need to do some taxes this week but I should probably have some time to play this weekend.
Somewhere in this forum I introduced the ‘||’ notation.
Where ‘|’ separates the tens digits from the ones, ‘||’ separates the hundreds from the ones.
So 65 || 25 = 6500 + 25 = 6525.
In your example:
(a,d)(a,c) = a(a+1) || dc
Which - imho - makes for a slightly nicer way of presenting the end result.
The commas do the same - I could write
a(a+1), ,dc
and sometimes do. I thought it looked nicer with the zero there as a clear marker. It’s a good sign that that’s about the only difference between the schemes. It shows it has a ‘natural’ logic and should appeal to others.
But I’m thinking I’ll experiment with both and perhaps also : maybe get some feedback from other readers as to what’s easiest. Perhaps the commas are too small and would be better in bold. I do like my commas but I’m more interested in having a uniform widely accepted convention so we all speak the same language. I’ll try the | symbol the next few times.
Similar decisions need to be made about the layout. It’s much easier if every entry is laid out in the same format, or in a predictable format. Again, I want to write the expressions in a way that makes it obvious what to calculate and which slot it goes in. I think this notation has the power to express the methods as calculation schemes
Yes, Anker definitely
Have you tried Doom Emacs or Spacemacs? They are pre-configured Emacs that use Vim keybindings. I think Org Mode comes with them (or can be installed by adding a line to the config), along with plugins that make it easier to work with files. There’s also an Org Mode thread.
I used Spacemacs for a while but then switched to a custom configuration after watching this video.
I’m not sure if it was already mentioned somewhere, but this post is interesting. I think it uses Vim and some custom scripts.
I use Vim. I wouldn’t be without it. I have :w trouble with this editor. But it’s a serious commitment to learn it. Takes about 4 weeks.
You can edit forum posts here with Vim if you install Tridactyl for Firefox. Then click in the textarea (or use the f
key to select it). Press ctrl-i
to open the draft in Vim. Then :wq
to send the draft back into the forum’s editor. If the preview doesn’t update, press a key in the draft to trigger the event listener.
There are some related threads about Vim here:
The esoterica is deep. As I think about breaking up my personal notebooks into multiple files it seems like a package like org-mode could capture my notes/thoughts on work items, rework, editing comments about the notebook with links into it rather than littering the actual notebook with things not to be published. MathJax being a subset of LaTeX is nice for publishing individual equations but when you want to publish from LaTeX to the internet there seems to be a gap. PDF, DVI, PostScript are not the linqua franca of the internet. Is there such a thing as a LaTeX to HTML converter. tex4ht does both LaTeX to HTML and LaTeX to Mathjax. Will likely give it a try this weekend.
Org Mode’s capture templates let you use custom note templates that can be opened with a keybinding.
I’m not sure if it’s what you’re looking for, but it’s possible to export whatever you write in Org Mode to many formats:
Pandoc could probably do the format conversion too. Or maybe a program like Kile.
Here’s fun little pattern for multiples of 11
(d|d)^{2} = d^{2} | 2xd^{2} | d^{2}
(setting the precedence of exponentiations
77^{2} = 7^{2} |2*7^{2} |7^{2}
= 49|98|49 = 5929
You can see the 11’s at work if you lay it out as
d^{2}| d^{2}| 0 |
0 | d^{2}| d^{2} |
I was just noodling on the old a^2 + 2ab + b^2 for 3 and 4 digit squares.
Spent my afternoon with the soroban and just practicing basic 2 digit mental multiplication and 1 digit soroban addition today however.
I’m not really seeing this as 11’s:
I think of squares of 3 and 4 digits as a “gentle” segue into 3 and 4 digit calculation as you start having to keep track of the zeros and subtotals but the actual calculation adds minimal load to the problem while you are getting accustomed to the complexity. My brain tends to rebel as I add on layer so I try to make it as simple as possible to adjust. I am still on my refresher of 2 digit and soroban addition so it will be a few weeks/months before I start working 100 - 999 and 100-999 * 1000-9999.
Hopefully, by then my little math notebook will be working on something other than properties of numbers and multiplication of integers.