Having a little fun with the basics tonight

I feel similar. I spent my career doing math and engineering work. As soon as I am working with symbols I am very comfortable. I used to be able to invert a 2x2 matrix in my head. I am also very quick with estimating numerical results. This is a trick engineers learn for ‘back of the envelope’ calculations. But none of this ever parlayed into numerical accuracy. I like numbers, they feel friendly in my head. They have personalities but I can’t keep them disciplined without a lot of work. Part of this is ADD and Dyslexia but it’s clear, I have no natural talent.

My comma notation is an attempt to manage this. I suspect my post looks like a page of dense math, but the basic idea is very simple, allow the positions representing the powers of 10 to accomodate more than one digit and use commas to separate them. I find this helps both in actual computation and also for describing computational patterns. You have the freedom to arrange the terms in a way that suggests the procedure.

Realize that a decimal representation such as 567 is NOT a finished result. It tells you how to calculate the result -=` take 5 100’s add to 6 10’s plus 7 ones. While it’s true we can go no further, the ‘number’ is just a sequence of coefficients to be used in a summing algorithm. They are, of course, coefficients for a polynomial in powers of 10.

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This is a fantastic method and I advocate using this highly.
On this forum I already use this, for example here:

Instead of a comma, I use the | character (vertical bar) which I find makes for better reading.
And indeed it has lots of advantages, like better carry handling.

Realising that 69 can be written as: 7 | -1 makes for much easier multiplication, like I wrote about here:


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Thankyou Kinma. It’s good to see someone knowledgeable uses this same idea. I was beginning to wonder if the idea made any sense at all. If the only thing I got wrong was to use , instead of | then I did OK :slight_smile:

One tech question. How do you handle the precedence of the separator over exponentiation and factorials.
does a|b2 = (a|b)2 or a|(b2) ?

Once you’ve done the anti differentiation, and most simple integrals can be done mentally with experience, then you are left with numerical evaluation of the limits where you have to subtract the two end values. If those are polynomials then all you need is the stuff we are talking about here but you will often get trig function, fractional power and exponentials and logs. I know you know your way around logarithms. Engineers usually know several reference points on these functions and use them to interpolate. Back of the envelope calculations are very important, they allow you to catch mistakes in complex calculations to quickly check possibilities for plausability before investing in deep calculations.

I have yet to read throughall the examples you listed. - thanks.

It absolutely does!

You did not get anything wrong! Whether we use a comma as a separator or a vertical bar is just a matter of taste IMHO.

a|b2 = (a|b)2
If a for example is 6 and b=9, then a|b = 69.
Of course 692 = (69)2 = (a|b)2

Risking sounding arrogant; you will like them.
Now in all fairness I did not came up with this myself.
It is described in Dead Reckoning: Calculating Without Instruments by Ronald W. Doerfler; a book I highly recommend.

About trig functions, I personally hardly ever use them and because of this I don’t have much to say about them using mental calculation.
Dead Reckoning however does and if you are interested them I again highly recommend this book.

Definitely will look through those examples.

I was joking about the ‘mistake’, of course. Not sure which way to go with that yet. Perhaps we can use the word ‘position’ or ‘position field’ as a notation agnostic way of referring to each power of ten.

I’m considering changing the precedence so a squared term in a position doesn’t need parens eg (m,n)2 = m2 ,2mn , n2
But I am nervous about fiddling with a well tried order.

I also avoid try to avoid implicit multiplication since this is ambiguous when also using single letters to denote a digit

is 2mn the generic form of some number in the 200’s range or is it 2xmxn?

I really like the way it makes negative coefficients seem so natural.

It makes it easy to work with numbers in compliment form which exposes some simple, useful
symmetries and the compliment structure is preserved in the answer

7 = 1,-3 7*13 =(1,-3)(1,3) = 1,0,-9 = 91


Sometimes I am too serious.

We can. Also; keep in mind that it is some times useful - when working with two digits at a time, to have 2 digits in a position. For example here:

I would see 2mn as 2 * m * n.

Love it.

I mainly use it for making multiplication easier; with less carrying.


79 X

Criss cross:

60 X 70= 4200
60 X 9 + 70 X 9 = 540 + 630 = 1170
9 X 9 = 81
4200 + 1170 + 81 = 5370 + 81 = 5451

Now instead of 60 + 9, think 70 - 1 and instead of 70 + 9, think 80 - 1:

70 -1
80 -1 X

70 X 80 = 5600
-1 X 80 + -1 X 70 = -150
-1 X -1 = 1
5600 - 150 +1 = 5451

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It’s funny how, at the slightest provocation, a discussion about numbers can break out. My in-laws are fairly careful not to engage me in a discussion of my hobby. It embarrasses me a bit to think that I have been playing with this for a few years now and essentially gotten not much further in my math skills than an intermediate grasp of integers and their relationships. Watched a recent Terence Tao video on primes distribution last night and noticed that he grew up. He’s still 30 years younger than me. I was hoping that would have improved by now.

When I look at the chicken scratches above, I find myself wondering why I can’t calculate multiple columns simultaneously. I mean, if I look at 2 peas in a pile and 3 peas in a pile beside them I don’t count or calculate. I simply look at them and see how much they are.

I saw a video recently on simultaneous Anzan. Unless I know a multiplication and/or algebraic solution by rote I effectively decompose the problem into some kind of matrix every time. Why can’t I do that mentally/visually) with a picture for “trivial” functions? … multiplication, addition, subtraction. I suspect the underlying issue is that I am not terribly bright but I keep seeing indications that these are generally trainable skills (although maybe those that don’t excel quit).

In those cases where I must engage reason to calculate I am slow and error-prone meanwhile the visual component of my brain seems so much more gifted. I’m not talking about being a geometer but rather focusing on numbers in this case. I am saying that we “should” be able to see and manipulate our number faster and with more accuracy with mental imagery than we do rationalizing a series of steps…BUT we don’t.

I generally play with finding ways of reading my numbers like words rather than calculating them where I can but maybe I have been wasting my time trying to make math a more accessible language for me. Maybe I should be thinking about my little abacus more and trying to see the numbers rather than think them.

Just another very small though of no consequence I had.


At some point, it would be nice to start playing with triangles

as an aside 11 is definitely not a number we fear especially if all we have to do is add it.

79 X

11 … 1
69 X 79

68 * 80 + 11 = 5451

I need a better layout language for these chats.

So true. For people who try to follow the thread and have no clue how my esteemed Robert can do this; here is the breakdown.

He is using the video’s this thread started with, using either ‘base’ 80 or 68. I put base in quotes because it is the term the video uses and maybe mathematically not correct.

So we get the following steps:
1: Measure total distance from 80. 69 - 80 = -11
2: 79 - 80 = -1
2a: total distance is -12.
(next we will multiply 80 with the number the distance away)
3: 80-12 = 68
4: 68 X 80 = 5440. I do 70 X 80 - 2 X 80 = 5600 - 160 = 5440.
5: 5440 + (-1 X -11) = 5440 +11 = 5451.

Or, using 68 as the ‘base’:
1: Measure total distance from 68. 69 - 68 = 1
2: 79 - 68 = 11
2a: total distance is 12.
(next we will multiply 68 with the number the distance away)
3: 68+12 = 80
4: 68 X 80 = 5440. I do 70 X 80 - 2 X 80 = 5600 - 160 = 5440.
5: 5440 + (1 X 11) = 5440 +11 = 5451.

Love to help you out anyway I can!
Maybe you want to do too many things at once?
Stick with this for a while. Make a sheet for training or use mine.

And just stick with it until you find yourself doing the calculation the moment you see one.

You did see I made a training sheet, just for you, right? :grinning::grinning::grinning:

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You know that you are comparing yourself to arguably the best mathematician in the world, right?

Hear, hear!

Do you mean formatting? (I hope I’m not misunderstanding.) Mathjax is supported.

Edit: I created a general #faqs page about how to edit text in the forum and moved the text from my reply there. I’ll keep adding to that one post. There’s a section on mathjax in the middle of it.


Thanks Josh,

Exactly what I was looking for. Had MathJax installed previously but as you know I haven’t been active in quite a while. Kinma never seems to take sabbaticals but I find life often interferes with my hobbies.


Re: Terence, fairly decent mathematician (Field Medal etc) Absolutely… I prefer to compare my progress to Euler or Newton but most people haven’t watched their youtube videos :wink:

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Lol! You crack me up.

Mostly it’s the exponents that are awkward

you can write M2 by doing M<sup>2</sup>

And you can mostly keep the Markup editors fingers off your text if you enclose it in grave ` quotes.

I’ll check out the HTML entities for special symbols after I’ve read Kinma’s notes on this

*“But I did sum the series…”

On that topic, I did some poking around for rendering math in web pages. Seems there is a MathML a set of tags for math eqns , in the W3 standard but only FireFox renders it.

It seems that Latex has dominated. People are writing eqns in Latex and it’s converted into HTML & CSS.

Not interested in learning Latex right now. I’ll stick with HTML for now.


Can do quite a bit - there must be an XKCD cartoon about this.

Here are the HTML entities which seem to work in this Markdown renderer


This page is interesting http://jkorpela.fi/math/

I don’t see why his eqn couldn’t be described in HTML

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