Having a little fun with the basics tonight

Mental Calculation, Mathematics, and Puzzles
1

I’m not sure whether or not base is the right term but I’m playing a bit with this
https://www.youtube.com/watch?v=Rgw9Ik5ZGaY and a bit of that https://www.youtube.com/watch?v=SV1dC1KAl_U .

Interesting how different distributions are so much easier than others. Sometimes this method is dramatically better than others but it’s not so easy for me to see when. I suspect with a little more practice it should read easier and then maybe I will pick my translation faster.

Getting down to sub-second multiplication is going to take me a lot more trust in the methods and a little better selection. It’s easy when you have the answer fully memorized but when you need to read the solution from the numbers it’s tough not to start calculating rather than reading the total and that always takes a couple of seconds.

By the time I get the method sorted, I will have probably memorized much of the subset but it’s certainly an expandable idea. Once the basics are mastered, you can always mix and match with bigger sets of numbers to make things faster. There are plenty of bases you can play with as well.

Not necessarily any better than anything else but it is a nice way to play with things differently.

A fun way to play in the low double digits for a bit while I get the bugs sorted out. Brings out a nice set of subtraction problems that forces you to work the complements fast. Darn complements keep stalling my speed. Every time I hit one I find myself proving the answer by also subtracting. If I know the answer starts with 1 has 2 in the middle and ends with 3 I should be able to answer “123” without doing the addition and subtraction to get me to the same answer but I lack confidence in my methods.

Hopefully, that will sort itself out in a few weeks and I will be able to use rather than practice it in short order. I’ve also forgotten a bunch of my 2 digit squares, I can imagine them by grid location and derive them but it is incredibly handy having the facts on hand instead of noodling on them. Again they’ll be back in a few weeks but right now I feel pretty rusty.

1 Like
2

Dear Robert,

Let’s analyse this algorithm.

If we multiply for example two two number digits close to a ‘base’ of 20, like 23 X 24 we do (23 + 4)x20 + 3x4. In. your head you go: 27 - 54 - 552 I hope.

If we do the criss cross we get 400, 540, 552.

Generally if we multiply two numbers close a base ‘a’, like “ab * ac”, we can write this as:

a | b
a | c *

‘Base’ sytem does:
(a+b+c)a + bc

If we expand the terms we get:
a^2 + ba + ca + bc

Criss cross multiplication does:
a^2 + ba + ca + bc

So they are both equal - mathematically.
Mentally they are not - imho.
If the 2 numbers are close to a ‘base’, then the base system might be preferable.

What do you think?

Can you give examples? When is it better , when not?
My suspicion is that it is better if the numbers are close to the base.

23 X 57 might lend itself better for criss cross or anker.

3

I haven’t yet watched your videos, Robert. But I’ve been playing around with this which might be useful.

I, developing a modified decimal system to represent numbers. Instead of confining each power of ten to a single column with a single digit, they are marked by commas which allows one overflows and also -ve numbers. Breaking free of the tyranny of the columns allows us to keep partial results in their original form as long as we wish and it allows us different ways to represent the same decimal number which can preserve structure that gets flattened by the column system.

This allows the same number to be represented different ways while still retaining a decimal form. In particular we can work with numbers in compliment form.

97 = 9,7 = 1,0,-3 or 10,-3 we are quite happy holding 10 tens in the second position

972 =(10,-3)2 = 1,0,0,-6,9 or = 100,-6,9

100,-6,9 =( take 1 from the 100 in the 3rd position 10+(6)=4) 99,49

The answer can be represented either way 9949 == 100,-6,9

Writing 97 as 1,0,-3 makes the calculation easier because there are hardly any carries.

(100+d) where |d| <=5 and d may be -ve : (1,0,d)2 = 1,0, 2d, ( d2)

d2 <= 25 and |2d| <= 10

1052 = 1,0,10,25 = 1 which normalizes to 1,1,2,5 = 1125

Two digit numbers with digits <= 5 also make for easy work

10a+b = a,b |a|&|b| <= 5 may be -ve

a,b 2 (commas have a higher precedence than exponentiation) = (a,b)2

a,b2 = (a2), 2ab, (b2)

again a2<25 & 2ab<50

432 = 16,24,9 = 18,29 = 1829

572 = 5,72 = 1,-4,-32

keeping the 43 as a single number 1,-432 = 1,0,-86,0,(432)

1,0,-86,(432) = 24,0,1849 = 32,0,49 = 32,4,9 = 3249

and it’s clear why both 432 & 572 have the same last digits.

762 & 242 should have the same last digits

24 = 576

762 = 1,0,-242= 1,0,-48,0,(242) = 52,0,576

52,0,576 = 5776

With mixed digits:

272 = 3,-32 = 9,-18,9 = 72,9 = 729

732 = 1,-272 = 1,-3,32

now we have all digits <=5 , keep the last two digits as a single number -3,3

1,-3,32 = 10, (-6,6), 0,(-3,32)
= 46, 0, (-3,32 )

Here it’s easiest to recognize that -3,3 = -27 & that -272 = +272 and we
have 46,729 = 5329

My meds leave my mind a bit cloudy at the moment so I need to check this. But I think this may offer a way to see patterns before they get normalized and flattened into single digit columns.

4

@Kinma don’t think I have anything insightful other than you have had a deeper trust of pairs than I. This 9 is 1, 1 is 9 thing always makes me second guess myself and I like my sums simple. If I don’t have to carry it just reads better in my head. I can read it rather than calculate it kind of thing. When I have to carry and sadly when I use complements I subtract and that just takes extra time. have to drill the basics. Criss Cross is a bit of a strain right now. Practiced a bit of that this evening but I need to subtotal rather than merge the numbers across by holding a few of them. Much rust, although truthfully I didn’t get much further than easy 4 digits, squares and some of the vedic stuff last time round. Never fully swallowed logs or division.

@zvuv I will play with your post in a couple of weeks. I am terribly out of practice and I need to get 20 hours doing bases, differences of squares, complements, shifting and remembering to do a checksum before I’m ready to stretch much further.

5

@Robert

If you ever feel like chewing through it, I’d appreciate the feedback. I think it’s useful for just the kind of problem you mention, managing the X product in the 2x2 which is most of the work.

It’s something I’m working on. I’m not a pure mathematician but I know enough to know that setting up any system, has to be done very carefully. A small ambiguity or inconsistency can bring the whole thing down. I would prefer to change the precedence of the exponential and the factorial so that they could be used within positions without parens. Also unsure about nested notation.

6

Right now everything old is new again for me. I really wish numeracy was something that was natural. Enough has stuck from my last round that I am catching back up fairly fast but working the bugs out of the pipes requires that I hit them with a heavy wrench repeatedly. It can take me as long to find my bugs as to fix them sometimes.

Talented folk just jump over this grind and move on. It’s hard for me to imagine people like Euler and sad to know that Calculus was essentially created on a summer break. I dabble like a 8 year child. As usual I will likely quit before I get terribly far but I enjoy the process each time and even though I don’t get very far up the hill it is a nice walk.

1 Like
7

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.

1 Like
8

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:

Exactly!

2 Likes
9

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.

10

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.

11

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

12

:joy::joy::joy:

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.

13

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

Example:

69
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

1 Like
14

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.

-R

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

15

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

69
79 X

11 … 1
69 X 79

(69-1)
68 * 80 + 11 = 5451

I need a better layout language for these chats.

16

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.

17

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

1 Like
18

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

20

Hear, hear!

21

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.

2 Likes