Mental Arithmetic 000

[fiat_knox]

I have been studying mental arithmetic for a good long while, now, and I thought I'd jot down some of the things I have picked up along the way.

I'm starting with the simple stuff to begin with, but I'll gradually make my way to the more involved maths later on.

I've looked at Vedic Mathematics and the Trachtenberg System, and I can tell you that they have taught me some beautiful things. I can also tell you not to fear numbers. Unlike people, they will not let you down.

So let's begin with something really basic: multiplying by 11.

A lot of people probably never got past the times tables at school. Times tables are probably something to fear for so many. And yet with systems such as Trachtenberg, you don't have to memorise any tables to master multiplication.

Here's a great example, and I might as well throw you in at the deep end with a big one.

Try multiplying 3,467,189 by 11.

Before you get out your calculators, let me show you how to do it quickly and efficiently, without having to do long multiplication.

First, add a zero to the start of the number:- 0 3 4 6 7 1 8 9

Now write it down, if you cannot do it in your head to start with:-

0 3 4 6 7 1 8 9

The first digit (the right hand digit) is always written down unchanged:

0 3 4 6 7 1 8 9
              9

Now take the next number of the left - 8 - and add the 9 to it:-

0 3 4 6 7 1 8 9
           ¹7 9

The result, 17, should be written down as above, with the 1 written in the top corner. Make a note of it.

The next operation with the 1 in the next column is performed in exactly the same way: add the 1 and the 8, only this time add the carried 1. This yields 1+8+1 = 10:-

0 3 4 6 7 1 8 9
          ¹0¹7 9

This carries on, adding the 7 to the 1 and the carried 1:-

0 3 4 6 7 1 8 9
        9¹0¹7 9

Then the 6 to the 7:-

0 3 4 6 7 1 8 9
     ¹3 9¹0¹7 9

Next, the 4 to the 6 and the carried 1, then continue all the way to the leftmost number, the zero:-

0 3 4 6 7 1 8 9
   ¹1¹3 9¹0¹7 9

0 3 4 6 7 1 8 9
  8¹1¹3 9¹0¹7 9

When you get to the zero, treat it as any other number. Zero, plus anything carried, plus the neighbour on the right, in this case the 3:-

0 3 4 6 7 1 8 9
3 8¹1¹3 9¹0¹7 9

The product is 38,139,079.

The basic rule, therefore, for multiplying by 11 is Add each number to its neighbour and any carries from the previous column.

Try it with some smaller numbers, like the ones below.

23447	13324	8368	14679	2149	22314	88217	88125
9947	23189	27592	193057	46837	35962	245827	36937
4102	1038	6295	19376	9385	20587	8628	59276

Answers in the next post.

Comments

[the-magician.livejournal.com]

To me, multiplying by 11 is easy, because it's multiplying by (ten plus one). And in decimal, multiplying by ten is a single shift left

so if I want to multiply 123456 by 11, it's (123456 x 10) plus (123456 x 1), 1234560 + 123456

And when you place one over the other, you can see why it is adding the neighbour digit plus carry as you've said:

1234560
0123456

Similarly multiply by 9 is multiply by ten then subtract the original number

I was helping a friends niece with maths homework and they had to use estimation ... but the little girl found it much easier to just do the maths in full.

The idea was that you should be able to get a good feel for what a right answer should be close to without worrying about whether it was exactly right.

So if you wanted to know how many square feet there were in a room that's 123 square metres, then you go "there's about ten square feet in a square meter, so probably around or over 1200 square feet" or "what's 1,023,476 times 98", it's roughly one million times 100, so 100 million is the estimate (and actually 100,300,648, so less than 1/3rd of 1% off) ... I remember doing that stuff and then working out that the first figure estimate was about 2% low and the second one was a little under 2% high, so when you multiply the estimates together, you'd be able to estimate how far your quick estimate was likely to be from the actual answer)

Estimation is an excellent tool for mental math(s). (Someone complained to Radio 4 today that "math" is American and "maths" is British and that the Radio 4 news should know better :-) )

[pete23.livejournal.com]

This.

11x = 10x + 1x

Triangular numbers next?

[fiat-knox.livejournal.com]

This is good to remember, but I'm showing you that you don't need that.

You only need to work on a couple of digits at a time - the digit you're working on, the one to the right, and anything carried over.

And the rest is like pulling up a zip.