Odd

Neither xkcd_rss nor girlgenius_feed have posted this morning on my friendlist page. They both usually posted on Monday, Wednesday and Friday about the same time.

It just seemed unusual not to see the usual links waiting for me first thing in the morning when I turned to the page.

Edit: The XKCD feed has appeared. No sign of the Girl Genius feed. I wonder if any of the other RSS feeds today failed, or came through late, for anybody else.

Mental Arithmetic 001

Here is the next lesson in mental arithmetic. This time, I will show you the technique for rapid multiplication by 12.

I need to offer you an explanation of where these techniques come from, and what it is I am trying to do by introducing such elementary arithmetical operations.

The methods I present here are taken from a method, known as the Trachtenberg Method. The inventor of this set of techniques, Jakow Trachtenberg, formulated them whilst detained in a Nazi prison camp during World War II.

The text which outlines these techniques is called The Trachtenberg Speed System of Basic Mathematics, translated and adopted by Ann Cutler and Rudolph McShane. The publisher is Souvenir Press, ISBN 978-0-285-62916-5. I will be referring to other books along these lines, such as Figuring: The Joy of Numbers by Shakuntala Devi. But that is for later.

And so on to this article's topic. And a further question. Why did I not start you off with something simpler like, oh, the five times table or some such? What's with this elementary stuff?

The answer is - I'm not teaching you the numbers or the multiplications as such.

I'm teaching you things you can do with the digits.

It's like learning how to boil. You start learning to cook by boiling an egg in a pan of water. The aim is not to get you just boiling eggs. The aim is to teach you how to learn the technique of boiling so you can apply the technique to other foods - carrots, cabbage, potatoes and so on.

So now you've learned how to add a number to its neighbour from the first lesson, you can practice the technique on a handful of digit pairs I will list for you below.

First, add the digit pairs together as quickly as you can, in your head. Instead of saying "Two plus two equals four" or "Two plus nine equals eleven," practice going straight from the first digit to the sum without the intermediary stage.

Rather than say "Three plus five equals eight," learn to say "Three, eight." Instead of "2 + 2 = 4," say "2, 4." Rather than "5 + 9 = 14" say in your head "5, 14."

Try it.

9 2	4 3	6 9	3 8	4 6	8 6	9 0	7 8	4 7	2 9
0 0	7 3	4 7	6 6	4 1	6 8	5 1	1 5	8 1	9 9

Doubling Digits

Before we move on to multiplying by 12, here is something new for you to try.

In order to multiply by 12, you will need to learn to double a digit quickly and as efficiently as possible, going straight to the product without the intermediate stage of saying "Twice X is Y."

Below is a list of individual digits. Don't just say "Twice four is eight," just go "4, 8." If you like, just say the doubled digit straight away in your head. No intermediate stages where possible; if you see 6, just go straight to 12.

8 4 2 7 5 3 5 0 3 6
6 7 4 5 1 0 9 7 9 5
9 8 5 8 0 2 1 5 5 9
8 4 7 2 0 3 7 2 8 1

Now try the following technique with the pairs of digits below:-

1) Double the left digit.

2) Add the right hand digit.

Try and skip through the intermediate stages. Don't look at "6, 8" and think "Twice 6 = 12, plus 8 = 20." Think "6, 12, 20" or "12, 20." With practice, you should be able to go straight to "20."

3 8	9 4	5 5	3 3	0 9	1 6	6 4	3 7	7 8	4 7
9 8	7 8	8 6	2 0	4 1	0 1	9 7	0 9	5 9	2 9

Now try it with the first table of digit pairs from the exercise above:-

9 2	4 3	6 9	3 8	4 6	8 6	9 0	7 8	4 7	2 9
0 0	7 3	4 7	6 6	4 1	6 8	5 1	1 5	8 1	9 9

Now you have hopefully got the hang of the technique of doubling each digit, then adding its neighbour on the right, let's try it on a number, for instance 763,487.

First, add a zero to the left hand column:-

0 7 6 3 4 8 7

With the right hand column, since it has no neighbour we shall just double the digit:-

0 7 6 3 4 8 7
            ¹4

Now try the rest of the digits, up to the leading 0 on the left:-

0 7 6 3 4 8 7
          ²4¹4

0 7 6 3 4 8 7
        ¹8²4¹4

0 7 6 3 4 8 7
      ¹1¹8²4¹4

0 7 6 3 4 8 7
    ¹6¹1¹8²4¹4

0 7 6 3 4 8 7
  ²1¹6¹1¹8²4¹4

0 7 6 3 4 8 7
 9²1¹6¹1¹8²4¹4

The number is 9,161,844.

Try this exercise on the following table. It may look familiar, and that is because these are the same numbers from the previous article. The point is, if the numbers are familiar you can concentrate on the techniques rather than on the exercise. Practice the technique of doubling the digits and adding the neighbour on these numbers.

hen you've finished with these, try coming up with random numbers, typically four or five digits long, and multiply them by 11 and by 12. Create strings of single digits and practice doubling them rapidly, and practice the techniques of doubling and adding the neighbour on random digit pairs.

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, as before, in the next post. Have fun.