fiat_knox: silhouette of myself taken at sunrise (Shadow person)
Today is 3.14 - Half-Tau Day.

Most people call this day "Pi Day," it being 3.14 and all; but Tauists - including myself - hold to a different philosophy.

The Tau Manifesto - No, Really, Pi Is Wrong!

The Tau Manifesto PDF


And here are the first 10,000 digits of tau.
fiat_knox: silhouette of myself taken at sunrise (Default)
Today is 3.14 - Half-Tau Day.

Most people call this day "Pi Day," it being 3.14 and all; but Tauists - including myself - hold to a different philosophy.

The Tau Manifesto - No, Really, Pi Is Wrong!

The Tau Manifesto PDF


And here are the first 10,000 digits of tau.
fiat_knox: silhouette of myself taken at sunrise (Default)
I like that people have such fondness for the circle constant π.

However, they kind of got it wrong a couple of thousand years ago.

Happy Half Tau Day!

π = τ/2 = 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 ...

τ = 6.28318 53071 79586 47692 52867 66559 00576 83943 38798 75021 16419 49889 18461 56328 12572 41799 72560 69650 68423 41359 64296 17302 65646 26564 ...

(You've probably been calling τ 2π all these years, never realising that the formula 2πr is even easier to remember as τr ...)

100,000 digits of τ

The Tau Manifesto

Download the PDF
fiat_knox: silhouette of myself taken at sunrise (Default)
This item is just to wrap up the first part of the articles on mental arithmetic - dealing with the basics of the Trachtenberg method by means of some simple multiplications.

In previous articles, you were shown techniques such as doubling each digit, halving a digit and adding 5 if the column is odd, taking the 9's complement and the 10's complement. Previous exercises set the tasks of multiplying by 5, 6, 7, 11 and 12. Now it's time to show you multiplications by 3, 4, 8 and 9.

Note

These techniques are mostly for show - I include them in order to demonstrate the versatility of a Vedic mathematics sutra (method) called Nikhilam, or "All from 9 and the last from 10."

These methods are listed in decreasing order of simplicity, each with one example. You will be encouraged to try out these techniques with some numbers in a table below.

Multiplication By 9
a) Take the 10s complement of the rightmost digit.
b) Take the 9s complement of each digit. Add the neighbour.
c) Reduce the left hand digit by 1. Write down this result.

Example: 9,415 x 9.
0 9 4 1 5 x 9
----------------------
5     


0 9 4 1 5 x 9
----------------------
13 5     


0 9 4 1 5 x 9
----------------------
713 5     


0 9 4 1 5 x 9
----------------------
4 713 5     


0 9 4 1 5 x 9
----------------------
8 4 713 5     


Multiplication By 8
8: a) Take the 10s complement of the rightmost digit and double.
b) Take the 9s complement of each digit, double it and add the neighbour.
c) Reduce the left hand digit by 2. Write down this result.

Example: 9,415 x 8.
0 9 4 1 5 x 8
----------------------
10     


0 9 4 1 5 x 8
----------------------
2210     


0 9 4 1 5 x 8
----------------------
132210     


0 9 4 1 5 x 8
----------------------
5132210     


0 9 4 1 5 x 8
----------------------
7 5132210     


Multiplication By 4
4: a) Take the 10s complement of the rightmost digit. Add 5 if the column is odd.
b) Take the 9s complement of each digit, and add half the neighbour. Add 5 if the column is odd.
c) Halve the left hand digit and reduce this by 1. Write down this result.

Example: 9,415 x 4.

0 9 4 1 5 x 4
----------------------
10     


0 9 4 1 5 x 4
----------------------
1610     


0 9 4 1 5 x 4
----------------------
61610     


0 9 4 1 5 x 4
----------------------
7 61610     


0 9 4 1 5 x 4
----------------------
3 7 61610     


Multiplication By 3
a) Take the 10s complement of the rightmost digit and double this. Add 5 if the column is odd.
b) Take the 9s complement of each digit and double this, and add half the neighbour. Add 5 if the column is odd.
c) Halve the left hand digit and reduce this by 2. Write down this result.

0 9 4 1 5 x 3
----------------------
15     


0 9 4 1 5 x 3
----------------------
2415     


0 9 4 1 5 x 3
----------------------
122415     


0 9 4 1 5 x 3
----------------------
8122415     


0 9 4 1 5 x 3
----------------------
2 8122415     


Finally, to summarise the techniques we have covered to date:-

3: a) Take the 10s complement of the rightmost digit and double this. Add 5 if the column is odd.
b) Take the 9s complement of each digit and double this, and add half the neighbour. Add 5 if the column is odd.
c) Halve the left hand digit and reduce this by 2. Write down this result.

4: a) Take the 10s complement of the rightmost digit. Add 5 if the column is odd.
b) Take the 9s complement of each digit, and add half the neighbour. Add 5 if the column is odd.
c) Halve the left hand digit and reduce this by 1. Write down this result.

5: Write down half each column's neighbour. Add 5 if the column is odd.

6: To each column, add half the neighbour. Add 5 if the column is odd.

7: Double each column and add half its neighbour. Add 5 if the column is odd.

8: a) Take the 10s complement of the rightmost digit and double.
b) Take the 9s complement of each digit, double it and add the neighbour.
c) Reduce the left hand digit by 2. Write down this result.

9: a) Take the 10s complement of the rightmost digit.
b) Take the 9s complement of each digit. Add the neighbour.
c) Reduce the left hand digit by 1. Write down this result.

11: Add each number to its neighbour and any carries from the previous column.

12: Double the column and add its neighbour.

These are the basic multiplications, and the most basic and essential techniques, you will ever learn - but for the simpler mathematical operations, you should by now be feeling a definite improvement in speed, accuracy and confidence.

You could multiply a 13-digit number by 12 in a matter of how long it takes for you to double each digit and add the neighbour, all the way along. Whether it be a number like 1,111,111,111,111 or 9,453,676,033,276 you can be confident that these simple processes will never be beyond you.

So, now, what about more involved multiplications, such as 114 x 109, or 97 x 88? You can't just use the techniques described above, but that's all right because there are other techniques I have to teach you.

But before that, we're taking a quick side trip to addition and subtraction. Because some of the techniques you have learned already will come in extremely handy there too.
fiat_knox: silhouette of myself taken at sunrise (Default)
So here we are, then, with the next stage of the lessons - bringing together all of those exercises in halving digits, doubling them and adding 5 in the column is odd that we did the last few times.

First, some digit exercises to get you going. Here's a table of digits. First, go through each one, double it and say out loud what the doubled digit is. Don't just say "Twice 4 is eight," just see 4 and say "Eight."

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


Now go back to the above table, only now halve the digit. See 4 and say "2;" see 9 and say "4."

Next, the table below has a list of odd numbers. Halve each number and add 5 to it; when you see 3, don't just say "half of 3 is 1, add 5 to yield 6." See 1, say "5;" see 3, say "6;" see 5, say "7," and so on.

5 3 9 1 7 9 7 3 5 1 9 5 1 7 3


Now here's the rule for multiplying by 5.

When multiplying each column by 5, just write down half its neighbour.
Add 5 if the column is odd.


Let us try it with the following multiplication - 345 x 5.

0 3 4 5 x 5


The first digit is easy - half of 0 (the neighbour) is 0, and add 5 because the column is odd. The first figure is 5:-

0 3 4 5 x 5
5


With the second column, write down half the neighbour - 5, yielding 2.

0 3 4 5 x 5
2 5


The rest of the calculation proceeds with the same rules:-

0 3 4 5 x 5
7 2 5


0 3 4 5 x 5
1 7 2 5


The product is 1,725.

Lastly, here's the rule for multiplying by 7:-

When multiplying each column by 7, double each column and add half its neighbour.
Add 5 if the column is odd.


Let us try it with the following multiplication - 345 x 7.

0 3 4 5 x 7


Let's take the first column, 5. Double that - 10 - and add half its neighbour. Since the neighbour doesn't exist, take it as 0, so the sum is now 10. Finally, add 5 because the column is odd, yielding 15:-

0 3 4 5 x 7
15


Now move on to the rest of the number:-

0 3 4 5 x 7
1115


0 3 4 5 x 7
141115


0 3 4 5 x 7
2141115


The answer is 2,415.

Now try the above on on the following numbers:-

08295 09922 01666 08602 09132 06556 07078 09709 02930 08240
02737 07688 03580 05584 05142 04683 05616 03905 08808 05275
06757 09965 09897 03830 06388 05187 01451 02731 05777 03304


So now you have learned the techniques you'd need to do basic multiplications by 5, 6, 7, 11 and 12. Halving and doubling digits, adding the product to the neighbour, adding 5 if the column is odd; these are simple enough.

Multiplying by other numbers - 3, 4, 8 and 9 - is a little more involved, a process requiring complements of 9 and of 10. We'll go into that in the next installment.

But first, something I've been wanting to demonstrate since the start of these lessons. And I'll start with a question. Without a calculator, can you tell me what 75 x 75 is?

Not only will I be able to show you how to work it out in a split second; I will show you a neat tool to allow you to work out the answers to similar calculations such as 45 x 45, 95 x 95 and even 113 x 117.

All these await in the interlude.

June 2017

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