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Mental Arithmetic 004
Oct. 22nd, 2010 08:44 pmThis 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.
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.
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.
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.
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.
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 | ||
| ---------------------- | ||||||||
| 7 | 13 | 5 | ||||||
| 0 | 9 | 4 | 1 | 5 | x | 9 | ||
| ---------------------- | ||||||||
| 4 | 7 | 13 | 5 | |||||
| 0 | 9 | 4 | 1 | 5 | x | 9 | ||
| ---------------------- | ||||||||
| 8 | 4 | 7 | 13 | 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 | ||
| ---------------------- | ||||||||
| 22 | 10 | |||||||
| 0 | 9 | 4 | 1 | 5 | x | 8 | ||
| ---------------------- | ||||||||
| 13 | 22 | 10 | ||||||
| 0 | 9 | 4 | 1 | 5 | x | 8 | ||
| ---------------------- | ||||||||
| 5 | 13 | 22 | 10 | |||||
| 0 | 9 | 4 | 1 | 5 | x | 8 | ||
| ---------------------- | ||||||||
| 7 | 5 | 13 | 22 | 10 | ||||
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 | ||
| ---------------------- | ||||||||
| 16 | 10 | |||||||
| 0 | 9 | 4 | 1 | 5 | x | 4 | ||
| ---------------------- | ||||||||
| 6 | 16 | 10 | ||||||
| 0 | 9 | 4 | 1 | 5 | x | 4 | ||
| ---------------------- | ||||||||
| 7 | 6 | 16 | 10 | |||||
| 0 | 9 | 4 | 1 | 5 | x | 4 | ||
| ---------------------- | ||||||||
| 3 | 7 | 6 | 16 | 10 | ||||
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 | ||
| ---------------------- | ||||||||
| 24 | 15 | |||||||
| 0 | 9 | 4 | 1 | 5 | x | 3 | ||
| ---------------------- | ||||||||
| 12 | 24 | 15 | ||||||
| 0 | 9 | 4 | 1 | 5 | x | 3 | ||
| ---------------------- | ||||||||
| 8 | 12 | 24 | 15 | |||||
| 0 | 9 | 4 | 1 | 5 | x | 3 | ||
| ---------------------- | ||||||||
| 2 | 8 | 12 | 24 | 15 | ||||
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.
