Mental Arithmetic Interlude - Squaring Numbers And Vedic Mathematics

And so we come to the interlude. Up till now, you've practiced adding a number to its neighbour, doubling digits, halving digits and adding 5 to a digit if the original number is odd.

You have multiplied by 5, 6, 7, 11 and 12, and each exercise has given you practice in these simple digit manipulation techniques.

But before moving on to the next phase, where you will learn the 9's complements and 10's complements of digits, here's a question for you. I asked you the question last time; here it is again.

Without resorting to a calculator, can you tell me what 75² is?

( Answer behind here )

What if I showed you how to calculate similar squares in a matter of seconds?

Yes, seconds. Not minutes or hours. And in your own head, not using anything resembling a calculator - not even a pen and paper.

And what if I showed you how to do a multiplication like 207 x 203 in so little time that you could work out the answer before you'd written down the numbers on the paper?

Read on.

There now follows a short introduction to a methodology which has, in recent years, begun spreading across the world from its homeland of India - a methodology which promises to revolutionise the way the general public uses mathematics. A methodology of which I am proud to call myself a student.

The methodology is known as Vedic Mathematics.

Between 1911 and 1918, His Holiness Sri Bharati Krsna Tirthaji (1884-1960) uncovered a total of sixteen Sutras, or word-formulae, based on research conducted on the Hindu sacred texts called the Vedas.

Vedic mathematics offers quick and simple solutions to many mathematical problems, because the sutras (methods) follow the way the mind functions, and are of great help in directing a student to working out the appropriate solution to a mathematical problem.

I will now show you one Sutra in particular. In Sanskrit (the language of the Vedas) the term is "Ekhadikena Purvena," but the English translation reads "By one more than the previous one," or perhaps "Given a value x, by (x+1)" if you prefer.

Here, "by" means "multiply by."

And so to the squaring.

This technique will work on any number that you want to square (to multiply by itself) whose last digit is 5. This works for 5, 75, 95, 205 and 1255. In fact, this will work for any number you want to square that ends in a 5 - though, for simplicity's sake, you will only see the simplest examples - two digit and small three digit numbers.

Here is the method for squaring a two digit number.

1. Take the first digit and separate it from the end digit. Where you see a "45," separate the 4 from the 5.

2. Add 1 to the tens digit. Add 1 to the 4 to yield 5.

3. Multiply this figure and the original tens digit. Multiplying 4 x 5 yields 20.

4. Add 25 to the end. To the 20, tack on 25 yielding a product of 2025.

Let's see what happens if we try it with two other examples: 15 and 85.

(15)² ==> (1 x 2)+25 ==>  225
(65)² ==> (6 x 7)+25 ==> 4225

How does this work? It's easy when you realise that you're not multiplying, say, 6 x 7.

You're multiplying 60 x 70.

And 60 x 70 = 4200.

To this, add 5², or 25, to yield the product.

This actually works, even on three digit numbers. Instead of working on one digit, however, you have a cluster of two digits - but add one to the truncated number and multiply as before, then add 25 to get the final product.

Here are two examples:

(115)² ==> (11 * 12)+25 ==> 13225
(205)² ==> (20 * 21)+25 ==> 42025

Try this on the following numbers.

25
35
55
95
105
125
155 (hint: 15 x 16 = (15)² + 15)
195

( Answers under here )

There's more to this sutra, however, than simplifying the squaring of numbers. This also works, with a little tweaking, for multiplying two different two digit numbers if they happen to have two things in common.

First, the lead digit (the tens digit) of both numbers is the same.

Second, the units digits added together must be equal to 10.

When multiplying two numbers such as, say, 23 and 27, use the same sutra as before, but modify the procedure as follows.

1. Take the first digit and separate it from the end digit.

2. Add 1 to the tens digit.

3. Multiply this figure and the original tens digit.

4. Multiply the units digits together, and tag the product to the end.

Let's try this on 23 x 27.

1. Add 1 to the tens digit and multiply that and the original tens digit together: 2 x 3 = 6.

2. Multiply the unit digits together and add them to the end: 3 x 7 = 21.

3. The product is 621.

This works for any similar multiplications. 24 x 26 = 624; 21 x 29 = 609 (note the use of the zero here).

This works because when both numbers add up to 10, basically you're multiplying (x + 1)(x - 1) (where X=5).

(x + 1)(x - 1) = x² - 1

You can extend this formula:

(x + 2)(x - 2) = x² - 4

(x + 3)(x - 3) = x² - 9

(x + 4)(x - 4) = x² - 16

Basically, when you're multiplying, say, 24 x 26, you're actually saying (25)² - 1, or 24.

You should already be familiar with this: 7 x 7 = 49, but 6 x 8 = 48; 6 x 6 = 36, but 5 x 7 = 35 and 4 x 8 = 32.

Now see what happens with small three digit numbers:

113 x 117 = 13221
101 x 109 = 11009

22 x 28
24 x 26
33 x 37
31 x 39
54 x 56
57 x 53
91 x 99
92 x 98
108 x 102
103 x 107
126 x 124
128 x 122
159 x 151
156 x 154
193 x 197
191 x 199

( Answers under here )

You have now been introduced to a universally - applicable, versatile and incredibly simple mathematical methodology that can, with practice, help you to calculate numbers with incredible speed, memory and concentration.

Andt's it for the interlude. Now back to the grind, and the next part of the lessons.

Before you start, here are a handful of digits to practice on. I want you to look at each digit, and work out its nines complement: 9 - the digit.

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

( Answers under here )

This will be essential for the next stage.

See you next time.

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( Behind the text )

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