1 - Examples of smart technique

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The simplicity and elegance of the ancient techniques is self-evident even in basic mathematical operations like multiplication and division, as will be shown here. Of course, Vedic mathematics goes beyond this, and covers algebra, trignometry, calculus and more. The discoverer of this system, the late Shankaracharya of Jagannatha Puri, His Holiness Bharati Krishna Tirthaji Maharaja, wrote 16 volumes on the subject matter but all the manuscripts were lost (details here), except for an introductory volume he wrote just before he attained samaddhi in 1960. Even in the sole existent volume, the Shankaracharya has given plenty of clues and pointers for further research. He lectured on this speedy system of mathematics and personally demonstrated its superior techniques before live audiences in India, America and the United Kingdom until he attained samaddhi in 1960. Here we give some simple examples of such technique, just to give an appetizer to a more comprehensive course on the subject. See how even large numbers can be easily and speedily juggled (multiplied/divided) by such technique. Without having to use a calculator at all! To illustrate each technique, some simple examples are given. If this is the first time you are encountering these techniques, they may seem too radically unconventional to be true. However the student who diligently tests the principles involved in each technique will discover that they work exceptionally well (definitely much better than the conventional methods!) and are applicable across the board, regardless of the size of the numbers involved. More technique, and the finer details of application, will be explained and reinforced with practical exercises in our classroom sessions. Click here to sign up for a classroom session.

2 - Long Multiplication reduced to one-line shortcut

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The Sanskrit sutra "Urdhva Tiryagbhyam", meaning "Vertically and Crosswise", gives a general technique for reducing long multiplication to a single-line shortcut. It works like this.

To multiply two numbers (of two or more digits), split each number into two parts. If the first number is a1 + b1 and the second number is a2 + b2, then the product of the two numbers is:
(a1 x a2) + (a1 x b2 + b1 x a2) + (b1 x b2)

The solution comprises three parts (as shown by the boxes and arrows above): the head, the middle, and the tail.

1. The digits on the right are multiplied vertically to get the tail part: b1 x b2 (excess carried over)
2. All digits are multipled crosswise and added together to get the middle part: a1 x b2 + b1 x a2 (excess carried over)
3. The digits on the left are multiplied vertically to get the head part: a1 x a2

Here is a simple example to illustrate this technique.
23 x 41 = 943

The steps are:

1. 3 x 1 = 3
2. 2 x 1 + 3 x 4 = 14, put down 4 and carry over 1
3. 2 x 4 = 8, plus the 1 carried over, is 9

The speed gain using this technique (over the conventional method of multi-line long multiplication) becomes more apparent when handling larger numbers. Here is another example involving excess carryover at each stage.
108 x 64 = 6912

The steps are:

1. 8 x 4 = 32, put down 2 and carry over 3
2. 10 x 4 + 8 x 6 = 88, plus the 3 carried over, is 91; put down 1 and carry over 9
3. 10 x 6 = 60, plus the 9 carried over, is 69

This powerful technique can be expanded upon to cover all cases of multiplication, not just two or three-digit numbers.

3 - Long Division reduced to one-line shortcut

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The Sanskrit sutras offer several techniques for division. Some of them work spectacularly to reduce to a single line certain cases of division that may take dozens of lines of calculation in the conventional method of long division taught in schools.

While there are special techniques reserved for special cases, we highlight here a general-purpose long division technique that will work for both small and large numbers. It is basically applying in reverse the Urdhva Tiryagbhyam (Vertically and Crosswise) principle used to simplify multiplication. Such is the power and elegance of this technique that we can easily take a large number and make short work of it by any divisor of two or more digits.

Take for example, 716769 ÷ 54. Yes, you too CAN work it out manually -- and in one line -- without having to reach for the calculator! You will instead rely on the calculator inside your head and the Dhvajanka sutra, meaning "on top of the flag". The trick is to reduce the divisor to a mentally manageable value by putting its other digits "on top of the flag". In this example, the divisor will be reduced to 5 (instead of 54) by pushing the 4 up the flagpost, as shown below. Corresponding to the number of digits flagged on top (in this case, one), the rightmost part of the number to be divided is split to mark the placeholder of the decimal point or the remainder portion.
Now observe carefully as we walk through the steps of this example:
716769 ÷ 54 = 13273.5

1. 7 ÷ 5 = 1 remainder 2. Put the quotient 1, the first digit of the solution, in the first box of the bottom row and carry over the remainder 2

2. The product of the flagged number (4) and the previous quotient (1) must be subtracted from the next number (21) before the division can proceed. 21 - 4 x 1 = 17
   17 ÷ 5 = 3 remainder 2. Put down the 3 and carry over the 2

3. Again subtract the product of the flagged number (4) and the previous quotient (3), 26 - 4 x 3 = 14
   14 ÷ 5 = 2 remainder 4. Put down the 2 and carry over the 4

4. 47 - 4 x 2 = 39
   39 ÷ 5 = 7 remainder 4. Put down the 7 and carry over the 4

5. 46 - 4 x 7 = 18
   18 ÷ 5 = 3 remainder 3. Put down the 3 and carry over the 3

6. 39 - 4 x 3 = 27. Since the decimal point is reached here, 27 is the raw remainder. If decimal places are required, the division can proceed as before, filling the original number with zeros after the decimal point
   27 ÷ 5 = 5 remainder 2. Put down the 5 (after the decimal point) and carry over the 2

7. 20 - 4 x 5 = 0. There is nothing left to divide, so this cleanly completes the division

While this example is easily solved, there are finer details in the application of technique that will be highlighted in subsequent examples later.

4 - Squaring

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First, a nifty shortcut! The square of a number ending in 5 is almost a no-brainer. If n is the number formed by the preceding digit/s (before the 5), get the product of n and n+1. Then just append 25 (i.e. 5 x 5) to this product.

For example, 75²:
7 x 8 = 56; therefore solution is 5625.

Another example, 115²:
11 x 12 = 132; therefore solution is 13225

For other cases of squaring, the same shortcut techniques used in multiplication may be utilised. Especially the general-purpose Urdhva Tiryagbhyam (Vertically and Crosswise) formula. To get the square of a number (of two or more digits), simplify by splitting it into at least two parts, a and b.

Thus (a + b)² = a² + 2ab + b²

The solution comprises three parts, neatly fitting the three boxes shown above. Just adjust for excess carry over.

1. the head: a²
2. the middle: crosswise multiplication and doubling a x b x 2
3. the tail: b²

Here is a simple example to illustrate this technique.
23² = 529

The steps are:

1. tail: 3² = 9, put it down in the rightmost box
2. middle: 2 x 3 x 2 = 12, put down the 2 in the middle box and carry over the 1
3. head: 2² = 4, plus the 1 carried over, is 5 in the left box

Another example.
108² = 11664

The steps are:

1. tail: 8² = 64, put down the 4 and carry over the 6
2. middle: 10 x 8 x 2 = 160, plus the 6 carried over, is 166; put down the 6 and carry over the 16
3. head: 10 x 10 = 100, plus the 16 carried over, is 116

The same technique can be expanded upon to handle the squaring of bigger numbers too.

5 - Cubing

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Extrapolating on the principles used for squaring, to get the cube of a number (of two or more digits), simplify by splitting the number into two parts, a and b.

Thus (a + b)³ = a³ + 3a²b + 3ab² + b³

The solution comprises four parts, neatly fitting the four boxes shown above. Just adjust for excess carry over.

1. a³
2. a² x b x 3
3. b² x a x 3
4. b³

A simple example to illustrate this technique.
23³ = 12167

The steps are:

1. 3³ = 27, put down the 7 in the rightmost box and carry over the 2
2. 3² x 2 x 3 = 54, plus the 2 carried over is 56, put down the 6 and carry over the 5
3. 2² x 3 x 3 = 36, plus the 5 carried over is 41, put down the 1 and carry over the 4
4. 2³ = 8, plus the 4 carried over, is 12 in the leftmost box

Another example.
108³ = 1259712

The steps are:

1. 8³ = 512, put down the 2 in the rightmost box and carry over the 51
2. 8² x 10 x 3 = 1920, plus the 51 carried over is 1971, put down the 1 and carry over the 197
3. 10² x 8 x 3 = 2400, plus the 197 carried over is 2597, put down the 7 and carry over the 259
4. 10³ = 1000, plus the 259 carried over, is 1259 in the leftmost box

That's a million-dollar figure worked out manually!

6 - To the fourth power

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Extrapolating on the principles used for cubing, to get the fourth power of a number (of two or more digits), simplify by splitting the number into two parts, a and b.

Thus (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

The solution comprises five parts, neatly fitting the five boxes shown above. Just adjust for excess carry over.

1. a⁴
2. a³ x b x 4
3. a² x b² x 6
4. b³ x a x 4
5. b⁴

A simple example to illustrate this technique.
23⁴ = 279841

The steps are:

1. 3⁴ = 81, put down the 1 in the rightmost box and carry over the 8
2. 3³ x 2 x 4 = 216, plus the 8 carried over is 224, put down the 4 and carry over the 22
3. 2² x 3² x 6 = 216, plus the 22 carried over is 238, put down the 8 and carry over the 23
4. 2³ x 3 x 4 = 96, plus the 23 carried over is 119, put down the 9 and carry over the 11
5. 2⁴ = 16, plus the 11 carried over, is 27 in the leftmost box

Another example.
108⁴ = 136048896

The steps are:

1. 8⁴ = 4096, put down the 6 in the rightmost box and carry over the 409
2. 8³ x 10 x 4 = 20480, plus the 409 carried over is 20889, put down the 9 and carry over the 2088
3. 10² x 8² x 6 = 38400, plus the 2088 carried over is 40488, put down the 8 and carry over the 4048
4. 10³ x 8 x 4 = 32000, plus the 4048 carried over is 36048, put down the 8 and carry over the 3604
5. 10⁴ = 10000, plus the 3604 carried over, is 13604 in the leftmost box

That's a hundred million-dollar figure worked out manually!