Consider the following three sutras:

1. "All from 9 and the last from 10," and its corollary: "Whatever the extent of its deficiency, lessen it still further to that very extent; and also set up the square (of that deficiency)."
2. "By one more than the previous one," and its corollary: "Proportionately."
3. "Vertically and crosswise," and its corollary: "The first by the first and the last by the last."

The first rather cryptic formula is best understood by way of a simple example: let us multiply 6 by 8.

1. First, assign as the base for our calculations the power of 10 nearest to the numbers which are to be multiplied. For this example our base is 10.
2. Write the two numbers to be multiplied on a paper one above the other, and to the right of each write the remainder when each number is subtracted from the base 10. The remainders are then connected to the original numbers with minus signs, signifying that they are less than the base 10.

        6-4
        8-2

3. The answer to the multiplication is given in two parts. The first digit on the left is in multiples of 10 (i.e. the 4 of the answer 48). Although the answer can be arrived at by four different ways, only one is presented here. Subtract the sum of the two deficiencies (4 + 2 = 6) from the base (10) and obtain 10 - 6 = 4 for the left digit (which in multiples of the base 10 is 40).

        6-4
        8-2
        4

4. Now multiply the two remainder numbers 4 and 2 to obtain the product 8. This is the right hand portion of the answer which when added to the left hand portion 4 (multiples of 10) produces 48.

        6-4
        8-2
        ----
        4/8

Another method employs cross subtraction. In the current example the 2 is subtracted from 6 (or 4 from 8) to obtain the first digit of the answer and the digits 2 and 4 are multiplied together to give the second digit of the answer. This process has been noted by historians as responsible for the general acceptance of the X mark as the sign of multiplication. The algebraical explanation for the first process is

        (x-a)(x-=x(x-a- + ab

where x is the base 10, a is the remainder 4 and b is the remainder 2 so that

        6 = (x-a) = (10-4)
        8 = (x- = (10-2)

The equivalent process of multiplying 6 by 8 is then

        x(x-a- + ab or
        10(10-4-2) + 2x4 = 40 + 8 = 48

These simple examples can be extended without limitation. Consider the following cases where 100 has been chosen as the base:

        97 - 3 93 - 7 25 - 75
        78 - 22 92 - 8 98 - 2
        ______ ______ ______

        75/66 85/56 23/150 = 24/50

In the last example we carry the 100 of the 150 to the left and 23 (signifying 23 hundred) becomes 24 (hundred). Herein the sutra's words "all from 9 and the last from 10" are shown. The rule is that all the digits of the given original numbers are subtracted from 9, except for the last (the righthand-most one) which should be deducted from 10.

Consider the case when the multiplicand and the multiplier are just above a power of 10. In this case we must cross-add instead of cross subtract. The algebraic formula for the process is: (x+a)(x+ = x(x+a+ + ab. Further, if one number is above and the other below a power of 10, we have a combination of subtraction and addition: viz:

        108 + 8 and 13 + 3
        97 - 3 8 - 2
        _______ ______

105/-24 = 104/(100-24) = 104/76 11/-6 = 10/(10-6) = 10/4

The Sub-Sutra: "Proportionately" Provides for those cases where we wish to use as our base multiples of the normal base of powers of ten. That is, whenever neither the multiplicand nor the multiplier is sufficiently near a convenient power of 10, which could serve as our base we simply use a multiple of a power of ten as our working base, perform our calculations with this working base and then multiply or divide the result proportionately.

To multiply 48 by 32, for example, we use as our base 50 = 100/2, so we have

    Base 50 48 - 2
    32 - 18
    ______

        2/ 30/36 or (30/2) / 36 = 15/36

Note that only the left decimals corresponding to the powers of ten digits (here 100) are to be effected by the proportional division of 2. These examples show how much easier it is to subtract a few numbers, (especially for more complex calculations) rather than memorize long mathematical tables and perform cumbersome calculations the long way.

In a class, a math professor claims that he can prove everything under the assumption that 1+1=1.
A student challenges him: "Then prove that you're the pope!"
He ponders for a moment and then replies: "I am one, and the pope is one. Therefore, the pope and I are one."
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Back in the old days - when slide rules were still the most sophisticated computing equipment available to scientists and engineers...
Engineering students are taking a math final. Of course, slide rules are not allowed. And, of course, someone is cheating and has brought a slide rule to the exam. He is hiding it under his desk, but the student sitting to his left - who is stuck on a difficult calculation - has noticed it.
"Hey", he whispers. "Can you help me? What's three times six?"
His classmate reaches for his slide rule, and after a few seconds replies: "Nineteen."
"Are you sure?"
The other student reaches again for his slide rule, and after another few seconds replies: "You're right. It's closer to eighteen - eighteen point three, to be precise."
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Two men are sitting in the basket of a balloon. For hours, they have been drifting through a thick layer of clouds, and they have lost orientation completely. Suddenly, the clouds part, and the two men see the top of a mountain with a man standing on it.
"Hey! Can you tell us where we are?!"
The man doesn't reply. The minutes pass as the balloon drifts past the mountain. When the balloon is about to be swallowed again by the clouds, the man on the mountain shouts: "You're in a balloon!"
"That must have been a mathematician."
"Why?"
"He thought long and thoroughly about what to say. What he eventually said was irrefutably correct. And it was of no use whatsoever..."
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Two men are sitting in the basket of a balloon. For hours, they have been drifting through a thick layer of clouds, and they have lost orientation completely. Suddenly, the clouds part, and the two men see the top of a mountain with a man standing on it.
"Hey! Can you tell us where we are?!"
The man doesn't reply. The minutes pass as the balloon drifts past the mountain. When the balloon is about to be swallowed again by the clouds, the man on the mountain shouts: "You're in a balloon!"
"That must have been a mathematician."
"Why?"
"He thought long and thoroughly about what to say. What he eventually said was irrefutably correct. And it was of no use whatsoever..."

There was a pilot flying a small single engine charter plane, with a couple of very important executives on board. He was coming into Seattle airport through thick fog with less than 10m visibility when his instruments went out. So he began circling around looking for landmark. After an hour or so, he starts running pretty low on fuel and the passengers are getting very nervous.

Finally, a small opening in the fog appears and he sees a tall building with one guy working alone on the fifth floor. The pilot banks the plane around, rolls down the window and shouts to the guy "Hey! Where am I?" To this, the solitary office worker replies "You're in a plane." 


The pilot rolls up the window, executes a 275 degree turn and proceeds to execute a perfect blind landing on the runway of the airport 5 miles away. Just as the plane stops, so does the engine as the fuel has run out.

The passengers are amazed and one asks how he did it. "Simple" replies the pilot, "I asked the guy in that building a simple question. The answer he gave me was 100 percent correct but absolutely useless, therefore that must be Microsoft's support office and from there the airport is just a while away."