Tuesday, June 30, 2009

Vedic Mathematics Lesson 2: 10's Complements

You can read about my interest in Vedic Mathematics in this earlier post and find a spectacular application of mental computation using Vedic Mathematics principles in Lesson 1 of this series of posts.

Today's lesson is going to be quite short. I am going to explain how to calculate the 10's complement of any number. Essentially, the 10's complement of a number tells you how far the number is below the next higher power of 10. For instance, 89 is 11 below 100. I call 11 the 10's complement of 89. Some texts refer to 11 as the "deficit" of 89 also.

The 10's complement of a number is very useful in a variety of Vedic Mathematical computations. Thus, learning to work out the 10's complement quickly and (preferably) mentally will come in handy when we proceed to later lessons in Vedic Mathematics.

The sutra that tells us how to compute the 10's complement of a number reads Nikhilam Navatascaramam Dasataha. Literally translated, it means All From 9 And The Last From 10.

The practical application of it is actually quite easy to follow directly from the translation of the sutra. Simply put, take your number and subtract each number from 9 as you go from left to right (All From 9). Put down the answers you get as the digits of the 10's complement going from left to right. When you get to the last digit (right-most digit) of the number for which you are finding the complement, subtract it from 10 (The Last From 10) and write this answer down as the last digit (right-most) of the answer.

Note that since all the subtractions are of single digits from 9 and the value being subtracted from is 9 (which is the highest single-digit number in the decimal system), there arises no question of borrowing digits or doing other mental gymnastics to get the individual digits of the answer. Hopefully, subtracting a single-digit number from 10 should not involve any extraordinary mental gymnastics either. However, that last subtraction can lead to a minor problem we deal with later in this lesson.

Let us apply this lesson to a simple example. Let us take 389,384,753 as the number for which we need the 10's complement. Note that the number has 9 digits, so we are looking for the difference between 1,000,000,000 (1 followed by 9 zeroes, making it a 10-digit number) and the given number (10 raised to the power of 9 is the next higher power of 10 for the given number).

Taking the first digit of the given number from left, we get 3. Subtract it from 9 to get 6. 6 is the first digit of the answer. The next digit of the answer is 9-8 = 1. The third digit is 9-9 = 0, and so on. When we get to the right-most digit of the given number, we find that the right-most digit of the answer has to be 10 - 3 = 7. Remember to subtract the last number from 10 rather than 9 to complete the answer. The answer in this case turns out to be 610,615,247. You can verify the answer in any calculator that can handle 10 digits or more. But the method will work for numbers with any number of digits, even numbers that can not be handled by any calculator because they have too many digits.

Note that we defined the 10's complement as the deficit from the next higher power of 10. This is not necessary for the method to work. This method can be put to work to find the difference between the given number and any power of 10 that has more digits than the given number. Let me illustrate with another example.

Suppose we need to find the difference between 1,000,000 and 98,567. Note that next power of 10 that is larger than the given number is 100,000. Thus, to simply find its 10's complement, we would apply the formula illustrated above and find the answer to be 1,433 (the left-most digit computes to a zero, and has therefore been dropped).

To find the deficit from a higher power of 10, first find the 10's complement. This time do not drop zeroes from the left of the answer. We find the 10's complement of the given number to be 01,433. Now pad the 10's complement to the left with 9's until it has one less digit than the power of 10 from which we are trying to find the deficit. Another way to express this is as follows: pad the 10's complement to the left with 9's until it has the same number of digits as the power of 10 has zeroes. In our case, the power of 10 we are finding the deficit from has six zeroes. So, padding out the 10's complement with 9's to the left so that the answer is 6 digits long gives us 901,433.

Yet another way to express this that may be more intuitive is: pad the given number with zeroes to the left so that it has as many digits as the power of 10 has zeroes. Then find the 10's complement of the padded number using the same rule as before. By this method, first we get our padded number as 098,567. Finding the 10's complement of this number using the rule we explained in the beginning, we get 901,433.

The only trouble you might encounter in the application of this method is if the last digit of the number is a zero. In that case, subtracting it from 10 gives you a 2-digit answer (10) rather than a single-digit answer. The way to deal with this is to then put down zero as the last digit of the answer and carry over the 1 to the left hand side (add it to the number you found earlier for that digit). If that carryover leads to the second digit becoming 10, repeat the procedure, carrying over extra digits to the left as long as is necessary.

That may sound confusing, so the easier way to deal with this is as follows: if the number consists of n zeroes at the end, leave them off initially. Find the 10's complement of the remaining number with respect to the power of 10 just above the left-over number. Then add n zeroes back to the right of the answer you get.

Let me illustrate by finding the 10's complement of 89,000. n, in this case, is 3. By dropping the 3 zeroes from the end of 89,000, we get 89. The 10's complement of 89 with respect to the next higher power of 10 (100) is 11. Therefore, the 10's complement of 89,000 is 11,000 (which is 11 with 3 zeroes added back to its right).

That is all there is to it! It should be easy to reel the 10's complements of any number off in seconds using the mental trick illustrated here. Remember to practice! Happy computing and good luck!!

Note that there are applications out there that define the 10's complement as one more than the 10's complement we have computed in this lesson. That kind of 10's complement is useful in some computations involving subtraction and addition. Remember to not get confused by this distinction between the different definitions of 10's complements.

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I taught my kids the trick to finding the reciprocal of any 2-digit number ending in 9. I thought it would be confusing, given their ages, but they picked it up right away and had no trouble applying either method 1 or method 2 to any given problem. They then went and demonstrated their new-found talent to their mother, who was very proud of them and very happy with me! Now, I have to test them in a few days and make sure they remember what I taught them. That will also give me a chance to make sure I remember what I taught them!

In other news, it looks like I will be taking two weeks off to visit and take care of my parents after my father's surgery starting sometime next week. I have convinced my wife that nothing will go wrong when I am not around. That does not mean that she will be entirely at peace, but for now, she seems confident about handling things during my absence.

Our plan to visit my brother-in-law to help with his mother's surgery and its aftermath turned into a bit of a disastrous fiasco because of various problems. We planned to go out there on Friday. But the aircraft on the flight we were supposed to take was replaced with one that had 70 fewer seats. So, we changed our plans and were accommodated on a flight that would take us where we wanted to go with a connection. But at the last minute, this aircraft developed a mechanical problem that delayed the flight so the connection would not work. We had to give up that day.

To make a long story short, we ended up trying repeatedly to get out during the weekend, and then my wife and kids tried on Monday, and today also. Ultimately, the airline managed to accommodate them today evening on a flight and they are off. Since she is going to her brother's later than originally planned, we are still not sure when she is going to come back. We had originally planned on her being back before I left to visit my parents. But now we might have to reevaluate that option.

The weed-killer application using the hose-end sprayer actually worked. The weeds shrivelled up and died, and the lawn looks much better now. So, I might be switching to the hose-end sprayer permanently and ditching the pressure sprayer altogether.

This last weekend saw me doing more yardwork as I had to dig a 1 foot x 1 foot x 1 foot hole in the ground to plant a rosebush. The rosebush unfortunately sat in a friend's car in the hot sun for a while after it was bought, and came to us with shriveled leaves. We watered it (in the pot) for a few days, but it still did not show much signs of life. I tried arguing that it was a waste of effort digging such a large hole for a plant that may already be dead. But my wife convinced me that all the plant needed was abundant sunshine and natural soil to thrive. So, the plant is in the ground, under more than abundant sunshine, and has been watered for a couple of days now. Still no signs of life, leave alone thriving. Oh well, at least I got some exercise...

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