21 October 2010

Vedic Mathematics: Multiplying 2-, 3- and 10-digit numbers in your head

Addition, division, subtraction and multiplica...Image via WikipediaI remember being teased in fourth game for being able to multiply 4-digit numbers in my head.

Actually, it wouldn't have helped anyway because, had I told them that I couldn't do such a ridiculous thing, they would have simply teased me for not being able to multiply 4-digit numbers in my head.

But that's all water under the bridge, including my not being able to multiply 4-digit numbers in my head. Want to learn how?

Many cases are not that difficult once you know a couple of mental math tricks related to "Vedic mathematics".

Vedic?



While some people like to play of the mysticism of the name, Vedic Math is just a system of mathematics that was created by the Vedas people hundreds/thousands of years ago, rediscovered on old Sanskrit manuscripts by Bharati Krsna Tirthaji in 1911 and decoded from Sanskrit into modern-day mathematical notation over the following seven years. So, while not super-mystical, it is friggin' awesome.

Multiplying 2-digit numbers in your head

First Approach

Take the two digit numbers and pick the power of 10 that they are both closest to. To start simple, let's do 99x99. The closest "10", then is 100.

Write down the numbers and beside them the difference between the numbers and their "10". Since we picked 100, 100 - 99 = 1, or the difference between 99 and it's closest power of ten (100) is 1.

99 1
99 1

Next, do a subtraction across on of the diagonals. 99 - 1 - 98. This will be the first two digits of the answer.

Finally, multiply the numbers in the right column together. 1*1 = 1. This will be the end of the number.

Here we would expect a four-digit number, so the answer is 9801.

Let's try another one. How about 96x94?

So, the closest power of 10 is 100, again. Let's do the subtraction and write down our table:

96 4
94 6

Do a subtraction along the diagonal: 96 - 6 = 90. This will be the beginning of our number.

Multiply the right-hand column: 4*6 = 24. This will be the end of our number.

So 96*94 = 9024

One more before adding a wrinkle: 89*90

89 11
90 10

89 - 10 = 79

11 * 10 = 110

So, here we get into a little quandry. The answer isn't 79110. With some visualization of what the trick is trying to accomplish, we can sort it out.

What you are basically doing when you do the diagonal subtraction is finding out the most significant digits in an x-digit number. When you multiply 2-digit numbers like 89 and 90, we will expect the result to be a four-digit number (consider 90*90 = 8100). So, 79 represents 7900 in the equation.

What you are basically doing when you do the right-column multiplication is finding out the rest that you need to add to the total. So, in this case, 110 added to 7900 is 8010, which you will find to be the correct answer.

What about 45*49, though? Those are kind of far away from 100. Let's pick 50, then instead. Now 50 isn't a power of ten, but it is half of a power of 10. We just need to make one change to our algorithm to accommodate it.

45 5 (remember 50-45=5)
49 1

Do a diagonal subtract: 45 - 1 = 44

(new step) Divide that by 2 since 50 is half of 100: 44/2 = 22

Right-column multiplication: 5 * 1 = 5

(no need to adjust the right-column step)

Add them together: 2205

Let's do the same thing with 37*34

37 13
34 16

Diagonal subtraction: 37 - 16 = 21

Adjust by half: 21/5 = 10.5 (So this would mean 1050 since well get a four-digit number)

Right-column multiplication: 13 * 16 = 208

Add the two results: 1050 + 208 = 1258

So, that was kind of hard because of the 13*16. Well, there's a way to simplify that, too.

Instead of picking 50, let's pick 40. Again, not a power of 10, but we can make the appropriate adjustment.

37 3
34 6

Diagonal subtraction: 37 - 6 = 31

(adjustment) Looking downward this time, it is 4 times bigger than 10 (as opposed to being 2/5 of 100). So, we'll multiply 31 by 4. 31 * 4 = 124. (We could have multiplied by 2/5, but that sounds hard.) Since we are filling in a four-digit number, this represents 1240.

Right-column multiplication: 3 * 6 = 18

Add them together: 1240 + 18 = 1258

Second Approach

Remember that with two-digit numbers, we are filling in a four-digit number like this: _ _ _ _. As we take each step, we are filling in those blanks.

Take the two numbers and write them down like so: (in this case, we'll do 71*68)

7 1
6 8

Multiply the left-column: 7 * 6 = 42. This represents the left-most digits, or the hundreds place, so it will fill in the left-most blanks: 42 _ _

Cross-multiply and add: 7*8 + 6*1 = 56 +6 = 62. This represents the middle two digits, or the tens place, so it will fill in the middle two blanks: 482 _

Multiply the right column: 1 * 8 = 8. This represents the right-most digits, or the ones place, so it will fill in the right-most blanks: 4828

Not too bad, eh?

Let's try 84*76 now.

8 4
7 6

Left-column: 8 * 7 = 56
Cross-multiply and add: 8*6 + 7*4 = 76
Right-column: 4*6 = 24

56 _ _
_ 76 _
+ _ _ 24
--------------
6384

So, 84*76 = 6384

Muliplying three-digit numbers in your head


Don't worry, the approaches are the same, but it's good to illustrate.

First Approach

986*994. The closest power of 10 is 1000.

986 14 (1000-986=14)
994 6

Diagonal subtraction: 986 - 6 = 980 (We are going to expect a 6-digit number here, so this means 980 _ _ _)

Right-column multiplication: 14 * 6 = 84

Add the two results: 980000 + 84 = 980084

How about 452*487?

452 48
487 13

Diagonal subtraction: 452 - 13 = 439 (Six-digit number, so 439 _ _ _)

Adjust: 439 / 2 = 219.5 (so 2195 _ _)

Right-column multiplication: 48*13 = 624 (The the second approach for this multiplication)

Add: 219500 + 624 = 220124

How about 256*282? Let's pick 300 for our target.

256 44
282 18

Diagonal subtraction: 256 - 18 = 238

Adjust: 300 is three times 100, so 238*3 = 714 (here, we'll have a five-digit number, think 300x300=90000, so 714 _ _ )

Multiply: 44 * 18 = 792

Add: 71400 + 792 = 72192

Okay, one more, and then the second approach.

110*114. This one is a bugger because it is a long ways away from the nearest power of 10 above it (1000) and the nearest multiple of 100 above it (200). Would it work to look downward and just switch the diagonal subtraction to addition?

110 10 (distance from the power of 10 below)
114 14

Add instead of subtract: 124 (so, 124 _ _)

Multiply: 10*14 = 140

Add: 12400 + 140 = 12540

Second Approach

The general idea with this approach is that every number must be multiplied by every other number. It gets challenging when we have 4 or more digits, but for 2 or 3 is works really well.

So, 213*768

2 1 3
7 6 8

Now, this will be a six-digit number. _ _ _ _ _ _

2*7 = 14, so 14 _ _ _ _
2*6 = 12, so _ 12 _ _ _
2*8 = 16, so _ _ 16 _ _
1*7 = 7, so _ 0 7 _ _ _
1*6 = 6, so _ _ 0 6 _ _
1*8 = 8, so _ _ _ 0 8 _
3*7 = 21, so _ _ 21 _ _
3*6 = 18, so _ _ _ 18 _
3*8 = 24, so _ _ _ _ 24

Writing it out like this makes it seem impossible. However, if you keep a running tally in your head, it isn't so bad with some practice.

So, 140000 + 12000 + 1600 + 7000 + 600 + 80 + 2100 + 180 + 24 = 163584

Multiplying 10 digit numbers in your head. Ew...

So, there is no magical way to make this easy, but the two approaches above definitely make it easier.

Just for fun, then, let's impress our friends by multiplying 9999999999 by 9999999999. We'll get a 20-digit number.

9999999999 1
9999999999 1

Subtract: 9999999998

Multiply: 1 * 1 = 1

Add: 99999999980000000000 + 1 = 99999999980000000001

Wow!
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