"This above does not even cover the tip of the tip of the iceberg of knowledge that can be gleaned from Vedas. Hoping to put on more of the stuff before long..."

Those are last lines from a post on my previous blog on Vedic Mathematics, something which I wrote way back in 2005. Wonder what happened in between that broke me off from the simple joys of living.

Ah! the simplicity of all such methods. I wonder why we were forced into learning by rote all those innumerable multiplication tables after encountering this side of knowledge. The only thing that was somehow included in a mathematics textbook that I can recall is synthetic division. And that too was used to divide linear algebraic factors. When it can be easily used to divide any two numbers. Just replace the 'x' in every factor with 10. You have the number and you have the result.

One thing that I knew before I started the post today is the decimals when a number is divided by 7.

For eg: 1/7 = 0.142857; 2/7 = 0.285714; 3/7 = 0. 428571; 4/7 = 0.571428;... essentially the decimals repeat themselves in the order 1 4 2 8 5 7, something which is not difficult to remember, if we notice that the number when grouped give us 14 (7 * 2), 28(14 * 2), 57(28 * 2 +1 ). Even this series follows a pattern.

Trying to recall how to calculate the square of any number. Let's take 298, which is some random number but easy to use in this case. Simple steps:

Those are last lines from a post on my previous blog on Vedic Mathematics, something which I wrote way back in 2005. Wonder what happened in between that broke me off from the simple joys of living.

Ah! the simplicity of all such methods. I wonder why we were forced into learning by rote all those innumerable multiplication tables after encountering this side of knowledge. The only thing that was somehow included in a mathematics textbook that I can recall is synthetic division. And that too was used to divide linear algebraic factors. When it can be easily used to divide any two numbers. Just replace the 'x' in every factor with 10. You have the number and you have the result.

One thing that I knew before I started the post today is the decimals when a number is divided by 7.

For eg: 1/7 = 0.142857; 2/7 = 0.285714; 3/7 = 0. 428571; 4/7 = 0.571428;... essentially the decimals repeat themselves in the order 1 4 2 8 5 7, something which is not difficult to remember, if we notice that the number when grouped give us 14 (7 * 2), 28(14 * 2), 57(28 * 2 +1 ). Even this series follows a pattern.

Trying to recall how to calculate the square of any number. Let's take 298, which is some random number but easy to use in this case. Simple steps:

- 300 (100 * 3) - 298 = 2
- 298 - 2 (from 1) = 296
- 296 (from 2) * 3 (300 / 100) = 888 (300 * 3 - 4 * 3)
- 2(from 1) ^ 2 ( you need to square atleast something, right?? ) = 4
- 298 ^ 2 = 88804 (first 3 digits from step3 and last 2 digits from step4)

Explanation: Step1 - for a n-digited number, first figure out the base in terms of a 1 followed by n-1 zeros. For eg: if the number were to be 378 the base would be 100, if the number were to be 4781 the base would be 1000. Then figure out the multiple of that, which considering the above two cases would be (400 / 100 = ) 4, and in the second case would be (5000 / 1000 = ) 5. Then find the difference between the multiplied base and the actual number, which is 300 - 298 = 2.

Step2 - Subtract the difference from the actual number. 298 - 2 = 296.

Step3 - Multiply the result from 2 with the multiple that was found while doing step 1.

Step4 - Square the difference found in step1.

Step5 - Put results from step3 and step4 in right places. Since the base is 100, the result from step3 is to be put down as 2 digits and the result from step3 needs to be prefixed to this. If the number is big from step3 such that it cannot be accommodated in two digits, add the hundred place digit to the last digit of the result from step4 and proceed.

More examples:

278 ^ 2

Step 1: 300 - 278 = 22

Step 2: 278 - 22 = 256

Step 3: 256 * 3 = 768

Step 4: 22 ^ 2 = 484

Step 5: 278 ^ 2 = 76(8+4)84 = 7(6+1)284 = 77284

417 * 2

Step 1: 417 - 400 = 17 (since 417 is closer to 400 than 500, to reduce the work to be done in step4)

Step 2: 417 + 17 = 434

Step 3: 434 * 4 = 1736

Step 4: 17 ^ 2 = 289

Step 5: 417 ^ 2 = 173(6+2)89 = 173889

To successfully wade through step 4, you can follow the same process, albeit here the base would be 10. So recursively the square of any number can be found. But, obviously at the time I actually learned this, I was short of time and took a short cut (which I regret to this day!). I memorized the squares of all numbers from 1 - 50.

... 49 ^ 2 = 2401; 48 ^ 2 = 2304; 47 ^ 2 = 2209; 46 ^ 2 = 2116; ... et. all. I wish I had not done this and had spent more time practicing the actual method.

The one real book on this endless topic can be viewed here. I don't know if this book has the method to square the number as I describe it. But it has a lot of other such tricks up on its pages, with enough detailed algebraic explanation to satisfy the curiosity of the cat.

Lament: Sitting in front of this monitor and keyboard, I forget to use those few grey cells that I have there ( yeah, I do have some!!! ) and instead do this.

Step2 - Subtract the difference from the actual number. 298 - 2 = 296.

Step3 - Multiply the result from 2 with the multiple that was found while doing step 1.

Step4 - Square the difference found in step1.

Step5 - Put results from step3 and step4 in right places. Since the base is 100, the result from step3 is to be put down as 2 digits and the result from step3 needs to be prefixed to this. If the number is big from step3 such that it cannot be accommodated in two digits, add the hundred place digit to the last digit of the result from step4 and proceed.

More examples:

278 ^ 2

Step 1: 300 - 278 = 22

Step 2: 278 - 22 = 256

Step 3: 256 * 3 = 768

Step 4: 22 ^ 2 = 484

Step 5: 278 ^ 2 = 76(8+4)84 = 7(6+1)284 = 77284

417 * 2

Step 1: 417 - 400 = 17 (since 417 is closer to 400 than 500, to reduce the work to be done in step4)

Step 2: 417 + 17 = 434

Step 3: 434 * 4 = 1736

Step 4: 17 ^ 2 = 289

Step 5: 417 ^ 2 = 173(6+2)89 = 173889

To successfully wade through step 4, you can follow the same process, albeit here the base would be 10. So recursively the square of any number can be found. But, obviously at the time I actually learned this, I was short of time and took a short cut (which I regret to this day!). I memorized the squares of all numbers from 1 - 50.

... 49 ^ 2 = 2401; 48 ^ 2 = 2304; 47 ^ 2 = 2209; 46 ^ 2 = 2116; ... et. all. I wish I had not done this and had spent more time practicing the actual method.

The one real book on this endless topic can be viewed here. I don't know if this book has the method to square the number as I describe it. But it has a lot of other such tricks up on its pages, with enough detailed algebraic explanation to satisfy the curiosity of the cat.

Lament: Sitting in front of this monitor and keyboard, I forget to use those few grey cells that I have there ( yeah, I do have some!!! ) and instead do this.