# Submit an Article to TRA

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You can submit an article to TRA by e-mailing Mr. Stolyarov at gennadystolyarovii@gmail.com.

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You can submit an article to TRA by e-mailing Mr. Stolyarov at gennadystolyarovii@gmail.com.

Hi Gen,

Not an article. I came across your IGG campaign and I’d like to help with video support. I’m the new Assistant Membership Secretary at LC and I have video and animation skills I’d like to lend to your project pro bono. I was thinking we could make an animated web comic out of it to reach additional audiences and give it a boost. Depending on how long it is, I would either do a handful of scenes to get people interested or the whole thing. It sounds like you have a decent mic (though you might need some coaching), you could do the narration or you can get professional narration at $80 for two minutes. I have subscriptions to everything else I’ll need to produce this, so there won’t be any other costs. Email me and I’ll link you to my portfolio.

Peter

Not an article, just a tidbit. I’m sure you’re familiar with the funny long division-like method for finding square roots. But what about cube roots, or nth roots? They’re all really extensions of the Binomial Theorem. (BTW: A prof I had circa 1986 assigned the task of coming up with the cube root procedure above and then proving — mathematically proving — that it works. No one could do it, and the person who had recently won the Fields Medal ‘had trouble with it’.) Here is my description of these methods:

FINDING NTH ROOTS BY HAND!

Review: Finding a SQUARE root by hand amounts to

1) Partitioning your number into two-digit intervals and finding an initial square closest to the first two digits as a base-ten number without going over. This is the first and only digit of your estimate so far.

2) Subtracting your estimate squared from those first two digits and bringing down the next two digits down next to that difference to form a new number, a modified difference. Or you multiply the difference by 102 and add the next two digits as a base ten number.

3) Doubling the estimate, multiplying it by 10 and finding a number from 0 to 9 which after being added to this amount forms a product with this amount plus itself that is less than or equal to the modified difference in 2) above. This becomes the second digit of your estimate so far.

4) Subtracting your doubled and multiply multiplied estimate from that modified difference and bringing down the next two digits down next to that new difference to form a new number, a new modified difference. Or you multiply the new difference by 102 and add the next two digits as a base ten number.

5) Doubling the estimate so far, multiplying it by 10 and finding a number from 0 to 9 which after being added to this amount forms a product with this amount plus itself that is less than or equal to the new modified difference in 4) above. This becomes the third digit of your estimate so far.

6) Keep repeating steps 4) and 5) to find more digits of your estimate.

In a nutshell: Double your last estimate (a) and multiply by 10, then find the digit (b) that you can add to that so that the same digit multiplies the resulting sum to less than or equal to the last difference (d) multiplied by 102 with the next two digits added as a two-digit base-ten number (d1d2).

OR

Find b such that (2a(10) + b)b is less than or equal to 100d + d1d2

(where a is the last estimate of the square root so far and d1d2 are the next

2 digits of your original number).

Prelude: Finding a CUBE root by hand amounts to

1) Partitioning your number into three-digit intervals and finding an initial cube closest to the first three digits as a base-ten number without going over. This is the first and only digit of your estimate so far.

2) Subtracting your estimate cubed from those first three digits and bringing down the next three digits next to that difference to form a new number, a modified difference. Or you multiply the difference by 103 and add the next three digits as a base ten number.

3) Tripling the estimate squared, multiplying it by 102 and finding a number from 0 to 9 which, after being squared and added to this amount and then multiplied by the estimate tripled and added again to the amount, forms a product with this final sum that is less than or equal to the modified difference in 2) above. This becomes the second digit of your estimate so far.

4) Subtracting your tripled and multiply multiplied estimate from that modified difference and bringing down the next three digits next to that new difference to form a new number, a new modified difference. Or you multiply the new difference by 103 and add the next three digits as a base ten number.

5) Tripling the estimate so far squared, multiplying it by 102 and finding a number from 0 to 9 which, after being squared and added to this amount and then multiplied by the estimate tripled and added again to the amount, forms a product with this final sum that is less than or equal to the new modified difference in 4) above. This becomes the third digit of your estimate so far.

6) Keep repeating steps 4) and 5) to find more digits of your estimate.

In a nutshell: Triple your last estimate (a) squared and multiply it by 102, then find the digit (b) that can be multiplied by 3a times 10 and added to that amount and also squared and added to the amount so that the same digit b multiplies the resulting sum to less than or equal to the last difference (d) multiplied by 103 with the next three digits added as a three-digit base-ten number (d1d2d3).

OR

Find b such that (3a2(102) + 3ab(10) + b2)b is less than or equal to

1000d + d1d2d3 (where a is the last estimate of the cube root so far and d1d2d3

are the next 3 digits of your original number).

Finding an nth root by hand amounts to

1) Partitioning your number into n-digit intervals and finding an initial number (a) raised to the nth power closest to the first n digits as a base-ten number without going over. This is the first and only digit of your estimate so far.

2) Subtracting your estimate to the nth power (an) from those first n digits and bringing down the next n digits next to that difference to form a new number, a modified difference. Or you multiply the difference by 10n and add the next n digits as a base-ten number.

3) Finding a number b such that (nC1 an – 1 (10n – 1) + nC2 an – 2 b(10n – 2 ) + . . . + nCn – 1 a bn – 2 (10 ) + nCn a0 bn – 1 (100 ) )b is less than or equal to the modified difference in 2) above (10n (d ) + d1d2. . . dn). This becomes the second digit of your estimate so far.

4) Subtracting the two quantities above in 3) and finding your new modified difference

5) Finding a new number b and using your estimate so far, a, such that (nC1 an – 1 b0(10n – 1) + nC2 an – 2 b1(10n – 2 ) + . . . + nCn – 1 a1 bn – 2 (10 ) + nCn a0 bn – 1 (100 ) )b is less than or equal to the new

modified difference in 4) above (10n (d ) + d1d2. . . dn). This becomes the third digit of your estimate so far.

6) Keep repeating steps 4) and 5) to find more digits of your estimate.

In a nutshell:

Find b such that

(nC1 an – 1 b0(10n – 1) + nC2 an – 2 b1(10n – 2 ) + . . .

+ nCn – 1 a1 bn – 2 (10 ) +nCn a0 bn – 1 (100 ) )b is less than or equal to

10n (d ) + d1d2. . . dn (where a is the last estimate of the nth root so far and d1d2. . . dn are the next n digits of your original number).

Note: This is basically a rolling binomial expansion minus the lead term, where the two terms involved are the prior estimate multiplied by 10 and the next digit to improve the estimate.

The sub- and superscripts are coming out as regular font above!

Better yet, see my wikiHow on finding nth roots:

http://www.wikihow.com/Find-Nth-Roots-by-Hand