User blog:Edwin Shade/Just Practice, That's All !

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Reverse Mathematics
Let $$$$ indicate the base-n reversal of the natural number $$x$$, where n is assumed to be 10. It is to be read "rev-x", as in the 'reverse of x'.

The following will demonstrate a method of finding the solution to the simple quadratic equation $$x^2+^2-f=0$$ when x is less than 1,000.

First it is necessary to write out $$x$$ as a number of the form $$100a+10b+c$$, such that $$$$ can be notated as $$100c+10b+a$$.

The next step is to write out a table, as below, which calculates the possible values of $$a^2+c^2 (mod\text{ }10)$$, or the last digit of $$f$$ in the above mentioned quadratic equation.

0 1  2  3  4  5  6  7  8  9  a    _  _  _  _  _  _  _  _  _  _ 0| 0 1  4  9  6  5  6  9  4  1 1| 1  2  5  0  7  6  7  0  5  2 2| 4  5  8  3  0  9  0  3  8  5 3| 9  0  3  8  5  4  5  8  3  0 4| 6  7  0  5  2  1  2  5  0  7 5| 5  6  9  4  1  0  1  4  9  6 6| 6  7  0  5  2  1  2  5  0  7 7| 9  0  3  8  5  4  5  8  3  0 8| 4  5  8  4  0  9  0  3  8  5 9| 1  2  5  0  7  6  7  0  5  2 b

Given this last digit of $$f$$, we are able to narrow down the possible values for a and b, and subsequently we are able to brute force check the possible values of a and b at almost a glance, until we are left with only two or three possible values, which we then check carefully.

After this we input the appropiate the values for both a and c, then calculate (100a+10b+c)^2+(100c+10b+a)^2, putting the b-terms off to one side, and the large number off to another. Set this equal to f, after which you should subtract f from both sides and then solve the resulting quadratic equation. This will give you the value of b, and then you will derive a solution for $$x^2+^2-f=0$$ !

Mental Mathematics
The following will go through some math problems that despite their daunting exterior are actually almost trivial to perform mentally, and with this list you will be able to impress your non-mathematical friends, (although I myself use this as a self-challenge, like practicing the multiplication table), or more humbly, build a repertoire of arithmetical shortcuts by which you may one day be able to circumvent the most circuitous problem most efficiently.

Extracting Roots Mentally
Perhaps the simplest of cases is the mental extraction of cube roots, or more specifically, two-digit cube roots. Firstly, memorize the following correspondence of digits, which are placed in the A-rack, and the N-rack respectively.

A 0  1  2  3  4  5  6  7  8  9 N 0  1  8  7  4  5  6  3  2  9

Next, become familiar with the first nine cubic numbers, beginning from 1^3, or just 1.

1^3  2^3   3^3   4^3   5^3   6^3   7^3   8^3   9^3   1     8    27    64   125   216   343   512   729

This is all that you need to know in order to extract two-digit cubic roots mentally ! Ask a friend to type in a two-digit number in a calculator, then read to you the result. Take the number in the thousand's place, and see which two cubic numbers it falls between. The cubic root of the lesser of these two numbers will be the first digit of the two-digit number. Lastly, take the last digit your friend reads to you and consider it as an entry in the N-rack, then match it up with it's corresponding entry in the A-rack; this is your second digit. That's all.

Squaring Numbers Mentally
I will first demonstrate a method to square two digit numbers, then show how it can be adapted to three digit numbers, and lastly exhibit a quick way I have found to square four-digit numbers.

Firstly, it is assumed you have knowledge of the first 5 squares, (e.g. 1, 4, 9, 16, and 25), and you are well-familiar with the multiplication table, (mainly for speed). To square a two-digit number we will be exploiting the algebraic property that $$(a+b)(a-b)=a^2-b^2$$.

When you are given a two-digit number by your friend, (or whomever happens to give it to you), you must first find the closest multiple of ten to that number, which we'll call B. Take the difference between the two-digit number and that closest multiple of ten, and subtract it from your two-digit number to yield a new number C.

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