Question

Is there a easier more efficient approach for the human brain to perform elementary mathematical computations (+-×÷) that differs from what is traditionally taught in school?

Answer 1

It depends...

Explanation:

There are various tricks and techniques to make it easier to perform mental arithmetic, but many involve memorising more things first.

For example, `(a-b)(a+b) = a^2-b^2`. Hence if you know a few square numbers you can sometimes conveniently multiply two numbers by taking the difference of squares. For example:

`17*19 = (18-1)(18+1) = 18^2-1^2 = 324 - 1 = 323`

So rather than memorise the whole "times table" you can memorise the 'diagonal' and use a little addition and subtraction instead.

You might use the formula:

`ab = ((a+b)/2)^2 - ((a-b)/2)^2`

This tends to work best if `a` and `b` are both odd or both even.

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For subtraction, you can use addition with `9`'s complement then add `1`. For example, the (`3` digit) `9`'s complement of `358` would be `641`. So instead of subtracting `358`, you can add `641`, subtract `1000` and add `1`.

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Other methods for multiplying numbers could use powers of `2`. For example, to multiply any number by `17` double it `4` times then add the original number.

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At a more advanced level, the standard Newton Raphson method for finding the square root of a number `n` is to start with an approximation `a_0` then iterate to get better approximations using a formula like:

`a_(i+1) = (a_i^2+n)/2`

This is all very well if you are using a four function calculator, but I prefer to work with rational approximations by separating the numerator and denominator of `a_i` as `p_i` and `q_i` then iterating using:

`p_(i+1) = p_i^2+n q_i^2`

`q_(i+1) = 2 p_i q_i`

If the resulting numerator/denominator pair has a common factor, then divide both by that before the next iteration.

This allows me to work with integers instead of fractions. Once I think I have enough significant figures I then long divide `p_i/q_i` if I want a decimal approximation.