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?

1 Answer
Feb 29, 2016

Answer:

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.