# 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?

Feb 29, 2016

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

$17 \cdot 19 = \left(18 - 1\right) \left(18 + 1\right) = {18}^{2} - {1}^{2} = 324 - 1 = 323$

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:

$a b = {\left(\frac{a + b}{2}\right)}^{2} - {\left(\frac{a - b}{2}\right)}^{2}$

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:

${a}_{i + 1} = \frac{{a}_{i}^{2} + n}{2}$

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.