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?

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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?

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Answer 1 · 2016-02-29T00:52:17

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

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

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:

$a b = {(\frac{a + b}{2})}^{2} - {(\frac{a - b}{2})}^{2}$ $ab = ((a+b)/2)^2 - ((a-b)/2)^2$

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

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

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Other methods for multiplying numbers could use powers of $2$ $2$ . For example, to multiply any number by $17$ $17$ double it $4$ $4$ 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 $n$ $n$ is to start with an approximation $a_{0}$ $a_0$ then iterate to get better approximations using a formula like:

$a_{i + 1} = \frac{a_{i}^{2} + n}{2}$ $a_(i+1) = (a_i^2+n)/2$

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 $a_{i}$ $a_i$ as $p_{i}$ $p_i$ and $q_{i}$ $q_i$ then iterating using:

$p_{i + 1} = p_{i}^{2} + n q_{i}^{2}$ $p_(i+1) = p_i^2+n q_i^2$

$q_{i + 1} = 2 p_{i} q_{i}$ $q_(i+1) = 2 p_i q_i$

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 $\frac{p_{i}}{q_{i}}$ $p_i/q_i$ if I want a decimal approximation.

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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

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