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...

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.
#color(white)()# 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#.
#color(white)()# 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.
#color(white)()# 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.
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Answer 2

One potential approach for the human brain to perform elementary mathematical computations more efficiently is through the use of mental math techniques such as approximation, estimation, and number sense. These techniques involve breaking down complex calculations into simpler steps, using patterns and relationships between numbers, and leveraging mental strategies to arrive at solutions quickly. Additionally, practicing mental math regularly can improve fluency and speed in performing mathematical operations.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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