Mental calculation
People may use mental calculation when computing tools are not available, when it is faster than other means of calculation (such as conventional educational institution methods), or even in a competitive context.
Many of these techniques take advantage of or rely on the decimal numeral system.
When multiplying, a useful thing to remember is that the factors of the operands still remain.
This method can be used to subtract numbers left to right, and if all that is required is to read the result aloud, it requires little of the user's memory even to subtract numbers of arbitrary size.
For any 2-digit by 2-digit multiplication problem, if both numbers end in five, the following algorithm can be used to quickly multiply them together:[1] As a preliminary step simply round the smaller number down and the larger up to the nearest multiple of ten.
In this case: The algorithm reads as follows: Where t1 is the tens unit of the original larger number (75) and t2 is the tens unit of the original smaller number (35).
To minimize the number of elements being retained in one's memory, it may be convenient to perform the sum of the "cross" multiplication product first, and then add the other two elements: i.e., in this example to which is it is easy to add 21: 281 and then 800: 1081 An easy mnemonic to remember for this would be FOIL.
For example: and where 7 is a, 5 is b, 2 is c and 3 is d. Consider this expression is analogous to any number in base 10 with a hundreds, tens and ones place.
FOIL can also be looked at as a number with F being the hundreds, OI being the tens and L being the ones.
This method can be adjusted to multiply by eight instead of nine, by doubling the number being subtracted; 8 × 27 = 270 − (2×27) = 270 − 54 = 216.
The product for any larger non-zero integer can be found by a series of additions to each of its digits from right to left, two at a time.
If a number sums to 10 or higher take the tens digit, which will always be 1, and carry it over to the next addition.
Finally copy the multipliers left-most (highest valued) digit to the front of the result, adding in the carried 1 if necessary, to get the final product.
In the case of a negative 11, multiplier, or both apply the sign to the final product as per normal multiplication of the two numbers.
Knowing that 152 is 225 and 22 is 4, simple subtraction shows that 225 − 4 = 221, which is the desired product.
This is because (x + 1)2 − x2 = x2 + 2x + 1 − x2 = x + (x + 1) x2 = (x − 1)2 + (2x − 1) Take a given number, and add and subtract a certain value to it that will make it easier to multiply.
For example, students who have memorized their squares from 1 to 24 can apply this method to any integer from 76 to 124.
The actual square root of 15 is 3.872983... One thing to note is that, no matter what the original guess was, the estimated answer will always be larger than the actual answer due to the inequality of arithmetic and geometric means.
Note that if n2 is the closest perfect square to the desired square x and d = x - n2 is their difference, it is more convenient to express this approximation in the form of mixed fraction as
Expanding yields If 'a' is close to the target, 'b' will be a small enough number to render the
out and rearrange the equation to and therefore that can be reduced to Alternatively, this approach to square root approximation can be viewed as a single step of Newton's method.
The first step in approximating the common logarithm is to put the number given in scientific notation.
Physical exertion of the proper level can lead to an increase in performance of a mental task, like doing mental calculations, performed afterward.
[2] It has been shown that during high levels of physical activity there is a negative effect on mental task performance.
[3] This means that too much physical work can decrease accuracy and output of mental math calculations.
Physiological measures, specifically EEG, have been shown to be useful in indicating mental workload.
[4] Using an EEG as a measure of mental workload after different levels of physical activity can help determine the level of physical exertion that will be the most beneficial to mental performance.
Previous work done at Michigan Technological University by Ranjana Mehta includes a recent study that involved participants engaging in concurrent mental and physical tasks.
This procedure, mostly used in cognitive experiments, suggests mental subtraction is useful in testing the effects maintenance rehearsal can have on how long short-term memory lasts.
It consists of four different standard tasks --- addition of ten ten-digit numbers, multiplication of two eight-digit numbers, calculation of square roots, and calculation of weekdays for given dates --- in addition to a variety of "surprise" tasks.
