User:Eanigro/mental math

comments:

See User_talk:Eanigro/mental_math page for comments.

ShaneTMueller (talk) 22:41, 17 February 2013 (UTC)

Approximating common logs (log base 10)
To approximate a common log (to at least one decimal point accuracy), a few log rules, and the memorization of a few logs is required. One must know:
 * log(a x b) = log(a) + log(b)
 * log(a / b) = log(a) - log(b)
 * log(0) does not exist
 * log(1) = 0
 * log(2) ~ .30
 * log(3) ~ .48
 * log(7) ~ .85

From this information, one can find the log of any number 1-9.
 * log(1) = 0
 * log(2) ~ .30
 * log(3) ~ .48
 * log(4) = log(2 x 2) = log(2) + log(2) ~ .60
 * log(5) = log(10 / 2) = log(10) - log(2) ~ .70
 * log(6) = log(2 x 3) = log(2) + log(3) ~ .78
 * log(7) ~ .85
 * log(8) = log(2 x 2 x 2) = log(2) + log(2) + log(2) ~ .90
 * log(9) = log(3 x 3) = log(3) + log(3) ~ .96
 * log(10) = 1 + log(1) = 1

The first step in approximating the common log is to put the number given in scientific notation. For example, the number 45 in scientific notation is 4.5 x 10^1, but we will call it a x 10^b. Next, find the log of a, which is between 1 and 10. Start by finding the log of 4, which is .60, and then the log of 5, which is .70 because 4.5 is between these two. Next, and skill at this comes with practice, place a 5 on a logarithmic scale between .6 and .7, somewhere around .653 (NOTE: the actual value of the extra places will always be greater than if it were placed on a regular scale. i.e., you would expect it to go at .650 because it is halfway, but instead it will be a little larger, in this case .653) Once you have obtained the log of a, simply add b to it to get the approximation of the common log. In this case, a + b = .653 + 1 = 1.653. The actual value of log(45) ~ 1.65321.

The same process applies for numbers between 0 and 1. For example, .045 would be written as 4.5 x 10^-2. The only difference is that b is now negative, so when adding you are really subtracting. This would yield the result .653-2, or -1.347.

Other systems
There are many other methods of calculation in mental mathematics. The list below shows a few other methods of calculating, though they may not be entirely mental.


 * Vedic mental mathematics - (Vedic Mathematics page has been co-opted by history of Indian mathematics page)
 * Trachtenberg system
 * Abacus system - As students become used to manipulating the abacus with their fingers, they are typically asked to do calculation by visualizing abacus in their head. Almost all proficient abacus users are adept at doing arithmetic mentally.
 * Chisanbop
 * Shi Fengshou

Mental Arithmetic as a Psychological Skill
Physical exertion of the proper level can lead to an increase in performance of a mental task, like doing mental calculations, performed afterward. It has been shown that during high levels of physical activity there is a negative effect on mental task performance. 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. Using an EEG as a measure of mental workload after different levels of physical activity can help determine the level if 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 study investigated the effects of mental demands on physical performance at different levels of physical exertion and ultimately found a decrease in physical performance when mental tasks were completed concurrently, with a more significant effect at the higher level of physical workload. This means that with too much psychical activity happening at the same time as the mental math computation, both activities cannot be done to their optimal performance level. The Brown-Peterson procedure is a widely known task using mental arithmetic. 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.

Mental Calculation World Cup
The first World Mental Calculation Championships (Mental Calculation World Cup) took place in 2004. They are repeated every second year. It consists of six different tasks: addition of ten ten-digit numbers, multiplication of two eight-digit numbers, calculation of square roots, and calculation of weekdays for given dates, calculation of cube roots plus some surprise miscellaneous tasks.