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.
The most notable technique outlined in the Trachtenberg system is cross-multiplication.
, write 5 (to the left of the first solved digit) and carry 4 to the next step.
As a result, the product of the two five-digit numbers has been identified as 838102050 using only elementary multiplication and addition.
Cross-multiplication --- with suitable adjustments --- can also be done left-to-right or can be done using blocks of more than one digit at a time.
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.
In the same way that cross-multiplication can be utilized to reduce a large multiplication question to a series of basic arithmetical computations, the inverse of cross-multiplication --- cross-division --- can be used to reduce a division question to a series of manageable calculations.
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.
An easy way to approximate the square root of a number is to use the following equation: The closer the known square is to the unknown, the more accurate the approximation.
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.
It consists of a range of different tasks such as addition, subtraction, multiplication, division, irrational and exact square roots, cube roots and deeper roots, factorizations, fractions, and calendar dates.
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.
