User blog:RomanescoLover355/Mental Arithmetic Techniques

For a while I was intrigued by the world of mental mathematics, and so I gave it a try. Nowadays I'm less active in that area, but I think I may want to retry it for the sake of having something to do when I'm bored, (e.g. squaring the numbers on license plates). Here is what I've learned thus far, both from personal experience and other people.

How To Mentally Square Two-Digit Numbers
This is very easy, and relies on the algebraic principle that \((x+a)(x-a)=x^2-a^2\). When given a two digit number, take the closest multiple of ten and find the different between that multiple of ten and your number. Then, find the two numbers which are an equal distance from the original two digit number according to the difference, multiply them together, and add the square of the difference. Here is an example if that explanation was confusing.

Suppose someone gives you the number 67. To find the square, find the closest multiple of 10, 70, and multiply that by the number which is 70-67 or 3 away from 67 in the opposite direction from 70, 64. 64 times 70 equals 4480, next we just add the square of the difference going either way, 32 or 9, and thus the answer is 4489, which is correct.

As an added time-saver, numbers which end with 5 can be squared by taking the digit string before the 5, s, calculating s(s+1), and appending a "25" at the end. So for instance, to square 75 you'd calculate 7(7+1)=56 then tack on a 25 to yield the answer 5625. Credit goes to my Chinese teacher who taught this to our class.

How To Mentally Square Three-Digit Numbers
Squaring three digit numbers can be done in a nearly identical process to that of squaring two digit numbers, which I will teach by example.

Suppose someone asks you to square the number 837 - it couldn't be easier! (Actually it would be if you used a calculator but the whole point is you're not supposed to.)

First find the closest multiple of 100, 800, extract the difference between 837 and 800, 37, and find what number lies 37 integers in the opposite direction from 837 than 800, (which would be 874). Now multiply 874 by 800, which is 699200, and add the square of the differences, 372, or 1369 to 699200 yielding 700569. With practice you should be able to have done that mentally in roughly the same amount of time it takes you to read the process out loud.

Note that to square the two digit number you can just put the three-digit square calculation on hold and calculate the square of the two-digit numbers, then push back out of the calculation and resume the three-digit calculation. Of course, this becomes unwieldy and cumbersome for four-digit squares and beyond, so it is better to commit the first 50 squares to memory, which are as follows, (I am doing this from memory so for the sake of accuracy please inform me of any errors):

Note it is not necessary to commit the 50th square to memory as if a three digit number is 50 away from a multiple of 100 it may just be squared like a two-digit number with two zeros appended to the end result.

How To Mentally Square Four-Digit Numbers
Up to this point I had had a lot of practice in what I'm sharing with you, but I have never been fast at squaring two digit numbers, nonetheless, the technique I discovered works if you practice it enough.

First, treat a four digit number in the form 100a+b, where a and b are the beginning and ending two-digit strings. Calculate the square of the four digit number by calculating a2, appending 4 zeros at the end of the result, then adding on b2. Next calculate 2ab, (which does involve multiplying two unique two-digit numbers, but I'll cover that next), tack on two zeroes and add it on to the final result, which is now 10000a2+200ab+b2

For an example, suppose someone wishes that you square 4367 mentally. First square 43, which is 1849, multiply by 10000 to get 18490000, and add the square of 67, which is 4489, to get a result of 18494489. Now store that number in your memory and multiply together 43 and 67, yielding 2881. Multiply 2881 by 200 to get 576200 and add it to 18494489 - this is arguably the most difficult part of four-digit squares due to the chance of dropping a digit accidentally - returning an answer of 19070689.

How To Mentally Multiply Two Two-Digit Numbers
Just cross-multiply.

Things Which I May Consider Adding To The Blog Post

 * Extracting thirteenth roots of 40-ish digit numbers.
 * Multiplying together two three digit numbers.
 * Multiplying together two five digit numbers

Do you have questions or would like clarification regarding any part of this blog post? Then call the toll free number at 1-800-


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