Can number theory help improve mental arithmetic?

[PeteyCoco]

I decided to try some of the problems in my Mechanics text without a calculator to see how well I could approximate the answers using differential equations and mental arithmetic. I was a bit slow, but I remembered some tricks I used as a kid in grade school that I hadn't used in ages. Does anyone have any tricks that they use for mental calculations? The most recent one I've learned was for cubing numbers near 50 which was from Surely You're Joking, Mr. Feynman

(I did a quick search of the forums and couldn't find a thread like this. If one exists I apologize)

 

[denjay]

I know there's a ton of strategies out there but ones I personally use... I'm not sure really. I think sometimes I use things like 9 x 8 is like 10 x 8 but minus an 8. Things similar to that. Oh and occasionally I'll multiply 3+ digits numbers from the left to right instead of the normallly taught (in America) right to left. Sometimes its easier sometimes it isn't.

So yeah I don't really use anything special just because I don't often come across actual numbers instead of variables nowadays.

 

[PeteyCoco]

I know there's a ton of strategies out there but ones I personally use... I'm not sure really. I think sometimes I use things like 9 x 8 is like 10 x 8 but minus an 8. Things similar to that. Oh and occasionally I'll multiply 3+ digits numbers from the left to right instead of the normallly taught (in America) right to left. Sometimes its easier sometimes it isn't.

So yeah I don't really use anything special just because I don't often come across actual numbers instead of variables nowadays.

Mmm, maybe this thread would make more sense in general engineering/physics.

 

[OMGCarlos]

I'm actually building a website which has an interactive "Mental Math" section which should be completed by this weekend (I'm releasing a demo tonight or tomorrow which has you solve random numbers by 11).

In the meantime, there's a great book called "Secrets of Mental Math" by Arthur Benjamin which covers several dozen tricks to do things as complex as finding the cubed root of 6 digit numbers, 5 digit squares, and figuring out the day of the week of any random date.

Here's an hour long video of him performing mental math and quickly explaining how he did it.

 

[Chip 201]

I know one!

When you multiply something with 11, you can do an easy calculation:
Let something be = 23459, then

11 x 23459 =

answer is easy,

the first number is the first of 23452, so 2
The second number is the sum of second and third number so, 2 + 3 = 5
The third.....sum of third and fourth, so 3 + 4 = 7,
The fourth... 4+5 = 9
The fifth.. 5 + 2 = 7,
the last is always the last number of which you multiplied 11, = 2

257972 !

11 x 22 =

first number of 22 = 2
second number is sum of the first number and the second of 22, = 4
third number is the last number of 22, = 2

11 x 22 =242

Also a funny site is calculationrankings.c, you can do a calculation game with clock to practice your numerical skills. So also have a rankings, so you how good you are..

Greetz Chip

 

[skeptic2]

To multiply two numbers, both of which are teens.
13 x 17
take 13 + 7 (or 17 +3) = 20.
multiply by 10. 20 x 10 = 200
add product of ones digits. 3 x 7 = 221

Square a number
37^2
Find the difference between the base number and the nearest number evenly divisible by 10. 40 - 37 = 3
Add that difference to the base number and get 40
Subtract the difference from the base number and get 34
Multiply 34 by 40 and get 1360.
Add the square of the difference to the previous result
1360 + 3^2 = 1369

As an RF engineer I work with decibels a lot and memorized the logarithms from 1 to 10 so I could convert back and forth to dB in my head. If you only memorize the logs for the prime numbers, you can derive the rest. e.g. log(15) = log(3) + log(5).

 

[lurflurf]

There are a lot, but calculators are cheaper enough that they are a waste of time. Like when they calculating roots by hand calculate one digit per step since by hand it is faster to do a few more steps than a few more digits per step. Write numbers as sums of common numbers to reuse steps. When adding many numbers you can pull separate them. like add the whole part then the fractional parts. When interpolating use a difference table. Use logs to multiply and extract roots. Use lots of identities to simplify expressions before computing them. And so on the, hand calculations are slow and error prone, even more so if you do not practice much. They do make for a fun Saturday night though.

 

[Curious3141]

One I use especially often is the trick in squaring numbers that end with '5'.

If a number is n.5 (where '.' is the place separator), the square is n(n+1).25

e.g. (0)5^2 = (0)25, 25^2 = 625, 75^2 = 5625, 125^2 = 15625

 

[HallsofIvy]

Personally, I have never considered "mental arithmetic" to have anything to do with "mathematics". Probably just because I am so bad at it!

 

[hddd123456789]

As far as approximating answers, my tried and true 'trick' is to always make a problem use 10 somehow. In any computation involving percents/multiplication/division, if you can make one of the numbers 10 to some power, then it's just a matter of moving around the decimal point.

What's 37% of 438? Well, 1% of 438 is 4.38, and 4.38*37=the answer. Well, I'm assuming doing multiplication by hand on paper is allowed to get to the actual answer.

 

[Michael Redei]

I do a lot of mental arithmetic, just for fun, not because I'm really interested in the answer. One "trick" I use is the following one for multiplying numbers that are close to each other:

Suppose you want to multiply 13 by 17. (That's "close".) Their average is 15, so 15² = 225 is close to the result you need. Both 13 and 17 are 2 units away from 15, so you subtract 2² from 225, and receive 221 = 13 × 17.

You need to know some squares by heart, but often they can be calculated easily, for instance if you need 47 × 53 or 195 × 205.

 

[dkotschessaa]

Personally, I have never considered "mental arithmetic" to have anything to do with "mathematics". Probably just because I am so bad at it!

I keep wavering on this. There is the joke that there's an inverse relationship between the time spent studying higher mathematics and ones ability to do arithmetic. For awhile I bought into this and gave up trying to be better at arithmetic. However, I've gone back to working on mental math a bit during my breaks. However, I'm now doing it simultaneously with a study of number theory. Doing this, I'm able to tie together the supposedly "purely mathematical" world of number theory with something immensely practical.

Really, mental math is about two things:

Being clever about, and understanding numbers (the number theory and abstract element)
The mental workout involved in getting good at the tricks. This involves a development of working memory and purely computational skill.

I'm focusing more on the first part. What are the tricks and why do they work? By doing this, we can get better at arithmetic - but working "smarter not harder" (sorry for the cliche).

I also want to stretch the definition of mental math a bit. It can also be "written math, but with a lot less writing." If you can be clever about numbers, you can reduce the amount of tedious paper calculation - and if you do this long enough, you may develop more of your mental ability.

Sorry if this post is too "meta" for this thread. Though I'll contribute more later.

-Dave K