some of you math people may appreciate this

[lane mcbride]

Some of you may be familiar with this already, but if not here goes. It's a little math trick you can do to impress your friends or make mental math a little easier.

the first one involves squaring numbers (forgive me if this is difficult to understand... it's not too easy to explain without a pencil and paper)... there is another one that works for multiplication not involving squares, if anybody is interested, I would be happy to explain that too.

for example: if you take 99 and you want to find out what it is squared (99 x 99) you take the difference from 100 of the number (which is 1 in this case) you then subtract that from the number so 99-1 = 98... now you square the difference you came up with (again 1) and put it on the end of the 98... so 99 squared is 9801.

another example 97 x 97. the difference from 100 is 3
subtract that from 97--> 97-3 =94
(now square the 3 and put it on the end of the 94)--> so 9409.

for numbers over 100 you add the difference. example:
103
difference from 100 is 3 103 + 3 = 106
now square the 3 and put it on the end 106(09)
so 10609 is 103 squared.

for numbers close to 50 you take the difference from fifty and divide by two before adding the suffix

example 48
difference from 50 is 2 48-2 =46 --> 46/2 = 23
now add the suffix to 23 so 23(04)
48 squared is 2304

53 difference from 50 is 3 53 + 3 = 56--> 56/2 = 28
now add the suffix (3 squared) to 28--> 28(09)
53 squared is 2809.

[Siegmund]

Yes, it's your old friend from Algebra I, the FOIL method, and its two most common special cases: (x+y)^2 = x^2+2xy+y^2 and (x+y)(x-y)=x^2-y^2.

For some reason they never point out in school that these shortcuts are very useful for arithmetic and not just equation-solving, when x is 50 or 100 or some other easy-to-handle number.

One of the more unusual books in my collection is one by a mathematician from India, advocating a return to pre-colonial methods of math teaching instead of the western approach, that involved a lot of shortcuts of this type: in fact he advocated only memorizing the times tables up to FIVE instead of the usual 10 or 12, and using "digits" -4 to +5 instead of 0 to 9 when convenient. That's an extension of sorts of your trick. Just as you observed that 97=100-3 and found it easier to calculate (97-3)*(97+3)+3*3 in your head than 97*97, he routinely taught schoolchildren to treat say 38x51 as (40-2)x51 so the intermediate steps in the long multiplication all involved small numbers, instead of holding back this secret until the 8th grade and then not telling them how to use it.

[lane mcbride]

what's the author's name? or the name of the book for that matter? I'd love to read it...

a similar trick can be used for numbers that aren't the same.

ex. 98 x 94
take the difference from 100 of one of them and then subtract from the other one.
so... either subtract 6 (difference of 94 from 100) from 98 to get 92
or... subtract 2 (difference of 98 from 100) from 94 to get 92...
then multiply the differences obtained for each together... and put them on the end. (so 6 x 2) = 12
92(12)

another example for clarification: 97 x 92
subtract 3 from 92--> 89
multiply 3 x 8--> 24
put it on the end 89(24)

[BruceZ]

[ QUOTE ]
One of the more unusual books in my collection is one by a mathematician from India, advocating a return to pre-colonial methods of math teaching instead of the western approach, that involved a lot of shortcuts of this type

[/ QUOTE ]

Do you have Vedic Mathematics by Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaja? That was given to me as a gift when I visited India.

[BruceZ]

This was posted on other topics.

[CitricAcid13]

Also, something like 13x19 can be made easy (16+3)(16-3) or 16 squared - 9. (247). When the average is not an integer, it still works but gets kinda messy. 17x22. 19.5squared - 2.5squared. (39squared/4)-(25/4) ((40*38+1)-25)/4=374. Don't know if that made any type of shortcut, but it is another simple FOIL method used to make multiplying tricky numbers together.