FAILING ALGEBRA RULES?

M. Mamikon, April 1, 2004

Recent cooperative research at UCLA and CALTECH has lead to a shocking discovery: the traditional multiplication rule "FOIL" (Firsts, Outers, Inners, Lasts):

(a + b)(c + d) = ac + ad +bc + bd

is NOT 'really' true when one of the numbers, say d, is negative.

In this case, a much simpler SHORTER rule is found:

(a + b)(c - d) = ac - bd

called "FaiL" meaning "Firsts and, indeed, Lasts".

Here are some examples (you can check them by hand or with calculator):

(3 + 2)(6 - 4) = 3.6 - 2.4 (Check: 5.2 = 18 - 8 = 10. Yes! )

(5 + 2)(10 - 4) = 5.10 - 2.4 (Check: 7.6 = 50 - 8 = 42. Yes!)

(6 + 5)(8 - 20/3) = 6.8 - 5.20/3 ( 11.4/3 = 48 - 100/3 = 44/3. Yes!)

Not only integers and rational numbers but real and complex ones also "FaiL".

MORE 'FaiL' EXAMPLES

Now educators are NOT very sure if FOIL always works. Are you?

Such simplified rules are possible for any algebraic formula. Experiments support them with infinitely many examples. Contact me if you want to know how they are generated. Of course, one can find some exceptions to these simplified rules, but according to two well known proverbs 'there are no rules without exceptions' and, moreover, 'exceptions prove the rule'.
 More Failing RULES
Students just love these new algebraic rules, because they are easy, natural and have an intuitive appeal to their early academic experiences.

Everyone is confused with these new developments - parents, teachers, educators. Scientists suspect that computers have gone wrong which will cause the manned Mars Mission to fail. Politicians blame mathematicians for misleading them to "War On Iraq", and economists expect the gas prices go up high. Dollar will fall below penny as the new algebra rules confirm. Meanwhile, school districts are seeking ways to keep millions of excited students out of streets.

The Secret of Failure

I have thoroughly investigated what could cause the failure of traditional rules in Theoretical and Practical Arithmetic and came to the conclusion that the problem is hidden in the roots of our numeration system. We have 10 fingers, hence our counting is in base-10 numeration system. Thus, calculations are NOT base-invariant !!! Therefore there should be NO universal formula in algebra. All depends on the base of the numeration system. So, in our case the base number 10 must be explicitly involved in the formulas.

Keeping this in mind, we can easily fix all formulas. For instance the formula for (a+b)2 should be fixed for any Base of numeration like this:

(a+b)2 = (Base).a + b2

Here are examples for Base=10, Base=20 used by Basques in Spain and former Soviet Georgians (they do not wear shoes and count fingers and toes), and Base=5 (when we use one hand if the other one is busy):
 Base 10: (a+b)2 = 10.a + b2 Base 20: (a+b)2 = 20.a + b2 Base 5: (a+b)2 = 5.a + b2 (8+1)2 = 10.8 + 12 = 81 (6+2)2 = 10.6 + 22 = 64 (4+3)2 = 10.4 + 32 = 49 (2+4)2 = 10.2 + 42 = 36 (12+-1)2 = 10.12 + -12 = 121 (14+-2)2 = 10.14 + -22 = 144 (18+1)2 = 20.18 + 12 = 361 (16+2)2 = 20.16 + 22 = 324 (14+3)2 = 20.14 + 32 = 289 (12+4)2 = 20.12 + 42 = 256 (10+5)2 = 20.10 + 52 = 225 You can easily continue down, even for negative numbers (3+1)2 = 5.3 + 12 = 16 (1+2)2 = 5.1 + 22 = 9 (7+-1)2 = 5.7 + -12 = 36 (9+-2)2 = 5.9 + -22 = 16 (11+-3)2 = 5.11 + -32 = 64 Warning: Chicken have 4 fingers, Goats have 2

I am working now on a universal computer program "BaseToFace" that will fix Algebra in all industrial computer systems before the year 2,999 DA (stands for new Dark Ages). Call me for your promotions.