Boreal B[l]og

I timed it right, entering Dave Kukla’s calculus class to kill an hour between volunteering in a couple of college algebra classes.  Dave is one of the smartest mathematicians I know, a wonderful man, who mentored me through his approach to teaching, and has a way to make the students love his classes.  He has received many awards, and he deserves every one of them.

About once a year, I can “scoop” Dave on something, if I am having a good year.  This time I had a good year.  Dave was talking to the students about the tangent to a curve at y=x^2 at the point (0,1).  As y got closer to 1 x^2 got closer to 1, but more slowly, since it was less than 1 and being squared.  As he started with 0.9 for y and 0.81 for x, he kept going with 0.99 for y and 0.9801 for x.  He then wrote down y=0.99999999 and said he would need the calculator for x^2.

“You don’t need a calculator,” I said from the back.  The place went deathly silent.  Dave looked at me and smiled.

“You want to tell me what it is?”

“Sure.”  And I wrote down the answer 0.9999999800000001.  The calculators couldn’t do it, for it had too many decimal places. I sat down.

“You were the one who gave me the idea,” I said from the back.

Dave looked at me, a quizzical expression on his face.  “What did I do?”

You brought it up last year, and I found the pattern.

square 9 and you get 81

square 99 and you get 9801  Notice the pattern.  One less nine in front, 8 the same number of zeros as 9s  and a 1 at the end.

square 999 and you get 998001.

“You had eight “9s” up there, so I wrote down 7 of them, put an 8 in the middle, followed with 7 “0”s and a one. ”  Pretty cool, right?  Dave nodded.  That’s what I like about him.  He appreciates this sort of stuff.  And he got me to think about it.  He’s brilliant.  I’m just a run of the mill pattern recognizer.

To multiply anything by 9, multiply by 10 first and then subtract the number.  After all, a number x (10-1) is the number times 10 minus the number x 1, or itself.  This is the distributive law in action!

It should follow, then, that multiplying by 99 is easy.  Multiply by 100 then subtract the number, for you are multiplying by (100-1).  So, 83 x 99 would be 8217:  (83 x100) – (83 x 1)= 8300-83=8217.

Multiplying by 11 can be done without great difficulty.  To multiply by 11, multiply by (10+1), which is multiplying by 10, then adding the number (which is multiplying by 1).  So, 123 x 11=(123 x 10) + (123 x 1)=1230 + 123 = 1353.

Multiplying by 50 is the same as dividing by 2 and multiplying by 100, since 100/2 is 50.  That means that 98 x 50=(1/2) x 98, or 49 x 100=4900.  In the same fashion as above, we can multiply by 51 or 49.  The neat thing about multiplying by 51, is that for an even number less than 100, it is half that number with the number appended.  Say what?  64 x 51 is half of 64 (that would be 32) with the number (64) appended.  So 64 x51=3264.

Multiplying by 25 is dividing by 4 and multiplying by 100:

87 x25 is 21 3/4 x 100.  But that is 2175, since 3/4 x 100 =75.

Try left to right multiplication:  23 x 7

Write 23 as 20 +3.  So 7 x 23 is 7 x 20, or 140, plus 7 x3, or 21.  Add 140 and 21 to get 161.  In my head, 43 x6=”6 times 40 is 24o; 6 times 3 is/are 18, and 240 plus 18 is/are 258. ”

354 x 6= “6 times 300 is 1800; 6 times 50 is 300, and 1800 plus 300 is 2100.  6 times 4 is 24.  Answer is 2124.

498 x 8= 3200 plus 720 plus 64 or 3984.  BUT, isn’t that also 8 x (500-2) or 8 x 500 (4000) minus 8 x 2, or 16?  Same answer.

This one is really cool:  Square any number  x, ending in 5.  The answer is (x)*(x+1), with 25 appended.

So, 85 squared is 8*9 with 25 appended, or 7225.

295 squared is 29 *30, with 25 appended, or 87025.

Multiply two numbers ending or beginning with the same digit, like 1:

31 x 31=  30*30 (or 900) +(3+6) x 10 (or 60) +1=961.

96 x97= 90 *90 (or 8100) + (6+7) 90 (or 1170) +(6*7)=9312.

This one came to me in the car on my way to a clinic in Safford:

The product of two numbers whose sum is an even number constant is maximal when the numbers are the same.  For every unit away from that maximum number, the sum decreases by the square of that unit:

So, two numbers whose sum is 100 have a maximal product of 50*50 or 2500 (this can be proven with simple calculus).

51*49=2499; 52*48=2496; 61*30=2500-11 squared or 2379.

It works in reverse:  84 x 78 is 81 squared minus 9 or 6561-9=6552.