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Thu, Jul 13 2006 100 Braces We use braces to hold teeth and other things together.
With acorns, we can brace the acorns with a bag to hold them together.
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If we had 3 in one hand and 5 in the other,
we put them in the bag,
we have 8 in the bag,
we give half to a friend, so we give out 4:
3+5 = 8
8/2 = 4
For this we don't need braces or a bag.
If we didn't count them first,
and did not know that a=3 and b=5,
we can assume we did by giving the numbers names:
3+5 => a+b
8/2 => (a+b)/2
We use the braces to show that a and b are
closer together then the 2,
and so this writing means:
first add a+b, then divide the result in 2.
We also agree that division and multiplication are always
closer than addition and subtraction so we don't have to write
braces all the time.
Also, while 2*3*4 is clearly 24,
2/3/4 could be 1/4 of 2/3 which is a 1/6,
or it could be 2 divided by 3/4 which is more than 2.
So we use braces:
(2/3)/4 = 1/6
2/(3/4) = 8/3 = 2 and 2/3
(exactly (4*4=16) times the first result, guess why?)

so, 3+5 is 8 and 2 times 8 is 16.
2*(3+5) = 16

When multiplying 2 added numbers in braces,
its like adding the multiplication of each of the numbers.
2*(3+5) = 2*3 + 2*5 = 6 + 10 = 16
the same is true if we had braces with two added numbers
on the left, like this:
6*8= 48
(4+2)*(3+5) = 4*3 + 4*5 +2*3 + 2*5 = 12 + 20 + 6 + 10 = 48
so:
(a+b)*(c+d) = a*c + a*d + b*c + b*d
and so:
(a+b)*(a+b) = a*a + a*b + b*a + b*b = a squared + 2*a*b + b squared

so 65 square is really just
(60+5)*(60*5) = 60*60 + 10*60 + 25 = 70*60 + 25 =
= 7*6*100 + 25 = 42*100 + 25 = 4200 + 25 = 4225
or, if a number ends with a five, to square it,
take the left part, multiply by the next number up,
write down the result from the already memorized
table of multiplication, and write down 25 to the right.
5*5 = 25
15*15 = 225
25*25 = 625
35*35 = 1225
45*45 = 2025
55*55 = 3025
65*65 = 4225
75*75 = 5625
85*85 = 7225
95*95 = 9025
105*105 = 11025

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