The Elegance of
Mathematics
Calculus for the 7th Grade
Keynotes at California Mathcounts 2003
Dr. Zhiqin Lu
03/22/2003
University of California, Irvine
1
The title is from
Brain Greene’s best selling book
The Elegant Universe
To professor Greene, the
universe is elegant. To us,
mathematics is elegant.
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2
Let’s begin with a problem
Find the value of the following:
1
12

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1
23

1
34
 
1
9899

1
99100
?
3
Idea?
1
1 2
 
1
2 3

1
1
1
2
1
2

1
3

1
98 99
1
99100
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


1
98
1
99

1
1
2
1
1
 12  12
 12
 12
1
2
 13 
3
23
2
 23

1
23

1
99
1
 100

1
98
 991 
1
99
1
 100

99
9899
98
 9899

100
99100
1
9899
99
 99100

1
99100
4
Then we have
1
12

1
23
1
    9899

1
99100
1
 ( 11  12 )  ( 12  13 )  ( 13  14 )      ( 981  991 )  ( 991  100
)
1
 11  100
99
 100
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What is the observation here?
In order to find the value of
a  1na      2a  1a
n
We write
a1  b1  b0
a2  b2  b1

an1  bn1  bn 2
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an  bn  bn1

6
Then we have
Theorem
a1  a2    an1  an
 (b1  b0 )  (b2  b1 )    (bn1  bn2 )  (bn  bn1)
 bn  b0

This is the idea of Calculus!
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Another Example
1
2

1
22
 
1
29

1
210
 (1 12 )  ( 12  212 )      ( 218  219 )  ( 219  2110 )
 1 2110

1023
1024
Back-to-the-envelope calculation
1
2k
1
 2 k1

2
2 k1
1
 2 k1

1
2 k1
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One more Example
1+2+…+99+100=?
Carl Friedrich Gauss
solved this problem
when he was six!
(but probably he didn’t use our method.)
Can you do it? I am sure.
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What is Calculus?
Calculus=Differentiation+Integration
Differentiation (or subtraction) : an  an  an1
an  a1  a2      an
Integration (or addition) :
The fundamental theorem of calculus states
the duality between
differentiation and integration.

Theorem If an  bn  bn1 , then
a1  a2      an  bn  b0
If a is the
differentiation of b,
Then b is the
integration of a.

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Go to Infinity
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b

f (x)dx a1  a2      an
a
f (x)  F'(x)  an  bn  bn1
Theorem (Fundamental theorem of Calculus)
b

f (x)dx  F (b)  F (a)
a
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The idea I have just showed
you can be used to prove the
Fundamental Theorem
of
Calculus
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Of course, we need to make the above setting precise.
However, the basic duality between differentiation and
integration roots in the very simple duality property
between subtraction and addition.
That is the elegance of mathematics.
You may be able to discover your own theorem by enjoying and
paying attention to the mathematics you are learning now.
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