Transcript The Power of 1 Keynote from Debbie Poss

The Power of 1
Debbie Poss
Lassiter High School
[email protected]
One Person’s Question…

What is the value of
48  2  9  3
One Person’s Question…

Should we teach
PEMA ?
The Power of 1
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
“One is the Loneliest Number”
“One More Day”
“We are #1!”
The Power of 1

Smallest Natural
Number

Greeks didn’t consider it
a number at all.
The Power of 1
 So
1 is not prime
because it doesn’t
have 2 natural
number factors.
The Power of 1
 Euclid
thought 1 was
powerful because it
guaranteed an infinite
number of primes…
The Power of 1
Let m and n be 1st two primes.
 Consider mn + 1
 Can it be factored?
 Then mn + 1 is also prime.
 Let m, n and p be 1st three
primes…
 Consider mnp + 1…

The Power of 1

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Important in our language,
though…
Unit
Unique
Unity
Universal
All based on Latin word for 1.
The Power of 1
 Multiplicative
Identity
 kX1 = 1Xk = k for all k
The Power of 1
Understood (or
Misunderstood) 1
 A + 3A = 4A

x  xy
1y
x
The Power of 1
1 is the only integer that
always produces more
by addition than by
multiplication.
 (I + k > k but 2 + k > 2k
isn’t always true.)

1 as a Power
1
 n =n
 POWERFUL!
The Power of 1
Most students see that
n
m
n+m
9 ∙9 =9
1/2
1/2
1
 So 9
∙ 9 =9
 Two equal numbers
whose product is 9…
1 as a Power
Therefore
1
2
9 3 9
1 as a Power
1
3
1
3

And 8  8  8

1
3
So
1
3
8
8 2 8
3
1
The Powers of 1
x
 1 =1
for all x.
 0 can’t be raised to
negative powers
 -1 raised to even powers
isn’t equal to -1
The Powers of 1
1 which means 1  1
 However, there are two
square roots of 1. The
principle square root is 1,
but the other square root is
-1, because both numbers
2
satisfy the equation x =1.
1/2
1 =
The Fourth Roots of 1
4
x
Solving = 1 can be
done intuitively.
 x = 1 or x = -1
 x = i or x = -i

The Third Roots of 1

Since x3 = 1 is cubic,
there are 3 cube roots of
1 and we can find them
all.
The Third Roots of 1
x 1  0
2
 x  1  x  x  1  0
3
x  1  0 or x  x  1  0
x  1 or
1  1  4 1  i 3
x

2
2
2
The Powers of 1

Let’s graph these roots in the
complex number plane…
imaginary
2
1
real
-3
-2
-1
0
-1
-2
1
2
3
4
imaginary
(cos θ , sin θ)
1
real
θ
The Powers of 1
Think about it. What is
the sum of the 5 fifth
roots of unity (i.e.
The 5 fifth roots of 1)?
The Powers of 1
ARML Question:
Find the sum of the four
non-real fifth roots of 1.
 -1
Find all 6 sixth roots of 1.
Obviously 1 and -1.
The angle between roots
is 360°/6 = 60°
cos 60° + isin 60 ° =
1
3

i
2
2
Find all 6 sixth roots of 1.
And by using the symmetry
of the graph…
1
3
 
i
2
2
1
3
 
i
2
2
1
3

i
2
2
1
3

i
2
2
Reflect Upon the Power of 1
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Is there 1 person who inspired
your love for mathematics?
Is there 1 person who inspired
you to be a mathematics teacher?
Is there 1 person who helped you
be the person you are today?
Reflect Upon the Power of 1
To the world you may be
just one person,
 But to one person, you may
be the world.

-Brandi Snyder
Reflect Upon the Power of 1

Go out and have
….“One Fine Day”