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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
“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 Important in our language,
though… Unit Unique Unity Universal All based on Latin word for 1.
The Power of 1
Multiplicative IdentitykX1 = 1Xk = k for all k
The Power of 1
Understood (or Misunderstood) 1
A + 3A = 4A
x xy1 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
n1=n
POWERFUL!
The Power of 1
Most students see that 9n∙9m=9n+m
So 91/2 ∙ 91/2=91
Two equal numbers whose product is 9…
1 as a Power
Therefore
1
29 3 9
1 as a Power
And
So
1 1 113 3 38 8 8 8
1338 2 8
The Powers of 1
1x=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
11/2= 1 which means However, there are two
square roots of 1. The principle square root is 1, but the other square root is -1, because both numbers satisfy the equation x2=1.
1 1
The Fourth Roots of 1
Solving x4 = 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 13x 1 0
2x 1 x x 1 0 2x 1 0 or x x 1 0
x 1or1 1 4 1 i 3
x2 2
The Powers of 1
Let’s graph these roots in the complex number plane…
imaginary
real -3 -2 -1 0 1 2 3 4
2
1
-1
-2
(cos θ , sin θ)
1 θreal
imaginary
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 32 2
i
Find all 6 sixth roots of 1.
And by using the symmetry of the graph…
1 32 2
i1 32 2
i
1 32 2
i1 32 2
i
Reflect Upon the Power of 1
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”
