the power of 1 debbie poss lassiter high school [email protected]
TRANSCRIPT
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
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.)
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…
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 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
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)?
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