The Power of 1

Debbie Poss

Lassiter High School

Deborah.poss@cobbk12.org

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”