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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 “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 Identity kX1 = 1Xk = k for all k The Power of 1 Understood (or Misunderstood) 1 A + 3A = 4A x xy 1y 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 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”