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The Power of 1

The Power of 1. Debbie Poss Lassiter High School Deborah.poss@cobbk12.org. One Person’s Question…. What is the value of. 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.

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The Power of 1

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  1. The Power of 1 Debbie Poss Lassiter High School Deborah.poss@cobbk12.org

  2. One Person’s Question… • What is the value of

  3. One Person’s Question… • Should we teach PEMA ?

  4. The Power of 1 • “One is the Loneliest Number” • “One More Day” • “We are #1!”

  5. The Power of 1 • Smallest Natural Number • Greeks didn’t consider it a number at all.

  6. The Power of 1 • So 1 is not prime because it doesn’t have 2 natural number factors.

  7. The Power of 1 • Euclid thought 1 was powerful because it guaranteed an infinite number of primes…

  8. 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…

  9. The Power of 1 • Important in our language, though… • Unit • Unique • Unity • Universal • All based on Latin word for 1.

  10. The Power of 1 • Multiplicative Identity • kX1 = 1Xk = k for all k

  11. The Power of 1 • Understood (or Misunderstood) 1 • A + 3A = 4A

  12. 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.)

  13. 1 as a Power • n1=n • POWERFUL!

  14. 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…

  15. 1 as a Power Therefore

  16. 1 as a Power • And • So

  17. 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

  18. 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.

  19. The Fourth Roots of 1 • Solving x4 = 1 can be done intuitively. • x = 1 or x = -1 • x = i or x = -i

  20. The Third Roots of 1 • Since x3 = 1 is cubic, there are 3 cube roots of 1 and we can find them all.

  21. The Third Roots of 1

  22. The Powers of 1 • Let’s graph these roots in the complex number plane… imaginary 2 1 -1 -2 real -3 -2 -1 0 1 2 3 4

  23. imaginary (cos θ , sin θ) 1 θ real

  24. 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)?

  25. The Powers of 1 ARML Question: Find the sum of the four non-real fifth roots of 1. • -1

  26. Find all 6 sixth roots of 1. Obviously 1 and -1. The angle between roots is 360°/6 = 60° cos 60° + isin 60 ° =

  27. Find all 6 sixth roots of 1. And by using the symmetry of the graph…

  28. 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?

  29. 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

  30. Reflect Upon the Power of 1 • Go out and have ….“One Fine Day”

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