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Why division as “repeated subtraction” works 2 1 2 7 8 2 6 1 1 1 3 8 3 5 rem 5 Why division as “repeated subtraction” works 2 1 2 7 8 2 6 1 1 1 3 8 3 5 rem 5 278 dividend = 2 · 10 · 13 + 18 = 2 · 10 · 13 + 1 · 1 · 13 + 5 = (2 · 10 + 1 · 1) · 13 + 5 = quotient · divisor + remainder Why division as “repeated subtraction” works 2x +1 2x 2 +7x +8 2x 2 +6x x x 1·x +3 +8 +3 5 rem 5 −3 | 2 7 8 | −6 −3 2 1 5 Why division as “repeated subtraction” works 2x +1 2x 2 +7x +8 2x 2 +6x x x 1·x +3 rem 5 −3 | 2 7 8 | −6 −3 2 1 5 +8 +3 5 2x 2 + 7x + 8 dividend = 2 · x · (1 · x + 3) + 18 = 2 · x · (1 · x + 3) + 1 · (1 · x + 3) + 5 = (2 · x + 1) · (1 · x + 3) + 5 = quotient · divisor + remainder Back to Complex Numbers ... Please work on the following problems: 1 2 3 (a) (b) (a) (b) (c) √ Find a number that, when added to √7, yields an integer. Find a number that, when multiplied 7,√yields an integer. Find a number that, when added to 2 + √7, yields an integer. Find a number that, when multiplied 2 + 7, yields an integer. Find√as many different numbers as possible that, when added to number. Do the same for numbers that 2 + 7, yield a rational √ multiply with 2 + 7 to yield a rational number. (If you have time) Find a number that, when multiplied with 2 + yields a rational number. √ 3 7, Today’s Class Themes: Representations: Graphical/Geometric, Algebraic/Symbolic Generalizing from Key Properties Topics: Various representations of complex numbers Complex roots of polynomials Two for the Price of One: the Complex Conjugate Theorem! Can we make the sale last a little longer? . . . Other Conjugates. Various representations of complex numbers geometric algebraic points in C or vectors in C coordinates (a, b) or “Cartesian” form a + bi . . . others? Complex roots of polynomials What does it mean to factor a polynomial “completely”? What if we only allow factors to be ones that we can graph on the xy -plane? Complex roots of polynomials What does it mean to factor a polynomial “completely”? What if we only allow factors to be ones that we can graph on the xy -plane? Degree n Theorem. A nonconstant, degree n polynomial has exactly n roots. Suppose the polynomial is f (x) = an x n + an−1 x n−1 + · · · + a1 x1 + a0 . Then f (x) can always be represented in the form f (x) = a(x − α1 )(x − α2 ) . . . (x − αn ), where α1 , α2 , . . . , αn are the roots of f , and they are complex – not necessarily real – numbers. How would we rephrase the Degree n Theorem if we only allowed polynomials with real coefficients? Complex conjugate: very special pairs The arithmetic of conjugate pairs is what makes them special as pairs, and what makes each complex number have a unique conjugate. Notation: α, Re (α), Im (α) Complex conjugate: very special pairs The arithmetic of conjugate pairs is what makes them special as pairs, and what makes each complex number have a unique conjugate. Notation: α, Re (α), Im (α) . . . Complex conjugate: very special pairs The arithmetic of conjugate pairs is what makes them special as pairs, and what makes each complex number have a unique conjugate. Notation: α, Re (α), Im (α) . . . α?? Complex conjugates and polynomials 1 (a) What is the simplest polynomial that you can think of that has i as a root? That √ has 1 − i as a root? √ (b) How about 7 as a root? That has 2 + 7 as a root? 2 How would your solutions to these problems change if we required: (a) . . . the polynomials have to have real coefficients? (b) . . . rational coefficients? Complex conjugates and polynomials 1 (a) What is the simplest polynomial that you can think of that has i as a root? That √ has 1 − i as a root? √ (b) How about 7 as a root? That has 2 + 7 as a root? 2 How would your solutions to these problems change if we required: (a) . . . the polynomials have to have real coefficients? (b) . . . rational coefficients? Complex conjugates and polynomials 1 (a) What is the simplest polynomial that you can think of that has i as a root? That √ has 1 − i as a root? √ (b) How about 7 as a root? That has 2 + 7 as a root? 2 How would your solutions to these problems change if we required: (a) . . . the polynomials have to have real coefficients? (b) . . . rational coefficients? 2(a) changes 1(a) . . . and not 1(b) 2(b) changes 1(b) . . . and not 1(a) Complex conjugates and polynomials 1 (a) What is the simplest polynomial that you can think of that has i as a root? That √ has 1 − i as a root? √ (b) How about 7 as a root? That has 2 + 7 as a root? 2 How would your solutions to these problems change if we required: (a) . . . the polynomials have to have real coefficients? (b) . . . rational coefficients? 2(a) changes 1(a) . . . and not 1(b) 2(b) changes 1(b) √ . . . and not 1(a) i is to R . . . as . . . 7 is to √ Q. C is to R . . . as . . . {a + b 7|a, b ∈ Q} is to Q Complex conjugates and polynomials i is to R . . . C is to R . . . as as √ . . . 7 is to √ Q. . . . {a + b 7|a, b ∈ Q} is to Q Both are square roots outside the home number system that we got by taking a square root of a number in the home system. Both can be roots of polynomials whose coefficient is in the home number system Complex conjugates and polynomials i is to R . . . C is to R . . . as as √ . . . 7 is to √ Q. . . . {a + b 7|a, b ∈ Q} is to Q Both are square roots outside the home number system that we got by taking a square root of a number in the home system. Both can be roots of polynomials whose coefficient is in the home number system . . . and come in pairs! In the case of complex numbers, the pairing looks like: The Conjugate Pair Theorem, a.k.a. Two for the Price of One! If f (x) is a real polynomial and a + bi is a root of f (x), then a − bi must also be a root of f (x). [Let’s work this out on the board. What are the conditions that we want to establish? What are the conditions that are given to use?] Complex Roots and Polynomials Why are roots of a polynomials called roots? Factor x 3 − 1 into linear factors Factor x 4 − 1 into linear factors When you are finished factoring them, plot their zeros as vectors on the complex plane. Complex Roots and Polynomials Why are roots of a polynomials called roots? Factor x 3 − 1 into linear factors Factor x 4 − 1 into linear factors When you are finished factoring them, plot their zeros as vectors on the complex plane. What about x 5 − 1? x 6 − 1? x 51 − 1 ???? Complex Roots and Polynomials Why are roots of a polynomials called roots? Factor x 3 − 1 into linear factors Factor x 4 − 1 into linear factors When you are finished factoring them, plot their zeros as vectors on the complex plane. What about x 5 − 1? x 6 − 1? x 51 − 1 ???? Next time: DeMoivre’s Theorem – The Holy Grail of Complex Numbers!