Quadratics. Shawn Godin. Cairine Wilson S.S Orleans, ON October 14, 2017

Transcription

1 Quadratics Shawn Godin Cairine Wilson S.S Orleans, ON October 14, 2017 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

2 Binary Quadratic Form A form is a homogeneous polynomial, that is a polynomial where each term has the same degree. Specifically, a binary quadratic form is a homogeneous polynomial in two variables of degree 2, that is a polynomial of the form f (x, y) = ax 2 + bxy + cy 2. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

3 Number Systems: In the Beginning Natural numbers, N = {1, 2, 3,... } Whole numbers, W = {0, 1, 2, 3,... } closed under addition (i.e. if x, y N then (x + y) N), not closed under subtraction (for example 2 5 N), closed under multiplication, not closed under division Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

4 Number Systems: Linear Equations ax + b = 0 N closed under + and, not under and W closed under + and, not under and Integers, Z = {..., 2, 1, 0, 1, 2,... } closed under addition, closed under subtraction, closed under multiplication, not closed under division not all equations ax + b = 0, with a, b Z have solutions in Z Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

5 Groups A group is a collection of elements, G, along with a binary operator,, that satisfy the following conditions: G is closed under (i.e. if x, y G then x y G), is associative, that is for x, y, z G, x (y z) = (x y) z, there exists a element,e, called the identity such that for any x G, e x = x e = x, each x G has an inverse, denoted x 1, that satisfies x x 1 = x 1 x = e. A group in which is also commutative, that is for all x, y G we have x y = y x, is called an Abelian group. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

6 Some Abelian Groups The following are all Abelian groups: If G is any of the sets: Z, Q, R, or C with regular addition. The identity is 0 and the inverse of an element x is its negative x. If G is any of the sets: Q \ {0}, R \ {0}, or C \ {0} with regular multiplication,. The identity is 1 and the inverse of an element x is its reciprocal 1 x. If G is the integers modulo n, Z n, with addition modulo n. The identity is 0 and the inverse of an element is its additive inverse modulo n. If G is Z p \ {0}, for some prime p, with multiplication modulo p. The identity is 1 and the inverse of an element is its multiplicative inverse modulo n. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

7 A Non-Abelian Group: The Symmetries of an Equilateral Triangle An equilateral triangle has 6 symmetries: counterclockwise rotation through 120 (r 1 ) or 240 (r 2 ), reflection in an axis of symmetry (l 1 ), (l 2 ), or (l 3 ) and do nothing (or rotate through 360 ) (e). l 1 l 2 l 3 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

8 Composition of Symmetries Transformations can be combined using composition. a b means to do transformation b then transformation a. Composing any two symmetries results in another symmetry. For example l 1 l 2 yields l 1 l 1 l 1 l 2 l 3 l2 l 2 l 3 l1 l 2 l 3 which is the same as r 1 l 1 l 1 l 2 l 3 r1 l 2 l 3 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

9 Composition of Symmetries Yet when we calculate l 2 l 1 we get l 1 l 1 l 1 l 2 l 3 l1 l 2 l 3 l2 l 2 l 3 which is the same as r 2 l 1 l 1 l 2 l 3 r2 l 2 l 3 Thus l 1 l 2 = r 1 r 2 = l 2 l 1, so composition is not commutative. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

10 Composition of Symmetries Using G = {e, r 1, r 2, l 1, l 2, l 3 } and = forms an non-abelian group called the dihedral group of order 6, D 6. e r 1 r 2 l 1 l 2 l 3 e e r 1 r 2 l 1 l 2 l 3 r 1 r 1 r 2 e l 3 l 1 l 2 r 2 r 2 e r 1 l 2 l 3 l 1 l 1 l 1 l 2 l 3 e r 1 r 2 l 2 l 2 l 3 l 1 r 2 e r 1 l 3 l 3 l 1 l 2 r 1 r 2 e Table: Table of composition of symmetries Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

11 Number Systems: Linear Equations ax + b = 0 N closed under + and, not for and W closed under + and, not for and Z closed under +, and, not, (Z, +) is a group Rational numbers, Q = { a b a, b Z, b 0} closed under addition, closed under subtraction, closed under multiplication, closed under division all equations ax + b = 0, with a, b Q have solutions in Q (Q, +) and (Q \ {0}, ) are groups Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

12 Rings and Fields A ring is a collection of elements, R, along with two binary operators, and, that satisfy the following conditions: R is closed under both and, (R, ) is an Abelian group, is associative, the distributive laws hold, that is for all x, y R we have (x y) z = (x z) + (y z) and x (y z) = (x y) (x z) A ring is called commutative if is also commutative. A ring is said to have an identity (or contain a 1) if there is an element 1 R such that 1 a = a 1 = a for all a R. A field is a commutative ring with identity in which all non-zero elements have a multiplicative inverse. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

13 Number Systems: Linear Equations ax + b = 0 N closed under + and, not for and W closed under + and, not for and Z closed under +, and, not, (Z, +) is a group, (Z, +, ) is a ring Rational numbers, Q = { a b a, b Z, b 0} closed under addition, closed under subtraction, closed under multiplication, closed under division all equations ax + b = 0, with a, b Q have solutions in Q (Q, +) and (Q \ {0}, ) are groups, (Q, +, ) is a ring, Q is a field Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

14 Measurement: The Square A = s 2 s Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

15 The Perfect Squares Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

16 Consecutive Squares = 25 = = 16 = = 9 = = 4 = = 1 = 1 2 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

17 Squares as Sums of Odd Numbers Thus 1 = = = (2n 1) = n 2.. Note n 2 (n 1) 2 = n 2 (n 2 2n + 1) = 2n 1 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

18 The Geoboard Problem How many different areas of squares are possible on an pin geoboard? 16 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

19 The Geoboard Problem 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

20 The Geoboard Problem 2, 8, 18, 32, 50 = 2 1, 2 4, 2 9, 2 16, 2 25 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

21 The Geoboard Problem A square with area of 13 square units. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

22 The Geoboard Problem A square with area of 13 square units. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

23 The Geoboard Problem A square with area of 13 square units. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

24 The Geoboard Problem A square with area of 13 square units. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

25 The Geoboard Problem A square with area of 13 square units. c a b Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

26 The Pythagorean Theorem If ABC is a right angled triangle with legs a and b, and hypotenuse c A b c C a B then a 2 + b 2 = c 2 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

27 Visual Proof of the Pythagorean Theorem a 2 b 2 c 2 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

28 Measurement: The Square Revisted A = s 2 s Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

29 Measurement: The Square Revisited A s = A Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

30 What About 2? Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

31 Continued Fraction Proof of Irrationality of 2 A little algebraic manipulation yields 2 = 1 + ( 1 + 2) = 1 + ( 1 + ( 1 + ) 2 2) = Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

32 Continued Fraction Proof of Irrationality of 2 Now we can substitute our expression into itself 2 = = Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

33 Continued Fraction Proof of Irrationality of 2 and again... 2 = = = Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

34 Continued Fraction Proof of Irrationality of 2 The convergents are 1 2 = Note that 1 1, 3 2, 7 5, 17 12, 41 29, 99 70, , , , 2 = = = 1.41 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

35 Hurwitz s Theorem For every irrational number α there are infinitely many relatively prime integers m and n such that α m < 1 n. 5 n 2 The convergents of the continued fraction expansion of α satisfy Hurwitz s theorem. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

36 Number Systems: Polynomial Equations N closed under + and, not for and W closed under + and, not for and Z closed under +, and, not ; (Z, +) is a group, (Z, +, ) is a ring Q closed under +,,, and ; (Q, +) and (Q \ {0}, ) are groups, (Q, +, ) is a ring, Q is a field. Some convergent sequences have limit outside Q. Some polynomials not solvable. Real numbers, R closed under addition, closed under subtraction, closed under multiplication, closed under division, (R, +) and (R \ {0}, ) are groups, (R, +, ) is a ring, R is a field, all convergent sequences in R has limit in R, all equations ax + b = 0, with a, b R have solutions in R, many polynomials (not all) unsolvable in Q, are solvable in R. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

37 The Quadratic Polynomial f (x) = ax 2 + bx + c Consider the polynomial function f (x) = ax 2 + bx + c, a, b, c R, a 0 then it is well known that the equation f (x) = 0 has solutions x = b ± b 2 4ac. 2a Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

38 The Discriminant The discriminant, D n, of the degree n polynomial function f (x) = a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0, a i R is a function of the coefficients D n (a 0, a 1,..., a n ) such that D n (a 0, a 1,..., a n ) = 0 if and only if f has at least one multiple root, if D n (a 0, a 1,..., a n ) < 0 then f has some non-real roots, if f has n distinct real roots then D n (a 0, a 1,..., a n ) > 0. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

39 The Discriminant of a Quadratic Polynomial In particular, for the quadratic polynomial f (x) = ax 2 + bx + c, a, b, c R, a 0 the discriminant is D = b 2 4ac, where if D > 0 then f has two distinct real roots, if D = 0 then f has a repeated real root, if D < 0 then f has no real roots, if D is a perfect square, then f has two distinct rational roots and f can be factored into two linear factors with rational or integer coefficients. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

40 The Polynomial f (x) = x Consider the polynomial f (x) = x 2 + 1, its roots are the solution to the equation for which there are no real roots. x = 0 x 2 = 1 Note: a = 1, b = 0, c = 1 so D = 0 2 4(1)(1) = 4. Thus there are degree n polynomials with real coefficients that do not have n real roots (counting multiplicities). Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

41 The Complex Numbers C If we define a number number i, the imaginary unit, such that i 2 = 1 then we can define a new number system C = {a + bi a, b R} called the complex numbers. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

42 Number Systems: Polynomial Equations a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 = 0 N, W closed under + and, not for and Z closed under +, and, not ; (Z, +) is a group, (Z, +, ) is a ring Q is a field; some convergent sequences have limit outside Q; some polynomials not solvable. R is a field; all convergent sequences have limit in R; some polynomials not solvable. Complex numbers, C is a field, all convergent sequences in C has limit in C, all polynomial equations a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 = 0, with a i C have n solutions in C (counting multiplicities). Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

43 The Graph of a Quadratic Function The graph with equation y = ax 2 + bx + c is a parabola y y = ax 2 + bx + c x Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

44 Conic Sections Consider the double cone sliced by various planes. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

45 Conic Sections Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

46 The Circle A circle is the locus of points that are a fixed distance, called the radius of the circle, from a fixed point called the centre of the circle. radius centre Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

47 The Ellipse An ellipse is the locus of points such that the sum of the distances to two fixed points, called the foci (singular focus), is a constant. P PF 1 + PF 2 = constant minor axis focus F 1 major axis focus F 2 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

48 The Parabola A parabola is a locus of points such that the distance from a point on the parabola to a fixed point, called the focus, is equal to the distance to a fixed line, called the directrix. PF = PD F focus P directrix D Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

49 The Hyperbola A hyperbola is the locus of points such that the difference of the distances to two fixed points, called the foci, is a constant. asymptote PF 1 PF 2 = constant asymptote P minor axis F 1 focus major axis F 2 focus Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

50 Equations of Conic Sections The equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 describes a (possibly degenerate) conic section. The discriminant D = B 2 4AC tells us the conic is an ellipse if D < 0 (and a circle if A = C and B = 0), a parabola if D = 0, a hyperbola if D > 0. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

51 Binary Quadratic Form A form is a homogeneous polynomial, that is a polynomial where each term has the same degree. Specifically, a binary quadratic form is a homogeneous polynomial in two variables of degree 2, that is a polynomial of the form f (x, y) = ax 2 + bxy + cy 2. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

52 The Discriminant of a Binary Quadratic Form Multiplying the binary quadratic form f (x, y) = ax 2 + bxy + cy 2 by 4a and completing the square yields 4af (x, y) = 4a 2 x 2 + 4abxy + 4acy 2 = (2ax) 2 + 2(2a)(by) + (by) 2 (by) 2 + 4acy 2 = (2ax + by) 2 (b 2 4ac)y 2 = (2ax + by) 2 y 2 where = b 2 4ac is called the discriminant. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

53 Properties of the Discriminant of a Binary Quadratic Form Since we have = b 2 4ac and hence 0, 1 (mod 4). b 2 4ac (mod 4) b 2 (mod 4) Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

54 Existence of a Form with a Given Discriminant If 0 (mod 4) then 4 is an integer, and x 2 ( ) y 2 4 is a binary quadratic form with discriminant. Similarly, if 1 (mod 4) then 1 4 is an integer, and ( ) 1 x 2 + xy y 2 4 is a binary quadratic form with discriminant. Hence, for every 0, 1 (mod 4) there exists at least one binary quadratic form with discriminant. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

55 Existence of a Form with a Given Discriminant: Examples Some binary quadratic forms with given discriminant: Case 1: 0 (mod 4) ( 20 if = 20: x 2 4 ( 44 if = 44: x 2 4 ) y 2 = x 2 5y 2, ) y 2 = x y 2, Case 2: 1 (mod 4) ( 5 1 if = 5: x 2 + xy 4 ( 11 1 if = 11: x 2 + xy 4 ) y 2 = x 2 + xy y 2, ) y 2 = x 2 + xy + 3y 2. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

56 Representation of n by a Binary Quadratic Form We say that a binary quadratic form f (x, y) = ax 2 + bxy + cy 2 represents an integer n, if there exists integers x 0 and y 0 such that f (x 0, y 0 ) = n. If gcd(x 0, y 0 ) = 1 then the representation is called proper, otherwise it is called improper. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

57 Representation Problems The following representation problems are of interest: Which integers do the form f represent? Which forms represent the integer n? How many ways does the form f represent the integer n? Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

58 Types of Binary Quadratic Forms A binary quadratic form f (x, y) = ax 2 + bxy + cy 2 can be one of three types. Indefinite if f takes on both positive and negative values. This happens when > 0. Semi-definite if f (x, y) 0 (positive semi-definite) or f (x, y) 0 (negative semi-definite) for all integer values of x and y. This happens when 0. Definite if it is semi-definite and the only solution to f (x, y) = 0 is x = y = 0. This happens when < 0 and thus a and c have the same sign. Thus we can have positive definite (if a, c > 0) or negative definite (if a, c < 0) forms. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

59 Improper Representation Suppose that n is represented by (x 0, y 0 ) with gcd(x 0, y 0 ) = d > 1, then x 0 = dx and y 0 = dy for some integers X and Y with gcd(x, Y ) = 1. Thus f (x 0, y 0 ) = n ax bx 0 y 0 + cy 2 0 = n a(dx ) 2 + b(dx )(dy ) + c(dy ) 2 = n d 2 (ax 2 + bxy + cy 2 ) = n which implies that d 2 n, and f properly represents n d 2. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

60 Example of Proper and Improper Representation Consider the binary quadratic form f (x, y) = x 2 + y 2 then x = 7, y = 1 is a proper representation of 50 since f (7, 1) = = 50 and gcd(1, 7) = 1, yet x = y = 5 is an improper representation of 50 since f (5, 5) = = 50 and gcd(5, 5) = 5 = d > 1. Hence d 2 = 25 50, so x = y = 5 5 = 1 is a proper representation of = 2 as f (1, 1) = = 2. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

61 Solution Set to x 2 + y 2 = 50 y x Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

62 Solution Set to x 2 + y 2 = 50 y x Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

63 Solution Set to x 2 + y 2 = 50 y x Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

64 Solution Set to x 2 + y 2 = 50 y x Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

65 Solution Set to x 2 + y 2 = 50 y x Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

66 Solution Set to x 2 + y 2 = 50 y x 2 + y 2 = 2 x Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

67 Forms Representing 0 If is a perfect square, or 0, then is a positive integer and 4af (x, y) = (2ax + (b + )y)(2ax + (b )y). Thus our form is factorable, and so f (x, y) = 0 has many solutions. If is a not perfect square, nor 0, then the only solution to f (x, y) = 0 is x = y = 0. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

68 Examples of Forms Representing 0 If = 16 = 4 2, then f (x, y) = x 2 has the given discriminant and hence ( ) 16 y 2 = x 2 4y 2 4 f (x, y) = (x + 2y)(x 2y) so any solution to x + 2y = 0 or x 2y = 0 satisfies f (x, y) = 0, that is f (±2k, k) = 0, k Z. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

69 Solution Set to x 2 4y 2 = 0 y x Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

70 More on Forms Representing 0 If we want to find all integer solutions to f (x, y) = x 2 4y 2 = 21 then factoring yields (x + 2y)(x 2y) = 21. Since x, y Z, then (x + 2y), (x 2y) Z, so (x + 2y) 21 and (x 2y) 21. Each pair of factors of 21 yields a system of equations which yield a solution to the original equation. For example, using 3 7 = 21 gives x + 2y = 3 (1) x 2y = 7 (2) which has solution x = 5, y = 1. The full solution set is (x, y) {(±5, ±1), (±11, ±5)}. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

71 Solution Set to x 2 4y 2 = 21 y x Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

72 Equivalence of Binary Quadratic Forms Consider the form f (x, y) = 7x 2 + 3y 2 which represents 103 four ways as f (±2, ±5) = 103. Consider the new form g defined by g(x, y) = f (2x + y, x + y) = 7(2x + y) 2 + 3(x + y) 2 = 31x xy + 10y 2. Solving the system 2x + y = 2 x + y = 5 yields x = 3, y = 8, which implies f (2, 5) = g( 3, 8) = 103 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

73 Equivalence of Binary Quadratic Forms Looking at all the representations of 103 we get f (2, 5) = g( 3, 8) = 103 f (2, 5) = g(7, 12) = 103 f ( 2, 5) = g( 7, 12) = 103 f ( 2, 5) = g(3, 8) = 103 y x Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

74 Linear Transformation of a Binary Quadratic Form Starting with the form f (x, y) = ax 2 + bxy + cy 2 if we define a new form f (x, y) = f (αx + βy, γx + δy) = a x 2 + b xy + c y 2 then a = aα 2 + bαγ + cγ 2 b = b(αδ + βγ) + 2(aαβ + cγδ) c = aβ 2 + bβδ + cδ 2. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

75 Linear Transformation of a Binary Quadratic Form The discriminant of the new form will be = b 2 4a c = (αδ βγ) 2 (b 2 4ac) = (αδ βγ) 2 so that if then (αδ βγ) 2 = 1 =. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

76 Equivalent Forms If two forms, f and g, are related by a transformation of the same type with αδ βγ = +1, then the forms are called properly equivalent and we write f g. If two forms are equivalent, they have the same discriminant and they represent the same integers. From our example 7x 2 + 3y 2 31x xy + 10y 2. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

77 Reduced Positive Definite Forms A positive definite form f (x, y) = ax 2 + bxy + cy 2, a, c > 0, b 2 4ac < 0 is called reduced if a < b a c, with b 0 if c = a. For example 7x 2 + 3y 2 and 31x xy + 10y 2 are unreduced forms but is reduced. 3x 2 + 7y 2 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

78 The Reduction Algorithm If f (x, y) = ax 2 + bxy + cy 2 is a positive definite form then we can find an integer δ such that b + 2cδ c then ax 2 + bxy + cy 2 a x 2 + b xy + c y 2 where b a and a = c b = b + 2cδ c = a bδ + cδ 2. If a c you are done, if not repeat the process. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

79 Example: Reducing 31x xy + 10y 2 To reduce 31x xy + 10y 2, we need a δ such that (10)δ 10 which is satisfied by δ = 2, thus we get a = c = 10 b = b + 2cδ = (10)(2) = 6 c = a bδ + cδ 2 = 31 34(2) + 10(2) 2 = 3 so 31x xy + 10y 2 10x 2 + 6xy + 3y 2 which is unreduced. If we perform the process one more time we get the reduced form 31x xy + 10y 2 10x 2 + 6xy + 3y 2 3x 2 + 7y 2. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

80 The Class Number For each discriminant < 0 there are a number of classes of equivalent forms. Each class contains a unique reduced form. The number of classes for a given discriminant < 0 is called the class number, h( ). For example, h( 84) = 4 so there are 4 equivalence classes of forms with discriminant 84. The reduced forms in the classes are x y 2, 2x 2 + 2xy + 11y 2, 3x 2 + 7y 2, 5x 2 + 4xy + 5y 2 Each class will represent its own set of numbers. The classes form an Abelian group called the class group where the group operation is called composition. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

81 Numbers Represented by the Form f (x, y) = x 2 + y Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

82 Numbers Represented by the Form f (x, y) = x 2 + y Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

83 Numbers Represented by the Form f (x, y) = x 2 + y Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

84 Numbers Represented by the Form f (x, y) = x 2 + y Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

85 Numbers Represented by the Form f (x, y) = x 2 + y Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

86 Numbers Represented by the Form f (x, y) = x 2 + y Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

87 Sums of Squares Modulo 4 hi n n 2 (mod 4) m 2 + n 2 (mod 4) m\n Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

88 Writing n as a Sum of Two Squares Diophantus Brahmagupta Fibonacci identity: (a 2 + b 2 )(c 2 + d 2 ) = (ac bd) 2 + (ad + bc) 2 Theorem: If p 1 (mod 4) is a prime, then there exists positive integers a and b such that a 2 + b 2 = p. Theorem (Fermat): If n is factored into primes as n = 2 α i where p i and q j are primes with p i 1 (mod 4) and q j 3 (mod 4), for all i and j, then n can be expressed as a sum of two squares if and only if γ j is even for all j. p β i i j q γ j j Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

89 Examples of D-B-F Identity Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

90 Examples of D-B-F Identity Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

91 Examples of D-B-F Identity Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

92 Writing n as a Sum of Two Squares Diophantus Brahmagupta Fibonacci identity: (a 2 + b 2 )(c 2 + d 2 ) = (ac bd) 2 + (ad + bc) 2 Theorem: If p 1 (mod 4) is a prime, then there exists positive integers a and b such that a 2 + b 2 = p. Theorem (Fermat): If n is factored into primes as n = 2 α i where p i and q j are primes with p i 1 (mod 4) and q j 3 (mod 4), for all i and j, then n can be expressed as a sum of two squares if and only if γ j is even for all j. p β i i j q γ j j Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

93 Primes p 1 (mod 4) Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

94 Writing n as a Sum of Two Squares Diophantus Brahmagupta Fibonacci identity: (a 2 + b 2 )(c 2 + d 2 ) = (ac bd) 2 + (ad + bc) 2 Theorem: If p 1 (mod 4) is a prime, then there exists positive integers a and b such that a 2 + b 2 = p. Theorem (Fermat): If n is factored into primes as n = 2 α i where p i and q j are primes with p i 1 (mod 4) and q j 3 (mod 4), for all i and j, then n can be expressed as a sum of two squares if and only if γ j is even for all j. p β i i j q γ j j Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

95 Sum of Two Squares Example Since we have so n = = = , 13 = , 9 = = 2 13 = ( )( ) = ( ) 2 + ( ) 2 = ( 1) = = 26 9 = ( )( ) = ( ) 2 + ( ) 2 = Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

96 A Curious Result Suppose α = a + ib and β = c + id are two complex numbers (a, b, c, d R), then α β = (a + ib)(c + id) = ac + iad + ibc + (i 2 )bd = (ac bd) + i(ad + bc) Diophantus Brahmagupta Fibonacci identity: (a 2 + b 2 )(c 2 + d 2 ) = (ac bd) 2 + (ad + bc) 2 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

97 A Curious Result Suppose α = a + ib and β = c + id are two complex numbers (a, b, c, d R), then α β = (a + ib)(c + id) = ac + iad + ibc + (i 2 )bd = (ac bd) + i(ad + bc) Diophantus Brahmagupta Fibonacci identity: (a 2 + b 2 )(c 2 + d 2 ) = (ac bd) 2 + (ad + bc) 2 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

98 Modulus of a Complex Number Recall for a complex number z = x + iy, x, y R, the modulus of z, z, satisfies z 2 = z z = (x + iy)(x iy) = x 2 + y 2 or z = x 2 + y 2. I z = x + iy R Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

99 A Curious Result Revisited Suppose α, β C with α = a + ib and β = c + id, then αβ = (ac bd) + i(ad + bc) Thus α 2 = a 2 + b 2, β 2 = c 2 + d 2, α β 2 = (ac bd) 2 + (ad + bc) 2, so the Diophantus Brahmagupta Fibonacci identity tells us αβ 2 = α 2 β 2 which, since z 0, is equivalent to αβ = α β. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

100 A Curious Identity We can write where (x x 2 2 )(y y 2 2 ) = z z 2 2 z 1 = x 1 y 1 x 2 y 2 z 2 = x 1 y 2 + x 2 y 1 as a statement of X Y = XY where X, Y C with X = x 1 + ix 2 and Y = y 1 + iy 2. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

101 Another Curious Identity We can also write where (x x x x 2 4 )(y y y y 2 4 ) = z z z z 2 4 z 1 = x 1 y 1 x 2 y 2 x 3 y 3 x 4 y 4 z 2 = x 1 y 2 + x 2 y 1 + x 3 y 4 x 4 y 3 z 3 = x 1 y 3 + x 3 y 1 x 2 y 4 + x 4 y 2 z 4 = x 1 y 4 + x 4 y 1 + x 2 y 3 x 3 y 2 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

102 More on Sums of Squares Sums of Three Squares: Every positive integer n can be written in the form n = a 2 + b 2 + c 2, a, b, c Z except for those n of the form n = 4 a (8b + 7) where a and b are non-negative integers. Sums of Four Squares: Every positive integer n can be written in the form n = a 2 + b 2 + c 2 + d 2, a, b, c, d Z. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

103 Quaternions If we define three distinct new numbers, i, j, and k, that satisfy i 2 = 1 j 2 = 1 k 2 = 1 ij = k jk = i ki = j then if q = a + bi + cj + dk we call q a quaternion and the set of all quaternions is denoted H. Using the definitions of i, j, and k, we find that ji = k = ij ik = j = ki kj = i = jk ijk = 1 so multiplication of quaternions is not commutative. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

104 Yet Another Curious Identity We can also write (x1 2 + x x8 2 )(y1 2 + y y8 2 ) = z1 2 + z z8 2 where z 1 = x 1 y 1 x 2 y 2 x 3 y 3 x 4 y 4 x 5 y 5 x 6 y 6 x 7 y 7 x 8 y 8, z 2 = x 1 y 2 + x 2 y 1 + x 3 y 4 x 4 y 3 + x 5 y 6 x 6 y 5 x 7 y 8 + x 8 y 7, z 3 = x 1 y 3 + x 3 y 1 x 2 y 4 + x 4 y 2 + x 5 y 7 x 7 y 5 + x 6 y 8 x 8 y 6, z 4 = x 1 y 4 + x 4 y 1 + x 2 y 3 x 3 y 2 + x 5 y 8 x 8 y 5 x 6 y 7 + x 7 y 6, z 5 = x 1 y 5 + x 5 y 1 x 2 y 6 + x 6 y 2 x 3 y 7 + x 7 y 3 x 4 y 8 + x 8 y 4, z 6 = x 1 y 6 + x 6 y 1 + x 2 y 5 x 5 y 2 x 3 y 8 + x 8 y 3 + x 4 y 7 x 7 y 4, z 7 = x 1 y 7 + x 7 y 1 + x 2 y 8 x 8 y 2 + x 3 y 5 x 5 y 3 x 4 y 6 + x 6 y 4, z 8 = x 1 y 8 + x 8 y 1 x 2 y 7 + x 7 y 2 + x 3 y 6 x 6 y 3 + x 4 y 5 x 5 y 4. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

105 Vector Spaces A set V is said to be a vector space over a field F if (V, +) is an Abelian group and for each a F and v V there is an element av V such that: a(u + v) = au + av, (a + b)v = av + bv, a(bv) = (ab)v, 1v = v, for all a, b F and for all u, v V, where 1 F is the multiplicative identity. If v V, then v is called a vector. If a F, then a is called a scalar. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

106 Normed Algebras A ring R is called an algebra over a field F if R is a vector space over F and (au) (bv) = (ab)(u v) for all scalars a, b, F and all vectors u, v R, where represents multiplication within the ring. A norm,, of a vector space V over a field F, is a function : V R such that: 0 = 0, v > 0 for all v 0 V, av = a v for all a F and for all v V, u + v u + v. An algebra with a norm is called a normed algebra. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

107 Examples of Normed Algebras The complex numbers C with z = z for all z C, Three dimensional Euclidean vectors R 3 with the cross product with the Euclidean norm (x, y, z) = x 2 + y 2 + z 2, The quaternions H with a + bi + cj + dk = a 2 + b 2 + c 2 + d 2. The octonions O with a 0 + a 1 i a 7 i 7 = a a a2 7 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

108 Adding Something and Losing Something The real numbers R as a normed algebra, is an ordered set where is commutative and associative. The complex numbers C as a normed algebra, is a non-ordered set where is commutative and associative. The quaternions H as a normed algebra, is a non-ordered set where is non-commutative but is associative. The octonions O as a normed algebra, is a non-ordered set where is non-commutative and non-associative. Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

109 The Geoboard Problem How many different areas of squares are possible on an pin geoboard? 16 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110

110 The End Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, / 110