Quadratics Shawn Godin Cairine Wilson S.S Orleans, ON - - PowerPoint PPT Presentation

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Quadratics Shawn Godin Cairine Wilson S.S Orleans, ON - - PowerPoint PPT Presentation

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Quadratics Shawn Godin Cairine Wilson S.S Orleans, ON Shawn.Godin@ocdsb.ca October 14, 2017 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 1 / 110 Binary Quadratic Form A form is a homogeneous polynomial, that is a

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Quadratics

Shawn Godin

Cairine Wilson S.S Orleans, ON Shawn.Godin@ocdsb.ca

October 14, 2017

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

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SLIDE 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) = ax2 + bxy + cy2.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 2 / 110

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SLIDE 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, 2017 3 / 110

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SLIDE 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, 2017 4 / 110

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SLIDE 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, 2017 5 / 110

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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, Zn, with addition modulo n. The identity is 0 and the inverse of an element is its additive inverse modulo n. If G is Zp \ {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, 2017 6 / 110

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A Non-Abelian Group: The Symmetries of an Equilateral Triangle

An equilateral triangle has 6 symmetries: counterclockwise rotation through 120◦ (r1) or 240◦ (r2), reflection in an axis of symmetry (ℓ1), (ℓ2),

  • r (ℓ3) and do nothing (or rotate through 360◦) (e).

ℓ1 ℓ2 ℓ3

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 7 / 110

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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 ℓ1 ◦ ℓ2 yields ℓ1 ℓ2 ℓ3

− →

ℓ2 ℓ1 ℓ2 ℓ3

− →

ℓ1 ℓ1 ℓ2 ℓ3 which is the same as r1 ℓ1 ℓ2 ℓ3

− →

r1 ℓ1 ℓ2 ℓ3

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 8 / 110

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Composition of Symmetries

Yet when we calculate ℓ2 ◦ ℓ1 we get ℓ1 ℓ2 ℓ3

− →

ℓ1 ℓ1 ℓ2 ℓ3

− →

ℓ2 ℓ1 ℓ2 ℓ3 which is the same as r2 ℓ1 ℓ2 ℓ3

− →

r2 ℓ1 ℓ2 ℓ3 Thus ℓ1 ◦ ℓ2 = r1 = r2 = ℓ2 ◦ ℓ1, so composition is not commutative.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 9 / 110

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Composition of Symmetries

Using G = {e, r1, r2, ℓ1, ℓ2, ℓ3} and ⊕ = ◦ forms an non-Abelian group called the dihedral group of order 6, D6.

  • e

r1 r2 ℓ1 ℓ2 ℓ3 e e r1 r2 ℓ1 ℓ2 ℓ3 r1 r1 r2 e ℓ3 ℓ1 ℓ2 r2 r2 e r1 ℓ2 ℓ3 ℓ1 ℓ1 ℓ1 ℓ2 ℓ3 e r1 r2 ℓ2 ℓ2 ℓ3 ℓ1 r2 e r1 ℓ3 ℓ3 ℓ1 ℓ2 r1 r2 e

Table: Table of composition of symmetries

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 10 / 110

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SLIDE 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, 2017 11 / 110

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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, 2017 12 / 110

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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, 2017 13 / 110

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Measurement: The Square

s

A = s2

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

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SLIDE 15

The Perfect Squares

1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 65 75 85 95 6 16 26 36 46 56 66 76 86 96 7 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 60 70 80 90 100 1 4 9 16 25 36 49 64 81 100

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 15 / 110

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Consecutive Squares

02 + 1 = 1 = 12 12 + 3 = 4 = 22 22 + 5 = 9 = 32 32 + 7 = 16 = 42 42 + 9 = 25 = 52

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 16 / 110

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Squares as Sums of Odd Numbers

Thus 1 = 12 1 + 3 = 22 1 + 3 + 5 = 32 . . . . 1 + 3 + 5 + · · · + (2n − 1) = n2 . . . . Note n2 − (n − 1)2 = n2 − (n2 − 2n + 1) = 2n − 1

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 17 / 110

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The Geoboard Problem

How many different areas of squares are possible on an 11 × 11 pin geoboard?

16

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 18 / 110

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SLIDE 19

The Geoboard Problem

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 19 / 110

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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, 2017 20 / 110

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The Geoboard Problem

A square with area of 13 square units.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 21 / 110

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The Geoboard Problem

A square with area of 13 square units.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 22 / 110

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The Geoboard Problem

A square with area of 13 square units.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 23 / 110

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The Geoboard Problem

A square with area of 13 square units.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 24 / 110

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SLIDE 25

The Geoboard Problem

A square with area of 13 square units.

c a b

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 25 / 110

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The Pythagorean Theorem

If ABC is a right angled triangle with legs a and b, and hypotenuse c C A B a b c then a2 + b2 = c2

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 26 / 110

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Visual Proof of the Pythagorean Theorem

a2

b2 c2

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 27 / 110

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Measurement: The Square Revisted

s

A = s2

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 28 / 110

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Measurement: The Square Revisited s = √ A

A

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 29 / 110

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What About √ 2?

1 1 √ 2

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 30 / 110

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Continued Fraction Proof of Irrationality of √ 2

A little algebraic manipulation yields √ 2 = 1 + (−1 + √ 2) = 1 + (−1 + √ 2)

  • 1 +

√ 2 1 + √ 2

  • = 1 +

1 1 + √ 2

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 31 / 110

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Continued Fraction Proof of Irrationality of √ 2

Now we can substitute our expression into itself √ 2 = 1 + 1 1 + √ 2 = 1 + 1 1 + 1 +

1 1+ √ 2

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 32 / 110

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Continued Fraction Proof of Irrationality of √ 2

and again . . . √ 2 = 1 + 1 1 + √ 2 = 1 + 1 2 +

1 1+ √ 2

= 1 + 1 2 +

1 1+1+

1 1+ √ 2 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 33 / 110

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Continued Fraction Proof of Irrationality of √ 2

√ 2 = 1 + 1 2 +

1 2+

1 2+ 1 2+ 1 2+···

The convergents are 1 1, 3 2, 7 5, 17 12, 41 29, 99 70, 239 169, 577 408, 1393 985 , · · · Note that √ 2 = 1.41421 . . . 99 70 = 1.41428 . . . 141 100 = 1.41

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 34 / 110

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SLIDE 35

Hurwitz’s Theorem

For every irrational number α there are infinitely many relatively prime integers m and n such that

  • α − m

n

  • <

1 √ 5 n2 . The convergents of the continued fraction expansion of α satisfy Hurwitz’s theorem.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 35 / 110

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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

  • utside 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, 2017 36 / 110

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The Quadratic Polynomial f (x) = ax2 + bx + c

Consider the polynomial function f (x) = ax2 + bx + c, a, b, c ∈ R, a = 0 then it is well known that the equation f (x) = 0 has solutions x = −b ± √ b2 − 4ac 2a .

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 37 / 110

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The Discriminant

The discriminant, Dn, of the degree n polynomial function f (x) = anxn + an−1xn−1 + · · · + a2x2 + a1x + a0, ai ∈ R is a function of the coefficients Dn(a0, a1, . . . , an) such that Dn(a0, a1, . . . , an) = 0 if and only if f has at least one multiple root, if Dn(a0, a1, . . . , an) < 0 then f has some non-real roots, if f has n distinct real roots then Dn(a0, a1, . . . , an) > 0.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 38 / 110

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The Discriminant of a Quadratic Polynomial

In particular, for the quadratic polynomial f (x) = ax2 + bx + c, a, b, c ∈ R, a = 0 the discriminant is D = b2 − 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, 2017 39 / 110

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The Polynomial f (x) = x2 + 1

Consider the polynomial f (x) = x2 + 1, its roots are the solution to the equation x2 + 1 = 0 x2 = −1 for which there are no real roots. Note: a = 1, b = 0, c = 1 so D = 02 − 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, 2017 40 / 110

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The Complex Numbers C

If we define a number number i, the imaginary unit, such that i2 = −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, 2017 41 / 110

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SLIDE 42

Number Systems: Polynomial Equations anxn + an−1xn−1 + · · · + a2x2 + a1x + a0 = 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 anxn + an−1xn−1 + · · · + a2x2 + a1x + a0 = 0, with ai ∈ C have n solutions in C (counting multiplicities).

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 42 / 110

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The Graph of a Quadratic Function

The graph with equation y = ax2 + bx + c is a parabola

x y y = ax2 + bx + c

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 43 / 110

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Conic Sections

Consider the double cone sliced by various planes.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 44 / 110

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SLIDE 45

Conic Sections

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 45 / 110

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The Circle

A circle is the locus of points that are a fixed distance, called the radius

  • f the circle, from a fixed point called the centre of the circle.

radius centre

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 46 / 110

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SLIDE 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. major axis minor axis P PF1 + PF2 = constant focus F1 focus F2

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 47 / 110

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SLIDE 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 directrix D focus F P

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 48 / 110

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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.

major axis minor axis

P focus F1 focus F2 |PF1 − PF2| = constant

asymptote asymptote

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 49 / 110

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SLIDE 50

Equations of Conic Sections

The equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 describes a (possibly degenerate) conic section. The discriminant D = B2 − 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, 2017 50 / 110

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SLIDE 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) = ax2 + bxy + cy2.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 51 / 110

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SLIDE 52

The Discriminant of a Binary Quadratic Form

Multiplying the binary quadratic form f (x, y) = ax2 + bxy + cy2 by 4a and completing the square yields 4af (x, y) = 4a2x2 + 4abxy + 4acy2 = (2ax)2 + 2(2a)(by) + (by)2 − (by)2 + 4acy2 = (2ax + by)2 − (b2 − 4ac)y2 = (2ax + by)2 − ∆y2 where ∆ = b2 − 4ac is called the discriminant.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 52 / 110

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SLIDE 53

Properties of the Discriminant of a Binary Quadratic Form

Since ∆ = b2 − 4ac we have ∆ ≡ b2 − 4ac (mod 4) ≡ b2 (mod 4) and hence ∆ ≡ 0, 1 (mod 4).

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 53 / 110

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SLIDE 54

Existence of a Form with a Given Discriminant

If ∆ ≡ 0 (mod 4) then ∆

4 is an integer, and

x2 − ∆ 4

  • y2

is a binary quadratic form with discriminant ∆. Similarly, if ∆ ≡ 1 (mod 4) then ∆−1

4

is an integer, and x2 + xy − ∆ − 1 4

  • y2

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, 2017 54 / 110

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SLIDE 55

Existence of a Form with a Given Discriminant: Examples

Some binary quadratic forms with given discriminant: Case 1: ∆ ≡ 0 (mod 4) if ∆ = 20: x2 − 20 4

  • y2 = x2 − 5y2,

if ∆ = −44: x2 − −44 4

  • y2 = x2 + 11y2,

Case 2: ∆ ≡ 1 (mod 4) if ∆ = 5: x2 + xy − 5 − 1 4

  • y2 = x2 + xy − y2,

if ∆ = −11: x2 + xy − −11 − 1 4

  • y2 = x2 + xy + 3y2.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 55 / 110

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SLIDE 56

Representation of n by a Binary Quadratic Form

We say that a binary quadratic form f (x, y) = ax2 + bxy + cy2 represents an integer n, if there exists integers x0 and y0 such that f (x0, y0) = n. If gcd(x0, y0) = 1 then the representation is called proper, otherwise it is called improper.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 56 / 110

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SLIDE 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, 2017 57 / 110

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SLIDE 58

Types of Binary Quadratic Forms

A binary quadratic form f (x, y) = ax2 + bxy + cy2 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, 2017 58 / 110

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SLIDE 59

Improper Representation

Suppose that n is represented by (x0, y0) with gcd(x0, y0) = d > 1, then x0 = dX and y0 = dY for some integers X and Y with gcd(X, Y ) = 1. Thus f (x0, y0) = n ax2

0 + bx0y0 + cy2 0 = n

a(dX)2 + b(dX)(dY ) + c(dY )2 = n d2(aX 2 + bXY + cY 2) = n which implies that d2 | n, and f properly represents

n d2 .

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 59 / 110

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SLIDE 60

Example of Proper and Improper Representation

Consider the binary quadratic form f (x, y) = x2 + y2 then x = 7, y = 1 is a proper representation of 50 since f (7, 1) = 72 + 12 = 50 and gcd(1, 7) = 1, yet x = y = 5 is an improper representation of 50 since f (5, 5) = 52 + 52 = 50 and gcd(5, 5) = 5 = d > 1. Hence d2 = 25 | 50, so x = y = 5

5 = 1 is a

proper representation of 50

25 = 2 as

f (1, 1) = 12 + 12 = 2.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 60 / 110

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SLIDE 61

Solution Set to x2 + y 2 = 50

x y

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 61 / 110

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SLIDE 62

Solution Set to x2 + y 2 = 50

x y

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 62 / 110

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SLIDE 63

Solution Set to x2 + y 2 = 50

x y

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 63 / 110

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SLIDE 64

Solution Set to x2 + y 2 = 50

x y

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 64 / 110

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SLIDE 65

Solution Set to x2 + y 2 = 50

x y

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 65 / 110

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SLIDE 66

Solution Set to x2 + y 2 = 50

x y x2 + y2 = 2

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 66 / 110

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SLIDE 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, 2017 67 / 110

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SLIDE 68

Examples of Forms Representing 0

If ∆ = 16 = 42, then f (x, y) = x2 − 16 4

  • y2 = x2 − 4y2

has the given discriminant and hence 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, 2017 68 / 110

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SLIDE 69

Solution Set to x2 − 4y 2 = 0

x y

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 69 / 110

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SLIDE 70

More on Forms Representing 0

If we want to find all integer solutions to f (x, y) = x2 − 4y2 = 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, 2017 70 / 110

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SLIDE 71

Solution Set to x2 − 4y 2 = 21

x y

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 71 / 110

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SLIDE 72

Equivalence of Binary Quadratic Forms

Consider the form f (x, y) = 7x2 + 3y2 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 = 31x2 + 34xy + 10y2. Solving the system 2x + y = 2 x + y = 5 yields x = −3, y = 8, which implies f (2, 5) = g(−3, 8) = 103

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SLIDE 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 x y

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 73 / 110

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SLIDE 74

Linear Transformation of a Binary Quadratic Form

Starting with the form f (x, y) = ax2 + bxy + cy2 if we define a new form f ′(x, y) = f (αx + βy, γx + δy) = a′x2 + b′xy + c′y2 then a′ = aα2 + bαγ + cγ2 b′ = b(αδ + βγ) + 2(aαβ + cγδ) c′ = aβ2 + bβδ + cδ2.

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SLIDE 75

Linear Transformation of a Binary Quadratic Form

The discriminant of the new form will be ∆′ = b′2 − 4a′c′ = (αδ − βγ)2(b2 − 4ac) = (αδ − βγ)2∆ so that if (αδ − βγ)2 = 1 then ∆′ = ∆.

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SLIDE 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 7x2 + 3y2 ∼ 31x2 + 34xy + 10y2.

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SLIDE 77

Reduced Positive Definite Forms

A positive definite form f (x, y) = ax2 + bxy + cy2, a, c > 0, b2 − 4ac < 0 is called reduced if −a < b ≤ a ≤ c, with b ≥ 0 if c = a. For example 7x2 + 3y2 and 31x2 + 34xy + 10y2 are unreduced forms but 3x2 + 7y2 is reduced.

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SLIDE 78

The Reduction Algorithm

If f (x, y) = ax2 + bxy + cy2 is a positive definite form then we can find an integer δ such that | − b + 2cδ| ≤ c then ax2 + bxy + cy2 ∼ a′x2 + b′xy + c′y2 where |b′| ≤ a′ and a′ = c b′ = −b + 2cδ c′ = a − bδ + cδ2. If a′ ≤ c′ you are done, if not repeat the process.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 78 / 110

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SLIDE 79

Example: Reducing 31x2 + 34xy + 10y 2

To reduce 31x2 + 34xy + 10y2, we need a δ such that | − 34 + 2(10)δ| ≤ 10 which is satisfied by δ = 2, thus we get a′ = c = 10 b′ = −b + 2cδ = −34 + 2(10)(2) = 6 c′ = a − bδ + cδ2 = 31 − 34(2) + 10(2)2 = 3 so 31x2 + 34xy + 10y2 ∼ 10x2 + 6xy + 3y2 which is unreduced. If we perform the process one more time we get the reduced form 31x2 + 34xy + 10y2 ∼ 10x2 + 6xy + 3y2 ∼ 3x2 + 7y2.

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SLIDE 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 x2 + 21y2, 2x2 + 2xy + 11y2, 3x2 + 7y2, 5x2 + 4xy + 5y2 Each class will represent its own set of numbers. The classes form an Abelian group called the class group where the group

  • peration is called composition.

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SLIDE 81

Numbers Represented by the Form f (x, y) = x2 + y 2

1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 65 75 85 95 6 16 26 36 46 56 66 76 86 96 7 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 60 70 80 90 100 1 4 9 16 25 36 49 64 81 100

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SLIDE 82

Numbers Represented by the Form f (x, y) = x2 + y 2

1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 65 75 85 95 6 16 26 36 46 56 66 76 86 96 7 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 60 70 80 90 100 1 4 9 16 25 36 49 64 81 100 2 5 10 17 26 37 50 65 82

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SLIDE 83

Numbers Represented by the Form f (x, y) = x2 + y 2

1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 65 75 85 95 6 16 26 36 46 56 66 76 86 96 7 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 60 70 80 90 100 1 4 9 16 25 36 49 64 81 100 2 5 10 17 26 37 50 65 82 8 13 20 29 40 53 68 85

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SLIDE 84

Numbers Represented by the Form f (x, y) = x2 + y 2

1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 65 75 85 95 6 16 26 36 46 56 66 76 86 96 7 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 60 70 80 90 100 1 4 9 16 25 36 49 64 81 100 2 5 10 17 26 37 50 65 82 8 13 20 29 40 53 68 85 18 34 45 58 73 90

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SLIDE 85

Numbers Represented by the Form f (x, y) = x2 + y 2

1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 65 75 85 95 6 16 26 36 46 56 66 76 86 96 7 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 60 70 80 90 100 1 4 9 16 25 36 49 64 81 100 2 5 10 17 26 37 50 65 82 8 13 20 29 40 53 68 85 18 34 45 58 73 90 32 41 52 65 80 97 61 74 89 72 85 98

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SLIDE 86

Numbers Represented by the Form f (x, y) = x2 + y 2

1 37 73 5 41 77 9 45 81 13 49 85 17 53 89 21 57 93 25 61 97 29 65 101 33 69 105 2 38 74 6 42 78 10 46 82 14 50 86 18 54 90 22 58 94 26 62 98 30 66 102 34 70 106 3 39 75 7 43 79 11 47 83 15 51 87 19 55 91 23 59 95 27 63 99 31 67 103 35 71 107 4 40 76 8 44 80 12 48 84 16 52 88 20 56 92 24 60 96 28 64 100 32 68 104 36 72 108 1 2 4 5 8 9 10 13 16 17 18 20 25 26 29 32 34 36 37 40 41 45 49 50 52 53 58 61 64 65 68 72 73 74 80 81 82 85 89 90 97 98 100 101 104 106

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SLIDE 87

Sums of Squares Modulo 4 hi n n2 (mod 4) 1 1 2 3 1 m2 + n2 (mod 4) m\n 1 2 3 1 1 1 1 2 1 2 2 1 1 3 1 2 1 2 .

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SLIDE 88

Writing n as a Sum of Two Squares

Diophantus–Brahmagupta–Fibonacci identity: (a2 + b2)(c2 + d2) = (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 a2 + b2 = p. Theorem (Fermat): If n is factored into primes as n = 2α

i

pβi

i

  • j

qγj

j

where pi and qj are primes with pi ≡ 1 (mod 4) and qj ≡ 3 (mod 4), for all i and j, then n can be expressed as a sum of two squares if and only if γj is even for all j.

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SLIDE 89

Examples of D-B-F Identity

1 37 73 5 41 77 9 45 81 13 49 85 17 53 89 21 57 93 25 61 97 29 65 101 33 69 105 2 38 74 6 42 78 10 46 82 14 50 86 18 54 90 22 58 94 26 62 98 30 66 102 34 70 106 3 39 75 7 43 79 11 47 83 15 51 87 19 55 91 23 59 95 27 63 99 31 67 103 35 71 107 4 40 76 8 44 80 12 48 84 16 52 88 20 56 92 24 60 96 28 64 100 32 68 104 36 72 108 1 2 4 5 8 9 10 13 16 17 18 20 25 26 29 32 34 36 37 40 41 45 49 50 52 53 58 61 64 65 68 72 73 74 80 81 82 85 89 90 97 98 100 101 104 106

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SLIDE 90

Examples of D-B-F Identity

1 37 73 5 41 77 9 45 81 13 49 85 17 53 89 21 57 93 25 61 97 29 65 101 33 69 105 2 38 74 6 42 78 10 46 82 14 50 86 18 54 90 22 58 94 26 62 98 30 66 102 34 70 106 3 39 75 7 43 79 11 47 83 15 51 87 19 55 91 23 59 95 27 63 99 31 67 103 35 71 107 4 40 76 8 44 80 12 48 84 16 52 88 20 56 92 24 60 96 28 64 100 32 68 104 36 72 108 1 2 4 5 8 9 10 13 16 17 18 20 25 26 29 32 34 36 37 40 41 45 49 50 52 53 58 61 64 65 68 72 73 74 80 81 82 85 89 90 97 98 100 101 104 106

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SLIDE 91

Examples of D-B-F Identity

1 37 73 5 41 77 9 45 81 13 49 85 17 53 89 21 57 93 25 61 97 29 65 101 33 69 105 2 38 74 6 42 78 10 46 82 14 50 86 18 54 90 22 58 94 26 62 98 30 66 102 34 70 106 3 39 75 7 43 79 11 47 83 15 51 87 19 55 91 23 59 95 27 63 99 31 67 103 35 71 107 4 40 76 8 44 80 12 48 84 16 52 88 20 56 92 24 60 96 28 64 100 32 68 104 36 72 108 1 2 4 5 8 9 10 13 16 17 18 20 25 26 29 32 34 36 37 40 41 45 49 50 52 53 58 61 64 65 68 72 73 74 80 81 82 85 89 90 97 98 100 101 104 106

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SLIDE 92

Writing n as a Sum of Two Squares

Diophantus–Brahmagupta–Fibonacci identity: (a2 + b2)(c2 + d2) = (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 a2 + b2 = p. Theorem (Fermat): If n is factored into primes as n = 2α

i

pβi

i

  • j

qγj

j

where pi and qj are primes with pi ≡ 1 (mod 4) and qj ≡ 3 (mod 4), for all i and j, then n can be expressed as a sum of two squares if and only if γj is even for all j.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 92 / 110

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SLIDE 93

Primes p ≡ 1 (mod 4)

1 37 73 5 41 77 9 45 81 13 49 85 17 53 89 21 57 93 25 61 97 29 65 101 33 69 105 2 38 74 6 42 78 10 46 82 14 50 86 18 54 90 22 58 94 26 62 98 30 66 102 34 70 106 3 39 75 7 43 79 11 47 83 15 51 87 19 55 91 23 59 95 27 63 99 31 67 103 35 71 107 4 40 76 8 44 80 12 48 84 16 52 88 20 56 92 24 60 96 28 64 100 32 68 104 36 72 108 1 2 4 5 8 9 10 13 16 17 18 20 25 26 29 32 34 36 37 40 41 45 49 50 52 53 58 61 64 65 68 72 73 74 80 81 82 85 89 90 97 98 100 101 104 106

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SLIDE 94

Writing n as a Sum of Two Squares

Diophantus–Brahmagupta–Fibonacci identity: (a2 + b2)(c2 + d2) = (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 a2 + b2 = p. Theorem (Fermat): If n is factored into primes as n = 2α

i

pβi

i

  • j

qγj

j

where pi and qj are primes with pi ≡ 1 (mod 4) and qj ≡ 3 (mod 4), for all i and j, then n can be expressed as a sum of two squares if and only if γj is even for all j.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 94 / 110

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SLIDE 95

Sum of Two Squares Example

n = 2 × 13 × 32 = 234 Since 2 = 12 + 12, 13 = 22 + 32, 9 = 02 + 32 we have 26 = 2 × 13 = (12 + 12)(22 + 32) = (1 × 2 − 1 × 3)2 + (1 × 3 + 1 × 2)2 = (−1)2 + 52 = 12 + 52 so 234 = 26 × 9 = (12 + 52)(02 + 32) = (1 × 0 − 5 × 3)2 + (1 × 3 + 5 × 0)2 = 152 + 32

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SLIDE 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 + (i2)bd = (ac − bd) + i(ad + bc) Diophantus–Brahmagupta–Fibonacci identity: (a2 + b2)(c2 + d2) = (ac − bd)2 + (ad + bc)2

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SLIDE 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 + (i2)bd = (ac − bd) + i(ad + bc) Diophantus–Brahmagupta–Fibonacci identity: (a2 + b2)(c2 + d2) = (ac − bd)2 + (ad + bc)2

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SLIDE 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) = x2 + y2

  • r

|z| =

  • x2 + y2.

ℜ ℑ z = x + iy

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SLIDE 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 = a2 + b2, |β|2 = c2 + d2, |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|.

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SLIDE 100

A Curious Identity

We can write (x2

1 + x2 2)(y2 1 + y2 2 ) = z2 1 + z2 2

where z1 = x1y1 − x2y2 z2 = x1y2 + x2y1 as a statement of |X||Y | = |XY | where X, Y ∈ C with X = x1 + ix2 and Y = y1 + iy2.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 100 / 110

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SLIDE 101

Another Curious Identity

We can also write (x2

1 + x2 2 + x2 3 + x2 4)(y2 1 + y2 2 + y2 3 + y2 4 ) = z2 1 + z2 2 + z2 3 + z2 4

where z1 = x1y1 − x2y2 − x3y3 − x4y4 z2 = x1y2 + x2y1 + x3y4 − x4y3 z3 = x1y3 + x3y1 − x2y4 + x4y2 z4 = x1y4 + x4y1 + x2y3 − x3y2

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SLIDE 102

More on Sums of Squares

Sums of Three Squares: Every positive integer n can be written in the form n = a2 + b2 + c2, a, b, c ∈ Z except for those n of the form n = 4a(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 = a2 + b2 + c2 + d2, a, b, c, d ∈ Z.

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SLIDE 103

Quaternions

If we define three distinct new numbers, i, j, and k, that satisfy i2 = −1 j2 = −1 k2 = −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 kj = −i = −jk ik = −j = −ki ijk = −1 so multiplication of quaternions is not commutative.

Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 103 / 110

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SLIDE 104

Yet Another Curious Identity

We can also write (x2

1 + x2 2 + · · · + x2 8)(y2 1 + y2 2 + · · · + y2 8 ) = z2 1 + z2 2 + · · · + z2 8

where z1 = x1y1 − x2y2 − x3y3 − x4y4 − x5y5 − x6y6 − x7y7 − x8y8, z2 = x1y2 + x2y1 + x3y4 − x4y3 + x5y6 − x6y5 − x7y8 + x8y7, z3 = x1y3 + x3y1 − x2y4 + x4y2 + x5y7 − x7y5 + x6y8 − x8y6, z4 = x1y4 + x4y1 + x2y3 − x3y2 + x5y8 − x8y5 − x6y7 + x7y6, z5 = x1y5 + x5y1 − x2y6 + x6y2 − x3y7 + x7y3 − x4y8 + x8y4, z6 = x1y6 + x6y1 + x2y5 − x5y2 − x3y8 + x8y3 + x4y7 − x7y4, z7 = x1y7 + x7y1 + x2y8 − x8y2 + x3y5 − x5y3 − x4y6 + x6y4, z8 = x1y8 + x8y1 − x2y7 + x7y2 + x3y6 − x6y3 + x4y5 − x5y4.

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SLIDE 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.

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SLIDE 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.

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SLIDE 107

Examples of Normed Algebras

The complex numbers C with z = |z| for all z ∈ C, Three dimensional Euclidean vectors R3 with the cross product with the Euclidean norm (x, y, z) =

  • x2 + y2 + z2,

The quaternions H with a + bi + cj + dk = √ a2 + b2 + c2 + d2. The octonions O with a0 + a1i1 + · · · + a7i7 =

  • a2

0 + a2 1 + · · · + a2 7

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SLIDE 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, 2017 108 / 110

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SLIDE 109

The Geoboard Problem

How many different areas of squares are possible on an 11 × 11 pin geoboard?

16

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SLIDE 110

The End

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