Meta- Complex Numbers Robert Andrews Complex Numbers Meta-Complex Numbers Dual Numbers Split- Complex Robert Andrews Numbers Visualizing Four- Dimensional Surfaces October 29, 2014 Table of Contents Meta- Complex Numbers Robert Andrews 1 Complex Numbers Complex Numbers Dual Numbers 2 Dual Numbers Split- Complex Numbers 3 Split-Complex Numbers Visualizing Four- Dimensional Surfaces 4 Visualizing Four-Dimensional Surfaces Table of Contents Meta- Complex Numbers Robert Andrews 1 Complex Numbers Complex Numbers Dual Numbers 2 Dual Numbers Split- Complex Numbers 3 Split-Complex Numbers Visualizing Four- Dimensional Surfaces 4 Visualizing Four-Dimensional Surfaces Quadratic Equations Meta- Complex Numbers Robert Andrews Recall that the roots of a quadratic equation Complex Numbers 2 Dual ax + bx + c = 0 Numbers Split- are given by the quadratic formula Complex Numbers p 2 Visualizing −b ± b − 4ac Four- x = Dimensional 2a Surfaces The quadratic formula gives the roots of this equation as p x = ± −1 This is a problem! There aren't any real numbers whose square is negative! Quadratic Equations Meta- Complex Numbers Robert Andrews What if we wanted to solve the equation Complex Numbers x2 + 1 = 0? Dual Numbers Split- Complex Numbers Visualizing Four- Dimensional Surfaces This is a problem! There aren't any real numbers whose square is negative! Quadratic Equations Meta- Complex Numbers Robert Andrews What if we wanted to solve the equation Complex Numbers x2 + 1 = 0? Dual Numbers The quadratic formula gives the roots of this equation as Split- Complex p Numbers x = ± −1 Visualizing Four- Dimensional Surfaces Quadratic Equations Meta- Complex Numbers Robert Andrews What if we wanted to solve the equation Complex Numbers x2 + 1 = 0? Dual Numbers The quadratic formula gives the roots of this equation as Split- Complex p Numbers x = ± −1 Visualizing Four- Dimensional Surfaces This is a problem! There aren't any real numbers whose square is negative! We make a new number that is defined precisely to solve this equation! Define the \imaginary number" i such that i2 = −1 Then i and −i are solutions to the equation x2 + 1 = 0 Quadratic Equations Meta- Complex Numbers Robert How do we deal with the fact that there is no real number x Andrews such that x2 + 1 = 0? Complex Numbers Dual Numbers Split- Complex Numbers Visualizing Four- Dimensional Surfaces Define the \imaginary number" i such that i2 = −1 Then i and −i are solutions to the equation x2 + 1 = 0 Quadratic Equations Meta- Complex Numbers Robert How do we deal with the fact that there is no real number x Andrews such that x2 + 1 = 0? Complex We make a new number that is defined precisely to solve Numbers Dual this equation! Numbers Split- Complex Numbers Visualizing Four- Dimensional Surfaces Quadratic Equations Meta- Complex Numbers Robert How do we deal with the fact that there is no real number x Andrews such that x2 + 1 = 0? Complex We make a new number that is defined precisely to solve Numbers Dual this equation! Numbers Define the \imaginary number" i such that Split- Complex 2 Numbers i = −1 Visualizing Four- Dimensional Then i and −i are solutions to the equation Surfaces x2 + 1 = 0 A complex number z is a number of the form z = a + bi 2 where a; b 2 R and i = −1. We call a the real part of the complex number and b the imaginary part. One important definition is that of the conjugate of z, denoted here by z∗. We define the conjugate of z as z∗ = a − bi In essence, we're just changing the sign of the imaginary part. Now that we've defined these new numbers, what can we do with them? Defining Complex Numbers Meta- Complex We use the imaginary unit i to build complex numbers. Numbers Robert Andrews Complex Numbers Dual Numbers Split- Complex Numbers Visualizing Four- Dimensional Surfaces One important definition is that of the conjugate of z, denoted here by z∗. We define the conjugate of z as z∗ = a − bi In essence, we're just changing the sign of the imaginary part. Now that we've defined these new numbers, what can we do with them? Defining Complex Numbers Meta- Complex We use the imaginary unit i to build complex numbers. Numbers A complex number z is a number of the form Robert Andrews z = a + bi Complex Numbers 2 Dual where a; b 2 R and i = −1. We call a the real part of the Numbers complex number and b the imaginary part. Split- Complex Numbers Visualizing Four- Dimensional Surfaces Now that we've defined these new numbers, what can we do with them? Defining Complex Numbers Meta- Complex We use the imaginary unit i to build complex numbers. Numbers A complex number z is a number of the form Robert Andrews z = a + bi Complex Numbers 2 Dual where a; b 2 R and i = −1. We call a the real part of the Numbers complex number and b the imaginary part. Split- Complex One important definition is that of the conjugate of z, Numbers denoted here by z∗. We define the conjugate of z as Visualizing Four- ∗ Dimensional z = a − bi Surfaces In essence, we're just changing the sign of the imaginary part. Defining Complex Numbers Meta- Complex We use the imaginary unit i to build complex numbers. Numbers A complex number z is a number of the form Robert Andrews z = a + bi Complex Numbers 2 Dual where a; b 2 R and i = −1. We call a the real part of the Numbers complex number and b the imaginary part. Split- Complex One important definition is that of the conjugate of z, Numbers denoted here by z∗. We define the conjugate of z as Visualizing Four- ∗ Dimensional z = a − bi Surfaces In essence, we're just changing the sign of the imaginary part. Now that we've defined these new numbers, what can we do with them? z + w = (a + c) + (b + d)i and define z − w as z − w = (a − c) + (b − d)i Addition and Subtraction Meta- Complex Numbers Robert Andrews Complex Let z = a + bi and w = c + di. Define z + w as Numbers Dual Numbers Split- Complex Numbers Visualizing Four- Dimensional Surfaces and define z − w as z − w = (a − c) + (b − d)i Addition and Subtraction Meta- Complex Numbers Robert Andrews Complex Let z = a + bi and w = c + di. Define z + w as Numbers Dual z + w = (a + c) + (b + d)i Numbers Split- Complex Numbers Visualizing Four- Dimensional Surfaces z − w = (a − c) + (b − d)i Addition and Subtraction Meta- Complex Numbers Robert Andrews Complex Let z = a + bi and w = c + di. Define z + w as Numbers Dual z + w = (a + c) + (b + d)i Numbers Split- Complex and define z − w as Numbers Visualizing Four- Dimensional Surfaces Addition and Subtraction Meta- Complex Numbers Robert Andrews Complex Let z = a + bi and w = c + di. Define z + w as Numbers Dual z + w = (a + c) + (b + d)i Numbers Split- Complex and define z − w as Numbers Visualizing Four- z − w = (a − c) + (b − d)i Dimensional Surfaces zw = (a + bi)(c + di) = a(c + di) + bi(c + di) = (ac − bd) + (ad + bc)i Multiplication Meta- Complex Numbers We can also easily define multiplication of complex numbers Robert Andrews using the idea of the distributive property. The distributive property simply says Complex Numbers Dual a(b + c) = ab + ac Numbers Split- Complex Keeping this idea in mind, if we let z = a + bi and Numbers w = c + di, then we define zw as Visualizing Four- Dimensional Surfaces Multiplication Meta- Complex Numbers We can also easily define multiplication of complex numbers Robert Andrews using the idea of the distributive property. The distributive property simply says Complex Numbers Dual a(b + c) = ab + ac Numbers Split- Complex Keeping this idea in mind, if we let z = a + bi and Numbers w = c + di, then we define zw as Visualizing Four- Dimensional zw = (a + bi)(c + di) Surfaces = a(c + di) + bi(c + di) = (ac − bd) + (ad + bc)i w∗ we multiply by w∗ . This allows us to define division as z z w∗ zw∗ (ac + bd) + (bc − ad)i = = = w w w∗ c2 + d2 c2 + d2 Note that division is undefined when c2 + d2 = 0, i.e. when w = 0. Division Meta- Complex Numbers Robert Andrews z Let z = a + bi and w = c + di. To define w , we use an Complex Numbers algebraic trick: Dual Numbers Split- Complex Numbers Visualizing Four- Dimensional Surfaces Note that division is undefined when c2 + d2 = 0, i.e. when w = 0. Division Meta- Complex Numbers Robert Andrews z Let z = a + bi and w = c + di. To define w , we use an Complex w∗ Numbers algebraic trick: we multiply by w∗ . This allows us to define Dual division as Numbers ∗ ∗ Split- z z w zw (ac + bd) + (bc − ad)i Complex = = = Numbers w w w∗ c2 + d2 c2 + d2 Visualizing Four- Dimensional Surfaces Division Meta- Complex Numbers Robert Andrews z Let z = a + bi and w = c + di. To define w , we use an Complex w∗ Numbers algebraic trick: we multiply by w∗ . This allows us to define Dual division as Numbers ∗ ∗ Split- z z w zw (ac + bd) + (bc − ad)i Complex = = = Numbers w w w∗ c2 + d2 c2 + d2 Visualizing Four- 2 2 Dimensional Note that division is undefined when c + d = 0, i.e. when Surfaces w = 0. Exponentiation Meta- Complex Numbers Robert To deal with complex exponents, we need to know a few Andrews Taylor series first. Recall that Complex Numbers 1 k x X x Dual e = Numbers k! k=0 Split- 1 Complex X x2k+1 Numbers sin x = (−1)k (2k + 1)! Visualizing k=0 Four- Dimensional 1 2k Surfaces X x cos x = (−1)k (2k)! k=0 Exponentiation Meta- Complex Numbers Robert Andrews If we expand the Taylor series for eix, we get Complex Numbers x x2 x3 Dual eix = 1 + i − − i + ··· Numbers 1! 2! 3! Split- x2 x4 x x3 x5 Complex = (1 − + + ··· ) + i( − + + ··· ) Numbers 2! 4! 1! 3! 5! Visualizing Four- = cos x + i sin x Dimensional Surfaces This equation, known as Euler's Formula, is essential for working with complex exponents.