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 √ 2 Visualizing −b ± b − 4ac Four- x = Dimensional 2a Surfaces The quadratic formula gives the roots of this equation as √ 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 √ 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 √ 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 ∈ 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 ∈ 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 ∈ 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 ∈ 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 ∞ k x X x Dual e = Numbers k! k=0 Split- ∞ Complex X x2k+1 Numbers sin x = (−1)k (2k + 1)! Visualizing k=0 Four- Dimensional ∞ 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. From this formula, we get the equation ea+bi = ea(cos b + i sin b)
Exponentiation
Meta- Complex Numbers
Robert Andrews Thanks to Taylor series, we get a very nice formula for
Complex dealing with imaginary exponents: Numbers
Dual ix Numbers e = cos x + i sin x
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces Exponentiation
Meta- Complex Numbers
Robert Andrews Thanks to Taylor series, we get a very nice formula for
Complex dealing with imaginary exponents: Numbers
Dual ix Numbers e = cos x + i sin x
Split- Complex This equation, known as Euler’s Formula, is essential for Numbers
Visualizing working with complex exponents. From this formula, we get Four- the equation Dimensional Surfaces ea+bi = ea(cos b + i sin b) the logarithm!
Exponentiation
Meta- Complex Numbers
Robert Andrews
Complex If we want to take exponents with a complex number as the Numbers base, we write Dual Numbers c+di log (a+bi) c+di Split- (a + bi) = (e ) Complex Numbers
Visualizing There’s one thing here that we have yet to define: Four- Dimensional Surfaces Exponentiation
Meta- Complex Numbers
Robert Andrews
Complex If we want to take exponents with a complex number as the Numbers base, we write Dual Numbers c+di log (a+bi) c+di Split- (a + bi) = (e ) Complex Numbers
Visualizing There’s one thing here that we have yet to define: the Four- Dimensional logarithm! Surfaces If we can write a complex number in this form, the logarithm becomes very easy to find. We use the polar form of a complex number to do this. Write a + bi as
a + bi = r(cos θ + i sin θ)
2 2 b where r = a + b and θ = arctan a . (Note that we have to make sure that θ is in the correct quadrant) From here, we can find that
log (a + bi) = log (r(cos θ + i sin θ)) = log r + θi
Logarithms
Meta- Complex Recall that Numbers ea+bi = ea(cos b + i sin b) Robert Andrews
Complex Numbers
Dual Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces We use the polar form of a complex number to do this. Write a + bi as
a + bi = r(cos θ + i sin θ)
2 2 b where r = a + b and θ = arctan a . (Note that we have to make sure that θ is in the correct quadrant) From here, we can find that
log (a + bi) = log (r(cos θ + i sin θ)) = log r + θi
Logarithms
Meta- Complex Recall that Numbers ea+bi = ea(cos b + i sin b) Robert Andrews If we can write a complex number in this form, the Complex logarithm becomes very easy to find. Numbers
Dual Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces Logarithms
Meta- Complex Recall that Numbers ea+bi = ea(cos b + i sin b) Robert Andrews If we can write a complex number in this form, the Complex logarithm becomes very easy to find. Numbers
Dual We use the polar form of a complex number to do this. Numbers Write a + bi as Split- Complex Numbers a + bi = r(cos θ + i sin θ) Visualizing Four- 2 2 b Dimensional where r = a + b and θ = arctan a . (Note that we have to Surfaces make sure that θ is in the correct quadrant) From here, we can find that
log (a + bi) = log (r(cos θ + i sin θ)) = log r + θi Suppose that the polar form of a + bi is given by reiθ. Then we can write this as
iθ (a + bi)c+di = e(c+di)(log (re )) = e(c log r−dθ)+i(d log r+cθ) = rce−dθ [cos (d log r + cθ) + i sin (d log r + cθ)]
Exponentiation
Meta- Complex Numbers Robert From before, we have Andrews c+di log a+bi c+di Complex (a + bi) = (e ) Numbers
Dual Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces Exponentiation
Meta- Complex Numbers Robert From before, we have Andrews c+di log a+bi c+di Complex (a + bi) = (e ) Numbers Dual iθ Numbers Suppose that the polar form of a + bi is given by re . Then
Split- we can write this as Complex Numbers c+di (c+di)(log (reiθ)) Visualizing (a + bi) = e Four- (c log r−dθ)+i(d log r+cθ) Dimensional = e Surfaces = rce−dθ [cos (d log r + cθ) + i sin (d log r + cθ)] Complex numbers are used in solving differential equations of the form
y00 + by0 + cy = 0
Electrical engineers use complex numbers to describe circuits And many more!
Importance of Complex Numbers
Meta- Complex Numbers
Robert Andrews The complex numbers form an algebraically closed field Complex Numbers
Dual Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces Electrical engineers use complex numbers to describe circuits And many more!
Importance of Complex Numbers
Meta- Complex Numbers
Robert Andrews The complex numbers form an algebraically closed field Complex Numbers Complex numbers are used in solving differential
Dual equations of the form Numbers
Split- 00 0 Complex y + by + cy = 0 Numbers
Visualizing Four- Dimensional Surfaces And many more!
Importance of Complex Numbers
Meta- Complex Numbers
Robert Andrews The complex numbers form an algebraically closed field Complex Numbers Complex numbers are used in solving differential
Dual equations of the form Numbers
Split- 00 0 Complex y + by + cy = 0 Numbers Visualizing Electrical engineers use complex numbers to describe Four- Dimensional circuits Surfaces Importance of Complex Numbers
Meta- Complex Numbers
Robert Andrews The complex numbers form an algebraically closed field Complex Numbers Complex numbers are used in solving differential
Dual equations of the form Numbers
Split- 00 0 Complex y + by + cy = 0 Numbers Visualizing Electrical engineers use complex numbers to describe Four- Dimensional circuits Surfaces And many more! 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 Dual numbers are written in the form
a + bε
where a, b ∈ R. We call a the real part and b the dual part. As with complex numbers, we define the conjugate z∗ of the dual number z = a + bε as
z∗ = a − bε
Defining the Dual Numbers
Meta- Complex Numbers The dual numbers are very similar to the complex numbers, Robert Andrews but in the place of i, we introduce an element ε such that 2 Complex ε = 0 but ε 6= 0. Numbers
Dual Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces As with complex numbers, we define the conjugate z∗ of the dual number z = a + bε as
z∗ = a − bε
Defining the Dual Numbers
Meta- Complex Numbers The dual numbers are very similar to the complex numbers, Robert Andrews but in the place of i, we introduce an element ε such that 2 Complex ε = 0 but ε 6= 0. Numbers Dual numbers are written in the form Dual Numbers a + bε Split- Complex Numbers where a, b ∈ . We call a the real part and b the dual part. Visualizing R Four- Dimensional Surfaces Defining the Dual Numbers
Meta- Complex Numbers The dual numbers are very similar to the complex numbers, Robert Andrews but in the place of i, we introduce an element ε such that 2 Complex ε = 0 but ε 6= 0. Numbers Dual numbers are written in the form Dual Numbers a + bε Split- Complex Numbers where a, b ∈ R. We call a the real part and b the dual part. Visualizing ∗ Four- As with complex numbers, we define the conjugate z of the Dimensional Surfaces dual number z = a + bε as
z∗ = a − bε and define subtraction as
z − w = (a − c) + (b − d)ε
Addition and Subtraction
Meta- Complex Numbers
Robert Andrews
Complex Let z = a + bε and w = c + dε. Define addition as Numbers Dual z + w = (a + c) + (b + d)ε Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces Addition and Subtraction
Meta- Complex Numbers
Robert Andrews
Complex Let z = a + bε and w = c + dε. Define addition as Numbers Dual z + w = (a + c) + (b + d)ε Numbers
Split- Complex and define subtraction as Numbers
Visualizing Four- z − w = (a − c) + (b − d)ε Dimensional Surfaces and define division as z ac + (bc − ad)ε = w c2 Note that division is not defined when c = 0.
Multiplication and Divison
Meta- Complex Numbers
Robert Andrews Keep z = a + bε and w = c + dε. Define multiplication as Complex Numbers zw = ac + (ad + bc)ε Dual Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces Multiplication and Divison
Meta- Complex Numbers
Robert Andrews Keep z = a + bε and w = c + dε. Define multiplication as Complex Numbers zw = ac + (ad + bc)ε Dual Numbers
Split- and define division as Complex Numbers z ac + (bc − ad)ε Visualizing = Four- w c2 Dimensional Surfaces Note that division is not defined when c = 0. From this, it is easy to verify that b log (a + bε) = log a + ε a Note that the logarithm is undefined when a = 0.
Exponentiation & Logarithms
Meta- Complex Numbers
Robert Andrews Using Taylor series in the same way we developed complex
Complex exponentiation, it can be shown that Numbers a+bε a a Dual e = e + e bε Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces Exponentiation & Logarithms
Meta- Complex Numbers
Robert Andrews Using Taylor series in the same way we developed complex
Complex exponentiation, it can be shown that Numbers a+bε a a Dual e = e + e bε Numbers
Split- Complex From this, it is easy to verify that Numbers Visualizing b Four- log (a + bε) = log a + ε Dimensional a Surfaces Note that the logarithm is undefined when a = 0. The final result for exponentiation is
bc (a + bε)c+dε = ac + ac d log a + ε a
Note that the base must have a non-zero real part.
Exponentiation & Logarithms
Meta- Complex Numbers
Robert Andrews Combining dual exponentiation with the dual logarithm Complex Numbers allows us to take exponents of arbitrary bases, similar to
Dual what we did with complex numbers. Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces Exponentiation & Logarithms
Meta- Complex Numbers
Robert Andrews Combining dual exponentiation with the dual logarithm Complex Numbers allows us to take exponents of arbitrary bases, similar to
Dual what we did with complex numbers. The final result for Numbers exponentiation is Split- Complex Numbers bc (a + bε)c+dε = ac + ac d log a + ε Visualizing a Four- Dimensional Surfaces Note that the base must have a non-zero real part. Uses of Dual Numbers
Meta- Complex Numbers
Robert Andrews
Complex Numbers
Dual Numbers We can use dual numbers to automatically differentiate Split- functions. Complex Numbers
Visualizing Four- Dimensional Surfaces If we evaluate f(x + ε), we get ∞ X f n(0) f(x + ε) = (x + ε)n n! n=0 ∞ X f n(0) = (xn + nxn−1ε) n! n=0 ∞ ∞ X f n(0) X f n(0) = xn + nxn−1 n! n! n=0 n=0 = f(x) + εf 0(x)
Uses of Dual Numbers
Meta- Consider some function f(x) and its Taylor series expansion Complex Numbers centered at 0 ∞ n Robert X f (0) n Andrews f(x) = x n! n=0 Complex Numbers
Dual Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces Uses of Dual Numbers
Meta- Consider some function f(x) and its Taylor series expansion Complex Numbers centered at 0 ∞ n Robert X f (0) n Andrews f(x) = x n! n=0 Complex Numbers If we evaluate f(x + ε), we get Dual Numbers ∞ X f n(0) Split- f(x + ε) = (x + ε)n Complex n! Numbers n=0 Visualizing ∞ n Four- X f (0) n n−1 Dimensional = (x + nx ε) Surfaces n! n=0 ∞ ∞ X f n(0) X f n(0) = xn + nxn−1 n! n! n=0 n=0 = f(x) + εf 0(x) 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 Split-complex numbers are written in the form
a + bj
where a, b ∈ R. We call a the real part and b the split part. As with complex numbers, we define the conjugate z∗ of the split-complex number z = a + bj as
z∗ = a − bj
Defining the Split-Complex Numbers
Meta- Complex Numbers The split-complex numbers are also similar to the complex Robert Andrews numbers, but in the place of i, we introduce an element j 2 Complex such that j = 1 but j 6= 1, −1. Numbers
Dual Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces As with complex numbers, we define the conjugate z∗ of the split-complex number z = a + bj as
z∗ = a − bj
Defining the Split-Complex Numbers
Meta- Complex Numbers The split-complex numbers are also similar to the complex Robert Andrews numbers, but in the place of i, we introduce an element j 2 Complex such that j = 1 but j 6= 1, −1. Numbers Split-complex numbers are written in the form Dual Numbers a + bj Split- Complex Numbers where a, b ∈ R. We call a the real part and b the split part. Visualizing Four- Dimensional Surfaces Defining the Split-Complex Numbers
Meta- Complex Numbers The split-complex numbers are also similar to the complex Robert Andrews numbers, but in the place of i, we introduce an element j 2 Complex such that j = 1 but j 6= 1, −1. Numbers Split-complex numbers are written in the form Dual Numbers a + bj Split- Complex Numbers where a, b ∈ R. We call a the real part and b the split part. Visualizing ∗ Four- As with complex numbers, we define the conjugate z of the Dimensional Surfaces split-complex number z = a + bj as
z∗ = a − bj and define subtraction as
z − w = (a − c) + (b − d)j
Addition and Subtraction
Meta- Complex Numbers
Robert Andrews
Complex Let z = a + bj and w = c + dj. Define addition as Numbers Dual z + w = (a + c) + (b + d)j Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces Addition and Subtraction
Meta- Complex Numbers
Robert Andrews
Complex Let z = a + bj and w = c + dj. Define addition as Numbers Dual z + w = (a + c) + (b + d)j Numbers
Split- Complex and define subtraction as Numbers
Visualizing Four- z − w = (a − c) + (b − d)j Dimensional Surfaces and define division as z (ac − bd) + (bc − ad)j = w c2 − d2
Note that division is not defined when c2 − d2 = 0.
Multiplication and Divison
Meta- Complex Numbers
Robert Andrews Keep z = a + bj and w = c + dj. Define multiplication as Complex Numbers zw = (ac + bd) + (ad + bc)j Dual Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces Multiplication and Divison
Meta- Complex Numbers
Robert Andrews Keep z = a + bj and w = c + dj. Define multiplication as Complex Numbers zw = (ac + bd) + (ad + bc)j Dual Numbers
Split- and define division as Complex Numbers z (ac − bd) + (bc − ad)j Visualizing = 2 2 Four- w c − d Dimensional Surfaces Note that division is not defined when c2 − d2 = 0. From this, we can set up a system of equations to relate x + yj to ea+bj, where a + bj = log (x + yj). This system is
ea+b + ea−b x = 2 ea+b − ea−b y = 2
Exponentiation & Logarithms
Meta- Complex Numbers Using Taylor series in the same way we developed complex Robert Andrews exponentiation, it can be shown that
Complex a+bj a Numbers e = e (cosh b + j sinh b) Dual Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces Exponentiation & Logarithms
Meta- Complex Numbers Using Taylor series in the same way we developed complex Robert Andrews exponentiation, it can be shown that
Complex a+bj a Numbers e = e (cosh b + j sinh b) Dual Numbers From this, we can set up a system of equations to relate Split- a+bj Complex x + yj to e , where a + bj = log (x + yj). This system is Numbers
Visualizing a+b a−b Four- e + e Dimensional x = Surfaces 2 ea+b − ea−b y = 2 This lets us easily define the logarithm!
1 1 a + b log (a + bj) = log ((a + b)(a − b)) + j log 2 2 a − b
Note that the split complex logarithm is not closed. For example, if (a + b)(a − b) < 0, we will get a complex number in our answer. To avoid this, we will only define the logarithm of a + bj when a > max (b, −b).
Exponentiation & Logarithms
Meta- Complex If we solve this system of equations, we get Numbers Robert 1 Andrews a = log ((x + y)(x − y)) 2 Complex 1 x + y Numbers b = log Dual 2 x − y Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces Note that the split complex logarithm is not closed. For example, if (a + b)(a − b) < 0, we will get a complex number in our answer. To avoid this, we will only define the logarithm of a + bj when a > max (b, −b).
Exponentiation & Logarithms
Meta- Complex If we solve this system of equations, we get Numbers Robert 1 Andrews a = log ((x + y)(x − y)) 2 Complex 1 x + y Numbers b = log Dual 2 x − y Numbers Split- This lets us easily define the logarithm! Complex Numbers Visualizing 1 1 a + b Four- log (a + bj) = log ((a + b)(a − b)) + j log Dimensional 2 2 a − b Surfaces Exponentiation & Logarithms
Meta- Complex If we solve this system of equations, we get Numbers Robert 1 Andrews a = log ((x + y)(x − y)) 2 Complex 1 x + y Numbers b = log Dual 2 x − y Numbers Split- This lets us easily define the logarithm! Complex Numbers Visualizing 1 1 a + b Four- log (a + bj) = log ((a + b)(a − b)) + j log Dimensional 2 2 a − b Surfaces Note that the split complex logarithm is not closed. For example, if (a + b)(a − b) < 0, we will get a complex number in our answer. To avoid this, we will only define the logarithm of a + bj when a > max (b, −b). d 2 c+dj a + b c (a + bj) = ((a + b)(a − b)) 2 (cosh k + j sinh k) a − b
c a + b d k = log + log a2 − b2 2 a − b 2
Exponentiation & Logarithms
Meta- Complex Numbers
Robert Andrews Combining the logarithm with exponentiation base e allows
Complex us to take exponents with an arbitrary base. After lots of Numbers nasty math, we can define exponentiation as Dual Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces Exponentiation & Logarithms
Meta- Complex Numbers
Robert Andrews Combining the logarithm with exponentiation base e allows
Complex us to take exponents with an arbitrary base. After lots of Numbers nasty math, we can define exponentiation as Dual Numbers d 2 Split- c+dj a + b c Complex (a + bj) = ((a + b)(a − b)) 2 (cosh k + j sinh k) Numbers a − b Visualizing Four- c a + b d Dimensional k = log + log a2 − b2 Surfaces 2 a − b 2 “Split-complex mulitplication has commonly been seen as a Lorentz boost of a spacetime plane.” 1
Uses of Split-Complex Numbers
Meta- Complex Numbers
Robert Andrews
Complex Numbers
Dual Split-Complex numbers have uses in physics: Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces
1en.wikipedia.org/wiki/Split-complex_number Uses of Split-Complex Numbers
Meta- Complex Numbers
Robert Andrews
Complex Numbers
Dual Split-Complex numbers have uses in physics: Numbers “Split-complex mulitplication has commonly been seen as a Split- 1 Complex Lorentz boost of a spacetime plane.” Numbers
Visualizing Four- Dimensional Surfaces
1en.wikipedia.org/wiki/Split-complex_number 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 the resulting surfaces are four-dimensional!
So What is My Project?
Meta- Complex Numbers
Robert Andrews
Complex Numbers I will be writing a program that can graph functions of these Dual Numbers different “meta-complex” numbers. There’s one big issue Split- that comes with doing this: Complex Numbers
Visualizing Four- Dimensional Surfaces So What is My Project?
Meta- Complex Numbers
Robert Andrews
Complex Numbers I will be writing a program that can graph functions of these Dual Numbers different “meta-complex” numbers. There’s one big issue Split- that comes with doing this: the resulting surfaces are Complex Numbers four-dimensional! Visualizing Four- Dimensional Surfaces Plot the output vectors as a vector field in the plane Project the four-dimensional surface into 3-space and allow four-dimensional rotations to be made
Four-Dimensional Surfaces
Meta- Complex Numbers
Robert Andrews
Complex How can we visualize four-dimensional results? There are Numbers several options: Dual Numbers Map outputs to different colors in RGB space Split- Complex Numbers
Visualizing Four- Dimensional Surfaces Project the four-dimensional surface into 3-space and allow four-dimensional rotations to be made
Four-Dimensional Surfaces
Meta- Complex Numbers
Robert Andrews
Complex How can we visualize four-dimensional results? There are Numbers several options: Dual Numbers Map outputs to different colors in RGB space Split- Complex Plot the output vectors as a vector field in the plane Numbers
Visualizing Four- Dimensional Surfaces Four-Dimensional Surfaces
Meta- Complex Numbers
Robert Andrews
Complex How can we visualize four-dimensional results? There are Numbers several options: Dual Numbers Map outputs to different colors in RGB space Split- Complex Plot the output vectors as a vector field in the plane Numbers
Visualizing Project the four-dimensional surface into 3-space and Four- Dimensional allow four-dimensional rotations to be made Surfaces Problem: colors cannot be consistently reproduced on different screens!
Color Mappings
Meta- Complex Numbers
Robert Andrews
Complex Numbers
Dual Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces A graph of the complex function (z3 − 1)(z − 1).2
2Image from www.pacifict.com/ComplexFunctions.html Color Mappings
Meta- Complex Numbers
Robert Andrews
Complex Numbers
Dual Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces A graph of the complex function (z3 − 1)(z − 1).2
Problem: colors cannot be consistently reproduced on different screens!
2Image from www.pacifict.com/ComplexFunctions.html Vector Fields
Meta- Complex Numbers
Robert Andrews
Complex Numbers
Dual Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces
A graph of the complex function (z + 2)(z − 2).3
3Image from www.pacifict.com/ComplexFunctions.html Projections into 3-Space
Meta- Complex Numbers
Robert Andrews
Complex Numbers
Dual Numbers
Split- Complex Numbers
Visualizing Four- Dimensional Surfaces Two projections of the four-dimensional complex exponential into 3-space.5
4Image from www.pacifict.com/ComplexFunctions.html 5Image from www.pacifict.com/ComplexFunctions.html