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