Monday 21st July
Dr Alex Fink
but why stop there?
Mind your is and js and ks
What happens when you go beyond complex numbers?
Answer: Going further means sacrificing some properties of numbers such as
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies the equation
In this expression, a is the real part and b is the imaginary part of the complex number. Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane (also called Argand plane) by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with real numbers alone.
A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. “Re” is the real axis, “Im” is the imaginary axis, and i is the imaginary unit which satisfies the equation
Charles Lutwidge Dodgson (27 January 1832 – 14 January 1898), better known by his pen name, Lewis Carroll, was an English writer, mathematician, logician, Anglican deacon and photographer who certainly did not like this new fangled algebra and poked fun at it in his famous books about Alice.
The 19th century was a turbulent time for mathematics, with many new and controversial concepts, like imaginary numbers, becoming widely accepted in the mathematical community. Putting Alice’s Adventures in Wonderland in this context, it becomes clear that Dodgson, a stubbornly conservative mathematician, used some of the missing scenes to satirise these radical new ideas.
Dodgson disliked non-commutative algebras since it contradicted the basic laws of arithmetic and opened up a strange new world of mathematics such as quaternions where multiplication is non-commutative, meaning that x × y is not the same as y × x, these ideas were even more abstract than that of the symbolic algebraists. He is believed to have poked fun of this in the Mad Hatter’s tea party by writing Alice’s answers to questions given by the Hare as equally non-commutative. When the Hare tells her to “say what she means”, she replies that she does, “at least I mean what I say – that’s the same thing”. “Not the same thing a bit!” says the Hatter. “Why, you might just as well say that ‘I see what I eat’ is the same thing as ‘I eat what I see’!”
Defining complex numbers
You can always simplify them down to a + bi where a and b are real.
Real numbers are one dimensional
Complex numbers are two dimensional. They do dilations and rotations around an axis.
Complex numbers are good for 2 dimensional geometry, +/- gives shifting and x/¸ gives scaling and rotation.
Real is 1 dimensional
Complexes 2 dimensional
With complex numbers we should be able to solve any polynomial equation.
There is bad news, however. Complex numbers are no longer ordered in a way we would like.
In mathematics, a hypercomplex number is a traditional term for an element of an algebra over the field of real numbers. In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature, added to the real and complex numbers. The concept of a hypercomplex number covered them all, and called for a discipline to explain and classify them.
What stops j = i?
There is a danger as j wants to be i but this won’t go down well especially with standard form.
Take a + bi + cj. We have to say what i.j is. It’s i.j as we can’t prove that it is anything else.
a + bi + cj + d(i.j). Is i.j = j.i?
(a + bi + cj + dij) x (e + fi + gi + hij) = w + xi + yj +zij
How to divide
There is no satisfactory answer
You want one of the js below to be zero but it can’t
Two non-zero numbers with product zero! Oh no!
(if i is undefined) but we can’t divide by i – j as ij might not equal ji
Four dimensional hypercomplexes lose division by some non-zero numbers
Save division by ij not = ji
A trade-off: save division at a cost of commutativity of multiplication i.e.
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors.
Rotation commute is 2 dimensional but not 3 dimensional.
Quaternion plaque on Brougham (Broom) Bridge, Dublin, which says:
Here as he walked by on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication
& cut it on a stone of this bridge
Sir William Rowan Hamilton (midnight, 3–4 August 1805 – 2 September 1865) was an Anglo-Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra.
So k is a name for ij so what is ji?
Quaternion multiplication table can be seen below left
The image above right shows Graphical representation of quaternion units product as 90°-rotation in 4D-space, ij = k, ji = −k, ij = −ji
This is still used for rotations in computer graphics.
Coming back to the Mad Hatter’s tea party it has to be said that Lewis Carroll wrote that the character Time had fallen out with the Hatter and wouldn’t let the clocks move past six pm. Reading this scene with Hamiltonian maths in mind, the members of the Hatter’s tea party represent three terms of a quaternion, in which the all-important fourth term, time, is missing. Without Time, we are told, the characters are stuck at the tea table, constantly moving round to find clean cups and saucers.
Their movement around the table is reminiscent of William Rowan Hamilton’s early attempts to calculate motion, which was limited to rotatations in a plane before he added time to the mix. Even when Alice joins the party, she can’t stop the Hatter, the Hare and the Dormouse shuffling round the table, because she’s not an extra-spatial unit like Time.