Hamilton, Boole and their Algebras Raymond Flood Gresham Professor of Geometry Hamilton, Boole and their Algebras

William Rowan Hamilton 18051865 George Boole 18151864 William Rowan Hamilton 1805 1865 William Rowan Hamilton 1805 1865

One thing only have I to regret in the direction of my studies, that they should be diverted or rather, rudely forced by the College Course from their natural bent and favourite channel. That bent, you know is Science Science

in its most exalted heights, in its most secret recesses. It has so captivated me - so seized on . My affections that my attention to Classical studies is an effort and an irksome one. Letter to his sister Eliza 1823

Trinity College, Dublin Dunsink Observatory Conical Refraction Hamiltons prediction of the behaviour of a ray of light passed through a prism with biaxial symmetry Humphrey Lloyd

180081 1 1 . Euler: Leonhard Of such numbers we may truly assert that they are neither nothing, nor greater than

nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible. 1 . Euler: Leonhard Of such numbers we may truly assert that they are neither nothing, nor greater than

nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible. Augustus de Morgan: We have shown the symbol to be void of meaning, or rather self-contradictory and Suppose that we try to calculate

with the symbol We find that addition is easy: (2 + 3) + (4 + 5) = 6 + 8 Suppose that we try to calculate with the symbol We find that addition is easy:

(2 + 3) + (4 + 5) = 6 + 8 And so is multiplication (replacing x whenever it appears by (2 + 3) x (4 + 5) = (2 x 4) + (3 x 4) + (2 x 5) + (15 x x ) = (8 15) + (12 +10)= 7 + 22 We call the object a + b a complex number

the number a is its real part, and the number b is its imaginary part Nowadays, we usually use the letter i to mean so that i2 = 1 Representing Complex numbers geometrically

Caspar Wessel in 1799 In this representation, called the complex plane, two axes are drawn at right angles the real axis and the imaginary axis and the complex number a + b is represented by the

point at a distance a in the direction of the real axis and at height b in the direction of the imaginary Is there any other algebraic representation of complex numbers that reveals all valid operations on them?

Is there any other algebraic representation of complex numbers that reveals all valid operations on them? We define a complex number as a pair (a, b) of real numbers. Is there any other algebraic representation of complex numbers that reveals all valid

operations on them? We define a complex number as a pair (a, b) of real numbers. They are added as follows: (a, b) + (c, d) = (a + c, b + d); Is there any other algebraic representation of complex numbers that reveals all valid

operations on them? We define a complex number as a pair (a, b) of real numbers They are added as follows: (a, b) + (c, d) = (a + c, b + d) They are multiplied as follows: (a, b) x (c, d) = (ac bd, ad + bc) Is there any other algebraic representation of

complex numbers that reveals all valid operations on them? We define a complex number as a pair (a, b) of real numbers They are added as follows: (a, b) + (c, d) = (a + c, b + d) They are multiplied as follows: (a, b) x (c, d) = (ac bd, ad + bc) The pair (a, 0) then corresponds to the real number

a the pair (0, 1) corresponds to the imaginary number i Triadic fancies Hamilton now looked at number triples such as (a, b, c) He wrote them as: a + bi + cj where i2 = j2 = 1

Addition was easy: (1 + 2i + 3j) + (4 + 5i + 6j) = (1 + 4) + (2i + 5i) + (3j + 6j) = 5 + 7i + 9j How to multiply triples? How could we define (1 + 2i + 3j) x (4 + 5i +

6j)? If we multiply them out in the analogous way to complex numbers and use i2 = j2 = 1 we obtain (4 10 18) + (5 + 8)i + (6 + 12)j + (12 + 15)ij What is this term ij? Every morning on my coming down to

breakfast, your brother William Edwin and yourself used to ask me. Well, Papa, can you multiply triplets? Whereto I was always obliged to reply, with a sad

Division law If D, E are triples with D non zero then there must be a triple X so that D multiplied by X equals E We can think of this as meaning that D divides E Examples: 3 x = 17 and (1 + i)x(1

i)/2 = 1 The law of the modulus The modulus of a + bi + cj is just Hamilton wanted his method of multiplication to have the following property:

the modulus of (a + bi + cj)(x + yi + zj) equals the modulus of (a + bi + cj) times the modulus of (x + yi + zj) i = j = k =ijk = 1 2 2

2 16th October 1843 Quaternions a + bi + cj + dk where i2 = j2 = k2 =ijk = -1 Quaternions

a + bi + cj + dk where i2 = j2 = k2 =ijk = 1 ijk = 1 ijkk = k ij = k ij = k Quaternions a + bi + cj + dk

where i2 = j2 = k2 =ijk = 1 ijk = 1 ijk = 1 ijkk = k jiijk = ji

ij = k jjk = ji ij = k k = ji (a + bi + cj + dk)(w + xi +

yj + zk) Clockwise ij = k, jk = i, ki = j Anticlockwise ji = k, ik = j, kj = i And how the One of Time,

of Space the Three, Might in the Chain of Symbols girdled be A quaternion has a scalar and vector part If q1 and q2 are two quaternions with no scalar terms then scalar part of q1q2 is the

negative of the dot product of the vectors q1 and q2 while the vector part of q1q2 is the vector product of the vectors q1 and q2 Vince, J (2011). Quaternions for Computer Graphics. 1st. ed. London: Springer Quaternions can be

used to achieve the transformation of any directed line in three dimensions to any other directed line which is why they are of use in computer graphics.

George Boole 181564 J.M.W. Turner, Cathedral Church at Lincoln, 1795 Appointed Professor of Mathematics at Cork, 1849 Booles Laws of Thought

Boole: A new algebra Let the symbol x denote the class of all white things And the symbol y denote the class of all sheep Then he used the compound symbol xy to denote the class of all white sheep

Boole: A new algebra Let the symbol x denote the class of all white things And the symbol y denote the class of all sheep Then he used the compound symbol xy to denote the class of all white sheep And if z denotes the class of all horned things

then zxy = all horned white sheep Boole: A new algebra Again if x = all white things And y = all sheep Then xy = yx Since the class of all white things that are sheep is the same as the class of

all sheep that are white Boole: A new algebra If every member of class x (say being a man) is also a member of class y (say being human) then xy = x and in the special case when x and y are the same

xx = x Boole: A new algebra If every member of class x (say being a man) is also a member of class y (say being human) then xy = x and in the special case when x and y are the same

xx = x or using the analogy of multiplication of numbers we can write x2 = x for all classes x Addition If x and y are classes, x+y

denotes the class of all objects which belong to class x or to class y. if x is the class of all women and y is the class of all men then x + y is the class of all humans. Then once again we have commutativity as x+y=y+x We have other results similar to what we see in arithmetic, for example z(x + y) = zx + zy

If z is the class of all Europeans then the left hand side is the class of all European humans while the right hand side is the class of European women or European men. Universal and empty class Boole also used the symbol 1 to denote the universal class, the symbol 0 for the empty class and he wrote 1 x for the class of all objects not in x.

If x = all white things Then (1- x) = all things not white Since objects cannot be white and not white at the same time x(1 x) = 0 or x x2 = 0 or x2 = x as before Symbolic Algebra

All As are B, all Bs are C: therefore all As are C Let a equal the class of all As Let b equal the class of all Bs Let c equal the class of all Cs In Booles notation the hypotheses are a = ab and b = bc Then by substitution a = ab = a(bc) = (ab)c = ac so a = ac so all As are C

Claude Shannon 19162001 Shannon says that the hindrance between two points a and b on a circuit is 0 if current can flow and 1 otherwise If the switch

between a and b is closed the hindrance is 0 and if open the hindrance is 1 The hindrance of two switches

X and Y in series is written X + Y The hindrance of two switches X and Y in parallel is written XY

Shannon says that the hindrance between two points a and b on a circuit is 0 if current can flow and 1 otherwise If the switch The The hindrance

between a and b hindrance of of two is closed the two switches switches X and hindrance is 0 X and Y in Y in parallel is

and if open the series is written XY Postulates: hindrance is 1in series written X +open Y circuit is an open circ

a closed circuit with an a closed circuit in parallel with an open circuit is a closed c Memorials Hamilton

http://www.maths.tcd.ie/pub/HistMath/ People/Hamilton/ Boole Des McHale, George Boole, A Prelude to the Digital Age, Cork University Press, 2015

1 pm on Tuesdays at the Museum of London Einsteins Annus Mirabilis, 1905 Tuesday 20 October 2015 Hamilton, Boole and their Algebras Tuesday 17 November 2015 Babbage and Lovelace Tuesday 19 January 2016 Gauss and Germain

Tuesday 16 February 2016 Hardy, Littlewood and Ramanujan Tuesday 15 March 2016 Turing and von Neumann Tuesday 19 April 2016