The Error of commutativity in Arithmetic Multiplication

In  order to answer the question "Why is -1 * -l  +1 ?',and why this may be considered in error, first of all we need to understand why we have negative numbers at all.In order to do this I will quote passages from works by other authors and comment on them.Because of this ,some concepts or mathematical ideas may be introduced prematurely.Where this happens I will seek to explain in square brackets[].

l. THE NEED FOR NEGATIVE NUMBERS 
The following passage is taken from Makers of Mathematics by Alfred Hooper,it initially contains some algebra, but this is what brought the negative number concept into focus, and I will explain it where necessary.

"Let us now see how mathematicians overcame the difficulty that caused Diophantus to declare that the equation 4X + 20 = 4 was absurd.  [The equation just means that 4 times some unknown number plus 20 is 4.It is seen here that if X was say 2,then 4 times 2 plus 20 would be 28 and not 4.The smallest positive number we can use is 1 but this gives 4 times 1 plus 20 is 24.Even if we use zero for X,the answer is still bigger than.4,namely 20,and this is why it is considered absurd here.If X is negative,though,we can have 4 times -4 plus 20 is 4,since 4 times -4 is -16 and 20 - 16 is 4.]
We shall see that the difficulty lay,not with the equation,but with the inability of the primitive number-scale to handle any but the numbers starting with zero and continuing through l,2,3,4....in an endless series.After being dubbed absurd for thousands of years,equations like the one just quoted at last came into their own,as mathematicians slowly came to realize that in many problems,the essential idea is not merely the magnitude,but the order in which things lie.If the non-mathematically minded reader will bear with an example that looks particularly trivial and childish,they may quickly grasp the idea of the new concept that was introduced into the primitive number-scale.By this new concept,the ideas of direction and relative position were linked to the number-scale,which was now extended into one having no beginning,as well as no end. Imagine we are a crowd of onlookers watching a large hotel on fire.We here a cry of distress.We see a would-be rescuer emerge from a sixth floor window on to the fire escape.He rushes up 18 steps,peers through a window,evidently sees nothing,and then climbs down 5 steps to look in at another window.Here we shall leave him,since,as mathematicians,we are,of course,supposed to be only interested in calculations.
Even a small child would know that the would-be rescuer is now 13 steps above his starting point.How would he know this? By working the calculation 18 - 5 = 13.Like Diophantus,he would say that the minus sign shows that 5 is to be subtracted from 18.
But there is another,and quite different interpretation possible The facts we have considered make it clear that in the statement 18 -5 = 13,the number 18 indicates an upward movement in this particular problem,and the number 5 a downward movement.So the minus sign may be regarded as indicating that the number following it is to be measured in a direction exactly opposite to that indicated by the number 18.
How would this idea have worked if our would be rescuer had run 5 steps up and then 18 steps down the fire escape? We should have indicated these movements by writing 5 - 18.This gives us the kind of problem that baffled ancient mathematicians.Obviously,if minus can only indicate subtraction,we,too,must remain baffled.But since we have agreed that a minus sign also indicates a change of direction,the statement makes sense,since it tells us that the climber,having gone up 5 steps,then took 18 steps  in the opposite direction.But can there be any answer to 5-18 ?
Well,let the would-be rescuer make his 18 downward steps in two movements,the first a downward movement of 5 steps.the second,another downward movement of the remaining 13 steps.We can now say that the number of steps from his starting point at which he finishes his three movements can be indicated by 5 - 5 - 13.Clearly,the first two of these movements bring him back to his starting point, so the 5 - 5 part of the statement can be ignored,leaving the answer -13.So 5 - 18 = -13.But what meaning can be attached to this -13 ?
Obviously it indicates the number of steps below his starting point at which the climber finishes.If we then agree that when calculating distances up and down a ladder,or any graduated scale,a minus sign may indicate either a downward movement or a position below some starting point,the expression 5 - 18 = -13 has a meaning which it could not have with the elementary number-scale of arithmetic.
Now let us consider another case.Suppose the climber had run up 4 steps,paused,then run up another 7 steps.We should then calculate his final position as 4 + 7 = ll.What does this + sign indicate? Diophantus would have answered "Addition."We should say "Yes,but it may also indicate an upward movement."So if we agree that a minus sign indicates a downward movement or a position below some starting point, a plus sign will indicate either an upward movement or a position above that same starting point.We can now label the starting point O,the steps above the starting point as +l,+2,+3,+4,....and the steps below the starting point as -l,-2,-3,-4,....
Numbers marked like this ,with a + or - sign are called directed,or signed numbers.A directed number having a + sign is called a positive number;one having a - sign,a negative number.
Let us now return to the equation 4X + 20 = 4,which Diophantus considered absurd.Using our new concept,it is clear that X must stand for a negative number,the number -4.A possible problem that would lead to this equation would be "What movement must be made by a man standing on a ladder,if four times that movement combined with twenty steps upward would bring him to a position that was four steps above his starting point?" Since the answer is " 4 downward steps," we see that the equation makes perfectly good sense after all.
These directed or signed numbers may be used whenever two movements occur in exactly opposite directions.Thus,on a horizontal scale,movements to the right,or positions to the right of a fixed starting point can be indicated by the positive numbers;movements to the left,or positions to the left of that starting point,by negative numbers.
Again,when a line rotates around a fixed point in a clockwise direction (the minute hand of a watch for example) it is customary to label the angle through which it passes as being a negative quantity.When a line rotates in a counter-clockwise direction,the angle through which it passes is labelled as a positive quantity.The whole concept of positive and negative is based on the idea of movements back and forth,or positions that result from such movements.For this reason,the developement of the idea of directed numbers opened out a great new field of research,a field that was necessarily closed to earlier mathematicians who did not have this concept"

This is,perhaps,an unnecessarily long winded explanation,but I hope it serves to demonstrate the need for being able to measure left and right or up and down with respect to some fixed point.We do this in banking for instance where our accounts can be in credit or deficit.

2. THE BASIC RULES OF ARITHMETIC
In arithmetic,it is commonly assumed that the following rules apply.They are taken from Complex Numbers by W Ledermann.
		I. A+B = B+A	(Commutative)
		II. (A+B)+C = A+(B+C)	(Associative)
	III. A+X=B  = X=B-A	(Subtractive)
		IV. AB = BA	(Commutative)  [This is the problem]
		V. (AB)C = A(BC)	(Associative)
		VI. AX=B = X=B/A	(Divisive)
	VII. (A+B)C  = AC + BC	(Distributive)

These are the intuitive concepts that we use every day when we add up or multiply.Rule I. is just stating that when we add two numbers together we can do this either way around ie 4+3=7 or 3+4=7. Rule IV. is saying the same thing for multiplication,4*3=12 or 3*4=12 If someone were to suggest that 3*4 was not 12,or that 3*4 was not the same as 4*3 we would think this intuitively wrong.
Before examining this I wish to illustrate why it is that we take this commutativity,as it is called,of multiplication for granted.
To do this I call on the help of Isaac Asimov's The Left Hand of the Electron. (see http://members.fortunecity.com/templarseries/l-hand1.html)

"I suspect that some prehistoric philosopher must have decided that there were two kinds of numbers:peaceful ones and warlike ones.The peaceful numbers were those of the type 2,4,6,8... while the numbers in between were warlike.
If there were 8 stone axes and two individuals possessing equal claim,it would be easy to hand 4 to each to make peace.If there were 7,however,you would have to give 3 to each and then toss away the remaining 1 (a clear loss of a valuable object) or let the two disputants fight over it.
The fact that the original property that marked out the significance of what we now call odd and even numbers was something like this is indicated by the very names that we give them. The word "even" means fundamental ly,flat, smooth ,without unusual depressions or elevations.We use the word in this sense when we say that a person says something "in an even tone of voice".An even number of identical coins,for instance,can be divided into two piles of exactly the same height.The two piles are even in height and hence the number is called even.The even number is the one with the property of "equal shares".
"Odd",on the other hand,is from an old Norse word meaning "point" or "tip".If an odd number of coins is divided into two piles as nearly equal as possible,one pile is higher than the other by one coin and therefore rears a point or tip into the air,as compared with the other.The odd number possesses the property of "unequal shares",and it is no accident that the expression "odds" in betting implies the wagering of unequal amounts of money by the two participants.
Since even numbers possess the property of equal shares,they were said to have "parity",from a latin word meaning "equal".Originally,this word applied (as logic demanded) to even numbers only,but mathematicians found it convenient to say that if two numbers were both even or both odd,they were,in each case "of the same parity". To see the convenience of this convention,consider the following:
If two even numbers are added,the sum is invariably even.(This can be expressed mathematically by saying that two even numbers can be expressed as 2m and 2n,where m and n are whole numbers and that the sum,2m+2n,is clearly divisible by two.[eg 2*3 + 2*4 = 6+8 = 14 which is even]However,we are friends,you and I, and I"m sure we can dispense with the mathematical reasoning and that I will find you willing to accept my word of honour as a gentleman,in such matters.Besides,you are welcome to search for two even numbers whose sum isn't even.)
If two odd numbers are added,the sum is also invariably even. If an odd number and an even number are added,however,the sum is invariably odd.
We can express this more succintly in symbols,with E standing for even and 0 standing for odd:
E+E = E
O+O = E
E+O = 0
O+E = 0

Or,if we are dealing with pairs of numbers only,the concept of parity enables us to say it in two statements,rather than four:
1. Same parities add to even.
2. Different parities add to odd.

A very similar state of affairs exists with reference to multiplication,if we divide numbers into two classes:positive numbers (+) and negative numbers(-).The product of two positive numbers is invariably positive.The product of two negative numbers is invariably positive [The two negatives being positive is the problem]The product of a positive and a negative number is invariably negative.In symbols:
	Rule A	+* + 	=	+
	Rule B	- * -	=	+	[The problem one]
	Rule C	+* - 	=	-
	Rule D	- * + 	=	-

Or,if we consider all positive numbers as having one kind of parity and all negative numbers another,we can say ,in connection with the multiplication of two numbers:
1. Same parities multiply to positive.
2. Different parities multiply to negative."

Asimov also utilizes the examples of magnetic poles which attract and repel according to a similar set of rules,opposites attract,like repel.The rules for multiplying positive and negative numbers (shown above) seem intuitively obvious.When used with respect to the man on the ladder we,could say the following:

1. He moves two steps upwards.He then moves to the point that is twice as far as his first move in the same direction.We know that he is four units above where he started. (Rule A)

2. He moves two steps downwards.He then moves to the point that is twice as far as his first move in the same direction.We know that he is four units below where he started.(Rule D)

3. He moves two steps upwards.He then moves to the point that is twice as far as his first move in the opposite direction.We know that he is four units below where he started. (Rule C)

4. He moves two steps downwards.He then moves to the point that is twice as far as his first move in the opposite direction.We know that he is four units above where he started. (Rule B)

All these statements are consistent.But the hardest one to grasp is that a minus times a minus is a plus.What I would like to do is alter the IV. commutative law and make multiplication work like this
	Rule A'	+ * + 	=	+
	Rule B'	- * -	=	-
	Rule C'	+ * - 	=	+
	Rule D'	- * + 	=	-

I hope that it is obvious that now the order in which one multiplies positive and negative numbers now makes a difference.
There is a reason and a problem with doing this,however.
These two things both depend on the idea of a square root.

3. SQUARE ROOTS,i,AND COMMUTATIVITY
First of all we need to consider a square.This familiar shape has the simple property of having all its four sides the same length.We know that we can find its area by multiplying any two together.For the sake of argument let's suppose we pick a length and a breadth.We could say:
Area of square = Length * Breadth
Or in symbols A  L * B,but since with a square L = B,we could also say:
Area of square = Length * Length
Or A = L * L.Mathematicians abbreviate anything multiplied by itself by writing a small number in the top right corner next to the symbol to be repeated,called an index or power.Thus L * L = L2.This is L squared and so we could write:
Area of square = L2
Equally well,we could know the area and want to find the length of a squares edge.we would be trying to find the square root,or the number that when multiplied by itself gave the area.
So we can see the following:
The square of 2 is 4 The square root of 4 is 2
There is often a confusion here that 2 is the square root because we are dividing by two and that the square root of 10 say would be 5,for this reason.However,two is the square root of four because it divides 4 two times,and the square root of 9 would be 3 (not 4½) because 3 divides 9, 3 times.
Now if we can extract a square root of a positive area square,why could we not find the square root of a negative area square.What do we mean by a negative area square? From the above arguments we can see that we mean one with area measured in the opposite direction.The precise meaning of this is not relevant here.Suffice to say that we are trying to find a number that when multiplied by itself gives a negative number.
But,from Rules A and B above we can see that a positive times a positive is a positive and a negative times a negative is a positive.In other words there aren't any numbers that when multiplied by themselves give negative numbers!
Stated simply there is no square root of a negative quantity.This is a consequence of the above rules.
Mathematicians get around this problem by inventing a root of minus 1 called i which is short for imaginary.In fact it is not really imaginary,but at right angles to our conventional -3,-2,-l,0,l,2,3... number-line~The whole number system then becomes more complicated by each number being a composite of two numbers,a Real and Imaginary part.
In my rule set A',B',C',D' the negative number would have a square root,it would be the negative version of the positive square root,but this forces us to accept that a positive number multiplied by a negative number does not have the same sign as a negative number multiplied by a positive one.
The reason for altering the commutativity rule is that it provides a simple answer to finding the square root of a negative quantity.The problem is that it takes away the need for a concept of i,and number pairs to find the solution,and replaces it with an idea of order of multiplying positive and negative numbers.
Why should we worry about this? Well,because ofthe conventional set of rules,mathematicians and physicists have had to keep comimg back to the idea of ordered multiplication,even though the ideas sprang from working with the number pairs used in complex (Real and Imaginary) numbers.
Let us see what has happened to them.

4.  QUATERNIONS ,COMPUTERS, FRACTALS , AND QUANTUM REALITY
In this section I will look at where commutativity has been broken,why it has been broken,and why sometimes we need it.This is the hardest part of this argument to grasp,but it shows the application of the idea and therefore any abstract mathematical mechanics of the source material should be overlooked in favour of the relevance of the argument.However,if you follow the formal reasoning in the symbols,so much the better.

[NB 29/9/2010- It is worth noting that the Mandelbrot Set [http://members.fortunecity.com/templarseries/Yahoo/Omegaman/fractal.html
http://members.fortunecity.com/templarseries/Yahoo/Omegaman/chaos.html
http://members.fortunecity.com/templarseries/mandel.html
http://members.fortunecity.com/templarseries/newmath]

points to the left – which is a consequence of the asymmetry in the roots of negative and positive numbers – if the commutative law was made symmetrical for both negative and positive numbers – the result maybe a more circular symmetric set – which currently has only top bottom – not left right symmetry – which would be more aesthetically pleasing and perhaps points to the idea that 
“i” as the square root of minus 1 is a kind of “fudge” solution because of the inherent imbalance from the commutative law.

If we look at the diagram of the Mandelbrot set upon which a cardioid has been superimposed we can see that in fact the rear part of the set has been reflected about the real axis. This is because of the way the imaginary terms change into real ones when squared. The two components that generate the next complex number in the formula z=z^2+c are:

Rn = (a^2-b^2) and In =  (2ab)

The b^2 term has been rendered REAL when it used to be IMAGINARY and so there is an asymmetry in the east/west (real) direction where there is no in the north/south (imaginary).
If the non-commutative arithmetic were used perhaps the Mandelbrot might either be a circular fractal or not actually be fractal at all. There would be no asymmetry. The fact that there is – perhaps (if “i” is in fact an aspect of reality) is perhaps intriguing given other “left handed” notions about the universe – such as chirality in molecules and at the sub atomic level,much as is indicated in “The left hand of the electron”.

http://members.fortunecity.com/templarseries/Yahoo/Omegaman/symmetry.html
http://members.fortunecity.com/templarseries/l-hand1.html

4a. QUATERNIONS AND THE LIBERATION OF ALGEBRA
We now follow the work of Sir William Rowan Hamilton and George Boole with the help of the BBC's Open University.
This calls back upon the notion of a complex number,but an explanation is provided in the text.

"Hamilton was interested in the formal algebra derived from complex
functions.A complex number has the form A+Bi [The A part is a number like the ones we use everyday,the B part is the same except for being multiplied by i,the square root of minus l],which satisfies the rule i2 = -l,complex numbers can be added thus:

(3+2i )+( 4- 1i )=( 3+4)+( 2i-1i )=7+1i

 they can also be multiplied thus:

		(3+2i)*(4 - 1i)=(3*4)+(-1i*3)+(2i*4)+(-1i*2)

because i2 = -1 this eventually becomes 12-3i+8i+2=15+5i
But the imaginary quantity was the source of some controversy[Their words,not mine!].The complex numbers can be interpreted as points in a plane [A plane can be thought of as like a sheet of graph paper],thus:

[This represents 3+2i,in a similar way to an X,Y graph that we are used to]

What Hamilton did was to dispense with the imaginary quantity [What I have already suggested to do],and consider number pairs	(couples).He defined rules for adding couples;(a,b)+(c,d)=(a+c,b+d) and for
	multiplication ; (a,b)*(c,d)= (ac-bd1ad+bc)
In particular,(0,1)*(0,l)=(-1,0),this is an exact parallel to addition and multiplication of complex numbers; (0+1i )*(0+1i)=(-1+0i) The logical conclusion was to consider number triples (a,b,c).Hamilton tried to extend the analogy from complex numbers.He interpreted
(a,b,c) as A+Bi+Cj,where both i2 and j2  = -1.

Therefore (3+2i+4j)+(1+4i-3j)= (3+1)+(2i+4i)+(4j-3j)= 4+6i+1j. The question now is,how do you multiply triples?
Multiplying the  above terms (3+2i+4j)*( 1+4i-3j)= (3-8+12)+(12+2)i+(4-9)j+(-6+16)ij
This gives an extra term in ij.What is ij?
Hamilton tried several approaches.
1.ij=1 or -1
2.ij=0
3.ij=k,ji=-k,k=0 [This one is interesting as i*j is not j*i]
4.(a+bi+cj) (x+bi+cj)

Part of his problem was to do with the modulus  of a triple.

Modulus (a+bi+cj) = Ö(a2+b2+c2)
He wanted the modulus of the product to be equal to the product of the moduli of individual terms.None of the values of ij allowed him to achieve this.On the 16th Oct 1843,Hamilton was walking to Dublin with his wife,on approaching Broom bridge,he felt a "galvanic surge" of inspiration.
Instead of triples,Hamilton considered adding a third distinct imaginary number Dk,also satisfying k2 = -1.
He now had quadruples A+Bi+Cj+Dk,or quaternions as he called them. To multiply quaternions we use i2 = j2 = k2 =  ijk = -1.
ij=k,jk=i,ki=j but ij is not the same as ji; ji=-k,kj=-i,ik=-j.
A good way to remember this is as a cycle,clockwise ij=k,ki=j,jk=i and anti-clockwise ik=-j, ji=-k,kj=-i.
With multiplication expressed like this,Hamilton had produced a consistent algebra of quadruples,in which multiplication was not commutative.This was the first non-commutative algebra,and is why Hamilton freed algebra from arithmetic.Hamilton was compelled to verse by his invention:

"And how the one of time and space the three might in the chain of symbols girdled be"

Hamilton saw the 4 elements of quaternions as representing the three space vectors,and the single numerical dimension of time."

Here we have been shown that Hamilton was forced to non-commutativity by the very invention of the symbol i for the square root of minus 1. We are now in the paradoxical situation of requiring the original rules of multiplication A,B,C,D to bring about i,so that the invention of quaternions is necessitated,but quaternions seem to require a rule as a consequence of this that is not arithmetical and follows the A',B',C',D' rules,that this argument is about.The paradox is that if we introduce them we would not need to introduce them!

Let's see what George Boole did with this idea.
4b. COMPUTERS AND THE LIBERATED BOOLEAN ALGEBRA 
"The following year (1844) saw applications of quaternions by Hamilton and others,but 1844 was an important year also for a little known schoolmaster.George Boole.[Get this....] Boole had no formal education,and was largely self taught.Boole published many texts,including some on quaternions,but he also published a mathematical analysis of logic,and the laws of thought.Of the laws of thought Boole says;
"I look upon the most valuable,if not the only valuable contribution that I have made,or am likely to make to science,and the thing by which I would desire,if at all,to be remembered hereafter."
Boole moved from Lincoln to the newly founded Queens college in Cork,as he had been elected 1st professor of mathematics. Boole was trying to develop a calculus of logic,and define the rules of the mind which govern reason.
He starts by giving a single letter name (X) to a class of individuals with some common trait.Thus,if an adjective such as "white" is employed,x stands for "all white things".We can then choose Y to represent "all sheep", XY can then mean "all white sheep".In like manner,if Z stands for "horned things" then ZXY="all white,horned sheep".Now consider the laws to which Z,X,Y etc are subject.It is evident that;
XY = YX, and that if X includes all things in Y that; XY = X,  hence,X*X=X or X2 = X,for all terms of X.
Boole developed this system,he introduced "+" for "or" so that if A="all men" and B="all women", A+B="all things men or women",however,this operation "+",can only be applied to sets which have no common members,at this stage Boole avoids the problem of adding overlapping sets. (see members.fortunecity.com/templarseries/fuzzylog.html & ptlytrue.html)
Boole thus established what we now call "Set Theory".But he developed these ideas further,he says:
If X="all white things" Then l-X="all things not white"
Since things cannot be both white and not white X*(l-X)=0, X-X2 =0, therefore X2 = X as before.
Yet,how could any element,multiplied by itself,remain just the same? Only two numbers in arithmetic (0 and 1) satisfy this condition.But in this algebra, X2=X for every member of the algebra.This was a clear break with arithmetic.De Morgan pointed out an interesting variation in the distributive law of Booles algebra.
We can demonstrate this with electric circuits.
A*(B+C)

       	1.                   

"*" means in series and "+" means in parallel.This circuit is equivalent to:

		2.

To allow current to flow,we can either switch A and B or A and C.The second arrangement expresses (A*B)+(A*C),this is the normal distributive law from arithmetic [See above].

[As an example of why these rules make sense even in arithmetic,let's substitute some numbers for A,B and C say 2,3 and 4.
We have A*(B+C) which is
2 * (3 + 4) = 2 * (7) = 14 And (A*B)+(A*C)  which is
(2 * 3 )+(2 * 4) = (6) + (8) = 14
Because they both give the answer 14 both rules are the same.] But now consider:

A+(B*C)

This is the same as:

Current flows if we either,close A by itself,or B and C together,the expression for the second circuit is (A+B)*(A+C),this is a second distributive law,exactly the same as the first,but with "+" and "*" swapped.The second law is nonsense in arithmetic.

[As an example of why these rules do not make sense in arithmetic,let's,as before,substitute some numbers for A,B and C say 2,3 and 4.
We have A+(B*C) which is

2 + (3 * 4) = 2 + (12) = 14 

And (A+B)*(A+C)  which is

(2 + 3 )*(2 + 4) = (5) * (6) = 30

Because they give different answers the rules are different.]

Quaternions inspired further developements in Quantum Physics and Space-Time,Boolean Algebra was taken up most effectively by Shannon,who set up the basis for modern digital electronic circuits.
Boole and Hamilton and other great mathematicians are celebrated on the Moon,their names being used to name craters."

In this section we are shown an arithmetical "nonsense" law,and the relevance of these ideas to modern technoloqy,such as,computing and Quantum Physics.Next we will see a support for commutativity (4c.) and another area where it is broken (4d.).These sections carry on the ideas of relevance to computing and quantum physics,and are necessarily mathematically indulgent.

4c.	COMPLEX NUMBERS AND FRACTALS
This section is meant to show that the consequences of the commutativity of multiplication is required in order to give rise to some modern revelations,based on the existence of i,the imaginary quantity.
First,we need a clearer explanation of a complex number,and then we will try to see why they are needed to produce fractal images,used in modern computer simulations of landscapes etc.
Roger Penrose will explain complex numbers to us (from The Emperors New Mind) and James Gleick will explain how a fractal is generated (from Chaos).

(see members.fortunecity.com/templarseries/enm3.html & enm9.html & enm10.html. Also pracfrac.html,mandel.html & members.fortunecity.com/templarseries /Yahoo/Omegaman/fractals.html)

"The real number system does not,as it turns out,have a monopoly with regard to mathematical power and elegance.There is still a certain awkwardness in that for example,square roots can be taken only of positive numbers (or zero) and not of negative ones [I hope I have already shown why!].From the mathematical point of view-and leaving aside,for the moment, any direct connection with the outside world-it turns out to be extremely convenient to be able to extract square roots of negative as well as positive numbers.Let us simply postulate,or "invent" a square root for the number -1.We shall denote this by the symbol "i",so we have:
i2 = -1
The quantity i cannot,of course,be a real number since the product of a real number with itself is always positive ( or zero,if the number itself is zero).for this reason the term imaginary has been conventionally applied to numbers whose squares are negative.However it is important to stress the fact that these "imaginary" numbers are no less real than the "real" numbers that we have become accustomed to.As I have emphasized earlier,the relationship between such "real" numbers and physical reality is not as direct or compelling as it may at first seem to be,involving,as it does,a mathematical idealization of infinite refinement for which there is no clear a priori [ known to be true independently of the experience of the subject matter] justification from Nature.
Having a square root for -1,it is now no great effort to provide square roots for all the real numbers.For if a is a positive real number,then the quantity
	i*Ö(a)	[Ö = square root]
is a square root of the negative real number -a.(There is also one other square root,
namely -i*Ö(a). )What about i itself?Does this have a square root? It surely does.For it is easily checked that the quantity
(1+i )/Ö( 2)
(and also the negative of this quantity)squares to i.Does this number have a square root? Once again the answer is yes;the square of
Ö((1+1/Ö(2))/2) + i(Ö((1-1/Ö(2))/2)) or its negative is indeed (1+i)/Ö(2).
[The salient point here is that the process is infinitely regressive,and is why i is needed as Ö(-l)]
Notice that in forming such quantities we have allowed ourselves to add together real and imaginary numbers,as well as to multiply our numbers by arbitrary real numbers (or divide by non-zero real numbers,which is the same thing as multiplying by their reciprocals) .The resulting objects are what are referred to as complex numbers.A complex number is a number of the form
	A+iB	[or equally A+Bi]
where A and B are real numbers,called the real part and the imaginary part respectively,of the complex number.The rules for adding and multiplying two such numbers follow the ordinary rules of (school) algebra,with the added rule that i2 = -1:
(A+iB)+(C+iD)=(A+C)+i(B+D)
(A+iB)*(C+iD)=(AC-BD)+i(AD+BC).
A remarkable thing now happens! Our motivation for this system of numbers had been to provide the possibility that square roots can always be taken.It achieves this task,though this is itself not yet obvious.But it also does a great deal more:cube roots,fifth roots,ninety-ninth roots,pi-th roots,(1+i)th roots etc. can all be taken with impunity(as the great 18th century mathematician Leonard Euler was to show)"

At this point Roger Penrose becomes even more complicated in his explanations than he already has been,and his definition of a fractal is more long-winded than is James Gleick's,so for the sake of brevity (not used very much so far!),I will use his explanation.The point being made here is that without the infinite regression born of using i as a solution to Ö(-l),fractals may have been a non-starter.

"A Mandelbrot set program needs just a few essential pieces.The main engine is a loop of instructions that takes its starting complex number and applies the arithmetical rule to it.For the Mandelbrot set,the rule is this: Z=Z2+C,where Z begins at zero and C is the complex number corresponding to the point being tested.So,take 0multiply it by itself,and add the starting number;take the result,the starting number;multiply it by itself,and add the starting number;take the new result,multiply it by itself,and add the starting number.Arithmetic with complex numbers is straightforward.A complex number is written with two parts:for example,2+3i(the address for the point 2 east and 3 north on the complex plane).To add a pair of complex numbers,you just add the real parts to get a new real part [as you would with conventional adding] and the imaginary parts to get a new imaginary part:
2 + 4i
+	9 - 2i
	-------------
11 + 2i

To multiply two complex numbers,you multiply each part of one number by each part of the other and add the four results together.Because i multiplied by itself equals -1 by the original definition of imaginary numbers,one term of the result collapses into another. [This is what separates it from algebra with real numbers]
	2	+3i
*	2 + 3i
	------------
6i + 9i2
4 + 6i
--------------
	4	+12i + 9i2
	=	4	+12i - 9
	= -5 + 12i	[Because 4-9=-5]

To break out of this loop,the program needs to watch the running total.lf the total heads off to infinity,moving farther and farther from the center of the plane,the original point does not belong to the set,and if the running total becomes greater than 2 or smaller than -2 in either it's real or imaginary part,it is surely heading off to infinity-the program can move bn.But if the program repeats the calculation many times without becoming greater than 2,then the point is part of the set.How many times depends on the amount of magnification.For the scales accessible to a personal computer,l00 or 200 is often plenty,and 1000 is safe.
The program must repeat this process for each of thousands of points on a grid,with a scale that can be adjusted for greater magnification.And the program must display it's result.Points in the set can be coloured black,other points white.Or for a more vividly appealing picture,the white points can be replaced by coloured gradations.If the iteration breaks off after ten iterations,for example,a program might plot a red dot;for twenty repetitions an orange dot;for forty repetitions a yellow dot,and so on.The choice of colours and cutoff points can be adjusted to suit the programmer's taste.The colours reveal the contours of the terrain just outside the set proper"

It is not easy to see how this process would evolve without i,the fact that fractals can resemble real world objects tends to support the commutativity of multiplication,with its construction of i as the Ö(-1).Now let's see an example of where non-commutativity affects the real world.

4d. QUANTUM REALITY AND MINKOWSKIAN SPACE
This section is meant to show that the consequences of the commutativityof multiplication is basically flawed because as with Hamilton and Boole, physicists have been forced into breaking the commutativity rule in order to explain reality.
Firstly Roger Penrose (Emperors New Mind) will explain the relevance of Hamiltons work with reqard to Space-Time,then John Gribbin (In Search of schrodingers Cat) ,will show non-commutativity as used in quantum physics.

"The names of many of the great mathematicians of the 18th and 19th centuries are associated with its developement: Euler, Lagrange, Laplace, Louiville , Poisson, Jacobi, Ostrogradski ,Hamilton .What is called "Hamiltonian Theory" summarizes most of this work,and a taste of this will be sufficient for our purposes here, The versatile and original Irish mathematician William Rowan Hamilton (l805-1865).-who was responsible for Hamiltonian circuits-had developed this form of the theory in a way that emphasized an analogy with wave propagation,This hint of a relation between waves and particles-and the form of the Hamilton equations themselves-was highly important for the later developement of quantum mechanics."

Now Roger Penrose explains the 4 dimensions of Minkowskian spacetime,although a lot of this is somewhat foreboding,and in some sense out of context,try to see the connection with Hamiltons quaternions and his poem that he was moved to write.

"Let us try to understand the basics of special relativity in terms of the magnificent space time of Minkowski.
One of the difficulties in coming to terms with the space-time concept is that it is four-dimensional,( see members.fortunecity.com/templarseries/tourist4a.html & dimens.html & mathex8.html)which makes it hard to visualize.However,having survived our encounter with phase-space ( see members.fortunecity.com/templarseries/beffect.html),we should have no trouble with a mere four dimensions. As before,we shall "cheat" and picture a space of smaller dimension-but now the degree of cheating is comparably less severe,and our picture will be correspondingly more accurate.Two dimensions (one space and one time) would suffice for many purposes,but I hope that the reader will allow me to be a little more adventurous and go up to three (two space and one time).This will. provide a very good picture,and it should not be hard to accept that in principle the ideas will extend,without much change,to the full four-dimensional situation.The thing to bear in mind about a space-time diagram is that each point in the picture represents an event[I will be omitting diagrams here,for they are irrelevant to the point I am making. If you need them they are on pages 251 & 253 of ENM)-that is,a point in space at a single moment,a point having only an instantaneous existence.The entire diagram represents the whole history,past present,and future.A particle,since it persists with time,is represented not by a point but by a line,called the world-line of the particle.This world-line,straight if the particle moves uniformly and curved if it accelerates (ie non-uniformly),describes the entire history of the particles existence.
In fig 5.16 (p251),I have depicted a space-time with two space and one time dimension.We imagine that there is a standard time coordinate t,measured in the vertical direction,and two space-coordinates x/c and z/c,measured horizontally (The reason for dividing the spatial coordinates by c,the speed of light,is in order that the world-lines of photons are angled at a convenient slope:45 degrees to the vertical).The cone at the centre is the (future)  light cone of the space-time origin O.To appreciate its significance,imagine an explosion taking place at the event 0. (Thus the explosion occurs at the spatial origin,at time t=0.)The history of the light emanating from the explosion is this light cone.In two-dimensional spatial terms,the history of the light flash would be a circle moving outwards,with the fundamental light speed c.In full three-dimensional space this would be a sphere moving outwards at speed c,the spherical wave front of thelight,but here we are suppressing the spatial direction y,so we just get a circle,like the circular ripples emanating from the point of entry of a stone thrown in a pond.We can see this circle in the space-time picture if we take successive horizontal slices through the cone moving steadily upwards.These horizontal planes represent different spatial descriptions as the time coordinate t increases.Now,one of the features of relativity theory is that it is impossible for a material particle to travel faster than light.
All the material particles coming from the explosion must lag behind the light.This means,in space-time terms,that the world-lines of all the particles emitted in the explosion must lie inside the light cone.It is often convenient to describe light in terms of particles  ,called  photons  ,rather than in terms of electromagnetic waves.For the moment we can think of a "photon" as a small "packet" of high-frequency oscillation of electromagnetic field.The term is physically more accurate in the context of the quantum descriptions that we shall be considering in the next chapter [Or in our case in the next section,courtesy of Mr Gribbin],but "classical" photons will also be helpful for us here.In free space,photons always travel in straight lines with the fundamental speed c.This means that in Minkowski's space-time picture the world-line of a photon is always depicted as a straight line tilted at 45 degrees to the vertical.The photons produced at our explosion at O describe the light cone centered at O.
These properties must hold generally at all points of the spacetime.There is nothing special about the origin;the point 0 is no different from any other point.Thus,there must be a light cone at every point of the space-time,with a significance the same as that for the light cone at the origin.The history of any light flash-or worldlines of photons,if we prefer to use the particle description of light,is always along the light cone at each point,whereas the history of any material particle must always be inside the light cone at each point.This is illustrated in fig 5.17 (p253).The family of light cones at all points may be regarded as part of the Minkowskian geometry of space-time.
(see memebrs.fortunecity.com/templarseries/space.html)
What is Minkowskian geometry? The light-cone structure is its most important aspect,but there is more to Minkowskian geometry than that.There is a concept of "distance" which has some remarkable analogies with distance in Eudlidean geometry [flat plane geometry of circles,squares, triangles etc.] . In three-dimensional Euclidean geometry,the distance r of a point from the origin in terms of standard Cartesian [ X,Y graph paper] coordinates,is given by
r2 = x2 + y2 + z2
(This is just the Pythagorean theorem,the two-dimensional case being perhaps more familiar.(The square of the hypotenuse is equal to the sum of the squares of the other two sides]).In our three-dimensional Minkowskian geometry,the expression is formally very similar,the essential difference bei~ng that we now have two{ minus signs{:
s2 = t2 - (x/c)2 - (z/c)2 More correctly we should have the four-dimensional Minkowskian geometry,of course, and the "distance" expression is
s2 = t2 -(x/c)2 - (y/c)2 - (z/c)2."

This last expression is what I wanted to formally identify,Compare it with Hamilton's Quaternion equation (a+bi+cj+dk).The equation here is for a description of space-time,and is a consequence of Hamiltons work which required i,itself a consequence of the inability to find Ö(-l).
Now let's look at how physicists have dealt with this situation when explaining matter.
We join John Gribbin explaining how non-commutativity became necessary in quantum physics.

"Born was happy enough to send Heisenberg's paper off to the Zeitschrift fur Physik,and almost immediately realized what it was that Heisenberg had stumbled upon.The mathematics involving two states of an atom couldn't be dealt with by ordinary numbers,but involved arrays of numbers [matrices],which Heisenberg had thought of as tables.The best analogy is with a chess board.There are 64 squares on the board,and in this case you could identify each square by one number,in the range 1 to 64.However,chess-players prefer to use a notation that labels the "columns" of squares across the board by the letters a,b,c,d,e,f,g and h,with the "rows" numbered l,2,3,4,5,6,7,8.Now,each square on the board can be identified by a unique pair of identifying labels:al is the home square of a rook;g2 is the home square of a knights pawn,and so on Heisenbergs tables,like a chess board,involved two-dimensional arrays of numbers,because he was doing calculations involving two states and their interactions . Those calculations involved, among other things,multiplying two such sets of numbers,or arrays together,and Heisenberg had laboriously worked out the right mathematical tricks to do the job.But he had come up with a very curious result,so puzzling that it was one of the reasons for his diffidence about publishing his calculations.When two of these arrays are multiplied together,the "answer" you get depends on the order in which you do the multiplication.
This is strange indeed [Not to me it isn't!].It is as if 2*3 is not the same as 3*2,or in algebraic terms a*b is not equal to b*a.Born worried at this peculiarity day and night,convinced that something fundamental lay behind it [I think it does,but not for the following reason].Suddenly,he saw the light.The mathematical arrays and tables of numbers,so laboriously constructed by Heisenberg,were already known in mathematics.A whole calculus of such numbers existed;they were called matrices,and Born had studied them in the early years of the 20th century,when he was a student in Breslau.It isn't really surprising that he should have remembered this obscure branch of mathematics more than twenty years later,for there is onefundamental property of matrices that always makes a deep impression on students when they first learn of it;the answer you get when you multiply matrices depends on the order in which you do the multiplying,or in mathematical language,matrices do not commute."

The point has been made here that physicists in understanding the atom have necessarily used non-commutative multiplication,but by using matrices;Is it possible that if arithmetic started out noncommutative,Born and Heisenberg wouldn't have had to look to matrices to solve the problem? We shall follow John Gribbin A little further in order to see both why this non-commutativity affects physical matter and to connect it with Hamiltons work.

 QUANTUM MATH
In the summer of 1925,working with Pascual Jordan,Born developed the beginnings of what is now known as matrix mechanics,and when Heisenberg returned to Copenhagen in September he joined them through correspondence in producing a comprehensive scientific paper on quantum mechanics.In this paper,far more clearly and explicitly than in Heisenbergs original paper,the three authors stressed the fundamental importance of the noncommutativity of quantum variables.Already,in his joint paper with Jordan,Born had found the relation pq - qp = h/i, where p and q are matrices representing quantum variables,the equivalent in the quantum world of momentum and position [note here that in conventional arithmetic pq-qp would always be 0].Plancks constant(h) appears in the new equation,along with i,the square root of minus 1 [this is part of the paradox, an equation that needs to be non-commutative but contains i,which is born out of a commutative arithmetic!);in what became known as the "three man paper",the Gottingen team stressed that this is the "fundamental quantum mechanical relation." But what did it mean in physical terms? Plancks constant was now familiar enough [6.6*10-34 joule seconds],and physicists knew equations involving i (a clue to what was to come,if they had but realized it,since such equations generally involve oscillations or waves).But matrices were totally unfamiliar to most mathematicians and physicists in 1925,and the noncommutativity seemed as strange to them as Plancks introduction of h had seemed at first sight to their predecessors in 1900.For those who could handle the mathematics,the results were dramatic.The equations of Newtonian mechanics [movements of billiard balls etc.] were replaced by similar equations involving matrices and,says Heisenberg,"It was a strange experience to find that many old results of Newtonian mechanics,like conservation of energy etc. could be derived also in the new scheme." In other words matrix mechanics included Newtonian mechanics within itself,just as Einstein's relatavistic equations include Newtons equations as a special case [Newton was subsumed into later theories, he wasn't wrong or corrected,he just described what he could see].Unfortunately,few people could understand the mathematics [Big surprise there!],and it was not immediately appreciated by most physicists just how significant a breakthrough Heisenberg and the Gottingen group had made.There was one exception,however,and that was in Cambridge,England.
Paul Dirac was a few months younger than Heisenberg,having been born on 8th August 1902.He is generally regarded as the only English theorist who can rank with Newton,and he developed the most complete form of what is now called quantum mechanics.Yet he did not turn to theoretical physics until after he graduated from Bristol University in 1921,with a degree in engineering.Unable to find a position in engineering,he was offered a studentship to study mathematics in Cambridge,but was unable to take it up through lack of money.Staying in Bristol and living with his parents,he took a 3 year course in mathematics  in only 2 years,thanks to his engineering degree,and completed a BA in applied mathematics in 1923.Now,at last,he could go to Cambridge to take up research,supported by a grant from the Department of Scientific and Industrial Research,and only on arriving in Cambridge did he learn for the first time about quantum theory.
So it was as an unknown and inexperienced research student that Dirac heard Heisenberg talk in Cambridge in July 1925.Although Heisenberg did not talk publicly about his new work then,he did mention it to Ralph Fowler,Diracs supervisor,and as a result he sent Fowler a copy of the proof of the paper in the middle of August,before it appeared in Zeitshrift.Fowler passed the paper on to Dirac,who had it in front of him before anybody outside Gottingen (except Heisenbergs friend Pauli) had had a chance to study the new theory.In this first paper,although he pointed out the noncommutativity of the variables in. quantum mechanics;the matrices;Heisenberg did not develop the idea but tried to fudge around it.When he got to grips with the equations,Dirac soon appreciated the fundamental importance of the simple fact that a*b is not equal to b*a.
Unlike Heisenberg,Dirac already knew of mathematical quantities that behaved in this fashion,and within a few weeks he was able to rework Heisenbergs equations in terms of a branch of mathematics developed by William Hamilton a century earlier [I told you it connected up!].By one of the most delicious scientific ironies,the Hamiltonian equations that proved so useful in the new quantum theory,which dispensed altogether with electron orbits,had been developed in the 19th century largely as an aid to the calculation of the orbits of bodies in a system,like the solar system,where there are several interacting planets.
(see members.fortunecity.com/templarseries/solarsys.html)

So Dirac discovered,independently of the Gottingen group,that the equations of quantum mechanics have the same mathematical structure as the equations of classical mechanics,and that classical mechanics is included within quantum mechanics as a special case,corresponding to large quantum numbers or to setting Plancks constant equal to zero.Following his own direction,Dirac developed yet another way of expressing the dynamics mathematically,using a special form. of algebra,which he called quantum algebra,involving the addition and multiplication of quantum variables,or "q numbers."These q numbers are strange beasts,not least because in this mathematical world that Dirac developed it is impossible to say which of the two numbers a and b is bigger,the concept of one number being bigger or smaller than another has no place in this algebra.But,again,the rules of that mathematical system exactly fitted the observations of the behaviour of atomic processes.Indeed,it is correct to say that quantum algebra includes matrix mechanics within itself,but does much more besides [This is how science and maths progresses;by new laws subsuming and adding to previous ones,until at some apex,or cusp,the whole idea has to be redrawn].
Fowler immediately appreciated the importance of Diracs work,and at his instigation it was published in the Proceedings of the Royal Society in December 1925.Among other things,the paper included,as an essential component of the new theory,the half-integral [half whole number] quantum numbers that had troubled Heisenberg a few years before.
Heisenberg, sent a copy of the manuscript of the paper by Dirac,was generous in his praise:"I have read your extraordinarily beautiful [aesthetics playing its part] paper on quantum mechanics with the greatest interest,and there can be no doubt that all your results are correct.....{the paper is) really better written and more concentrated than our attempts here." In the first half of 1926,Dirac carried this work through in a series of four definitive papers,with the whole package forming the thesis for which he was duly awarded his doctorate.While all this was going on,Pauli had used matrix methods to predict,correctly,the Balmer series for the hydrogen atom,and by the end of 1925 it had become clear that the splitting of some spectral lines into doublets could indeed be best explained by assigning the new property called spin to the electron.The pieces fit together very well indeed,and the different mathematical tools used by the different exponents of matrix mechanics were clearly just different aspects of the same reality.(In Dirac's version of quantum mechanics,a key expression in the Hamiltonian equations is replaced by the quantum mechanical expression (ab-ba)/iH [H here is standing for
h bar which is h/2 * pi, h being Plancks constant,and pi being 3.142 etc. the ratio of a circle's diameter to its circumference],which is just another form of the expression Born,Heisenberg and Jordan called the "fundamental quantum-mechanical relation,"in their three-man paper,written before Diracs first paper on quantum mechanics appeared,but published after Diracs paper.)
Again,the game of chess can help to make this clear.There are several ways of describing the chess game on the printed page.one way is to print a representative "chessboard" with all the positions of all  the pieces marked,but that would take up a lot of space if we wanted to record the whole game.Another way is to name the pieces being moved:"Kings pawn to Kings pawn four." And in the most concise algebraic notation the same move becomes simply "d2-d4." Three different descriptions provide the same information about a real event,the transition of a "pawn" from one state to another (and,just as in the quantum world,we know nothing about how the pawn got from one state to the other, a point that is even more clear if you consider the knights move).The different formulations of quantum mechanics are like this Diracs quantum algebra is the most elegant and "beautiful" in the mathematical sense [what is that sense?];the matrix methods developed by Born and his collaborators in the wake of Heisenberg are more clumsy but none the less effective.
(With characteristic,and genuine,modesty Dirac has described how easy it was to make progress once it was known that the correct quantum equations were simply classical equations put into Hamiltonian form.For any of the little puzzles that beset quantum theory,all that was necessary was to set up the equivalent classical equations turn them into Hamiltonians,and solve the puzzle."It was a game,a very interesting game one could play.Whenever one solved one of the little problems,one could write a paper about it.It was very easy in those days for any second-rate physicist to do first-rate work There has not been such a glorious time since then.It is very difficult now for a first-rate physicist to do second-rate work")
Some of Diracs most dramatic early results came when he tried to include special relativity in his quantum mechanics.quite happy with the idea of light as a particle ( the photon).Dirac was delighted to find that by including time as a q number along with all the rest in his equations he was inevitably led to the "prediction" that an atom must suffer a recoil when it emits light,just as it should do if the light is in the form of a particle carrying its own momentum,and he went on to develop a quantum-mechanical interpretation of the Compton effect.Diracs calculations went in two parts,first the numerical manipulations involving the q numbers,and secondly the interpretation of the equations in terms of what might physically be observed.This process exactly fits the way nature seems to "make the calculation" and then present us with the observed event,and electron transition ,say,but unfortunately instead of following this idea through completely in the years after 1926 physicists were seduced away from quantum algebra by the discovery of yet another mathematical technique that could solve the long-standing problems of quantum theory;wave mechanics.Matrix mechanics,and quantum algebra,had started out from the picture of an electron as a particle making a transition from one state to another.But what about de Broglies suggestion that electrons,and other particles,must also be thought of as waves?"

John Gribbin goes on to tell the story of wave mechanics and how Diracs algebra subsumed it as a special case,and how Hamilton,had he been around,would have been quick to spot the connection between matrix and wave mechanics.But for us the story is over,save for the summing up.

5. SUMMING UP-WHY BOTHER?
What this lengthy explanation has sought to do is justify one set of counter-intuitive rules (namely A',B',C',D') over the taken for granted ones (namely A,B,C,D)
	Rule A	+ *	+ =	+
	Rule B	- *	- =	+	[The problem one]
	Rule C	+ *	- =	-
	Rule D	- *	+ =	-
	Rule A'	+ *	+ =	+
	Rule B'	- *	- =	-	[The solution]
	Rule C'	+ *	- =	+
	Rule D'	- *	+ =	-

Rule B forces on us the concept of i for the Ö(-1),with all the above consequences. Rule B' gives the Ö(-1) as -1,which seems intuitively correct,but the second set of rules forces on us the concept of ordered multiplication that is used in the matrices of quantum physics,the second set of rules ( the ones I advocate) also suffer from the drawback of not having parity in the way Isaac Asimov -explained.In looking at these rules we should remember that there exists (in theory anyway) several other sets some,if not all of which don't make any sense.But,at least Rules A',B',C',D' are at least,in some sense,anyway, consistent with descriptions of reality if not intuitive arithmetic.
We should bother about this question because in having to resort to non-commutative multiplication to explain reality, physicists maybe showing us that arithmetic set out on the wrong foot.It may well be why Born felt that something fundamental lay behind it.
A second reason is that most people don't even seem to understand the multiplication that they use everyday, or that even within it's own rules they may be using it wrongly,let alone that they might be using the wrong rules and not even know it!

See members.fortunecity.com/templarseries/lorentz.html

0 GLOSSARY 0

COMPLEX NUMBER-A complex number is a two dimensional number comprised of two numbers A and B,such that A is Real and B is imaginary.

FRACTAL-Coined by Benoit Mandelbrot,the term fractal is comprised of an amalgamation of the words "fraction" and "dimensional.
It is used to describe anything that repeats the same pattern on smaller and smaller scales. eg a coastline

RECIPROCAL-The inverse of a number.This is created by dividing 1 by the number,creating a fraction. eg The reciprocal of 3 is 1/3.This can also be indicated using powers,with the use of -1.ie 36-16 = 1/3.

INTEGRAL-The word "integral" has two mathematical meanings.The on& used in this text refers to the set of whole numbers,both positive and negative and including zero.

MODULUS-The absolute value of a number.

PARITY-The property of being odd or even.

IMAGINARY NUMBER-A number multiplied by the quantity i (the Ö(-l)).

REAL NUMBER-A real number is an everyday number like 1,2,3,4 or -1,-
2,-3

ALGEBRA-The area of mathematics that generalises a methodology into symbols. eg X+Y means adding any two numbers together. 

EQUATION-An algebraically symbolic method of showing when quantities are in balance. eg X+Y = Y+X

EXPRESSION-A grouping of algebraic symbols with some meaning (there is no equals sign). eg 7D-T might mean 7 days pay less tax. 

VECTOR-A quantity with magnitude and direction.eg speed, two things can be travelling at the same speed,but in opposite directions. 

ATOM-The smallest part of a chemical element than can take part in a reaction.

PHOTON-Name given to a "packet" of light emitted in certain physical and chemical processes.

ELECTRON-Sub unit of an atom inhabiting the outer shells and having negative electric charge.

PHASE-SPACE-A multidimensional space in which coordinates represent the variables required to specify the state of a system. 

MINKOWSKI-SPACE-A mathematical space representing the 4-dimensional space-time of which the universe is said to consist.

SPACE-TIME-A unified idea in which the 3-dimensions of space ( up,down,left-right,and forward-backward),and the 1-dimension of time occupy a 4-dimensional whole.

MATRIX-MECHANICS-The methodology used to work out the interactions and arrangements of quantum particles.

WAVE-Physical description of a repeating series of oscillations.In quantum physics a wave is a description under certain circumstances of the behaviour of "particles" such as the electron,photon etc.

PARTICLE-Physical description of a lone object taking up finite space.In quantum physics the particle is one description of the behaviour of the sub-units of matter and energy.

LIGHT CONE-A conceptual model of how events would appear to be constrained in 4-d space-time.

QUANTUM PHYSICS-The physics in which wave/particles are said to only occupy certain energy states( defined by Plancks constant) ,and which suggests that the universe is statistical and intrinsically uncertain in its relationships, as shown by Heisenbergs uncertainty relation.

CLASSICAL (Or Newtonian) PHYSICS-The physics of everyday objects that suggests that if we could know starting conditions accurately enough ,the universe would be a simple clockwork scheme.

RELATIVITY THEORY-Albert Einsteins theory,based on the equivalence of observations of the same event from different frames of reference having different velocities and accelerations.

HEISENBERGS UNCERTAINTY THEORY-Werner Heisenbergs theory stating that there is a fundamental limit to the precision with which a position coordinate of a particle and its momentum in the same direction can be simultaneously known.

0 SYMBOLS 0

2 - "Squared" or "to the power 2". These are just ways of indicating a number multiplied by itself. As in 2 * 2, 3 * 3, 4 * 4 etc.These would be indicated by 22 , 32 , 42 respectively.

Ö- "Square root" is the opposite of squared (2). It means "the number that when multiplied by itself gives the number. As in 2 is the square root of four, 3 is the square root of nine,and 4 is the square root of 16.

Pi - The ratio (fraction) obtained by dividing the circumference (perimeter) of a circle by its diameter (the line that cuts it in half).It is an irrational number (it can't be represented by a fraction),but is approx. 22/7 or 3.1415     

h - "Planck's constant" is a physical constant relating to atoms

H - "h bar" is Plancks constant divided by 2 * Pi,and also relates to atoms.

i - is the symbol used to denote "the square root of minus one" or Ö(-l).It stands for the word "imaginary".