At the end of the previous article, we wanted to know when we can find solutions to the equation:
a * x = b
Well, let’s start with a very simple case. If I just give you the system of real numbers and ask, can you find the solution to ax = b when a ≠ 0? You’ll just multiply both sides by 1/a and say
x = b / a
Successful, System 1 : Non-zero numbers under multiplication
Now let’s go back to the system of symmetries of a rectangle we investigated in the second article. If we have the symmetries A and B and want to find a solution to:
A * X = B
At first we might try to do the same thing here. We know from the previous article, A has an inverse, denoted A-1. If we multiply both sides by it, it will cancel with A, giving R0. So we get:
R0 * X = A-1 * B
But as R0 is just the identity transform, it’s like multiplying by 1. So we get,
X = A-1 * B
Successful, System 2 : Symmetries of a rectangle under composition
So far so good, we seem to have found a method for finding X for these two systems. But, will it work for others? Let’s try it out!
System 3: Numbers (including 0) under multiplication
a * x = b
If a = 0 and b = 1, you can probably see that we can’t find any solutions because we get 0 = 1, which is of course complete nonsense. So, where do we go wrong in our method?
Well, in this case, there isn’t an inverse of a which we can multiply with on both sides, because 1/0 doesn’t exist!
Unsuccessful, System 3 : Real numbers (including 0) under multiplication
Reason: Inverse doesn’t exist for all elements
Let’s recall the definition of the inverse a-1:
a-1 * a = 1 when we’re talking about multiplication
A-1 * A = R0 when we’re talking about combining symmetries
We call elements like 1 or R0 the identity element of a system. They don’t really do anything to the other objects in the system, just like multiplying by 1 or doing an identity transform R0 does nothing.
Inverse and identity elements are closely linked together. The identity is what you get when you cancel out an element with its inverse. Like 1/4 * 4 = 1. We can’t really talk about an inverse existing if we don’t have an identity.
In our example, if I give you all the non-zero numbers but take out 1 from the system, you can’t really write down 1/4 * 4=1 because 1 doesn’t exist here.
Before we continue further, I want to point something out. So far we have studied different systems and how their elements interact with each other. We have tried combining different symmetries or multiplying different numbers together. This act of combining or multiplying is called an operation in maths. It’s like a rule telling you how two elements interact.
In our rectangle example, we see H and V combine to form R180. We write this as
H * V = R180
Here＊tells you how you should combine the symmetries H and V. You first do H (flip horizontally) then you do V (flip vertically).
Similarly, multiplication tells you how two numbers would interact in your system. If I give you 2 and 4, we get 2 * 4 = 8.
We can have many different operators. Something as familiar as multiplication or addition, to something very abstract, depending on the system we want to study. As a general notation, we denote an operator by ＊.
a * b = c tells you a and b combine under the operation ＊ to give c
So far we’ve seen that inverses and identity elements are very important pieces to our puzzle. Let us now try another system:
System 4 : 0 and all the odd numbers (…,-5,-3,-1,1,3,5…), under the operation of addition.
The equation becomes :
a + x = b
This system does have an identity, namely 0. Also, every element has an inverse, just its negative. For example 5 + (-5) = 0.
But if I give you a = 1, b =3, is there a solution? You might think, “Of course! 1+2=3, so x=2”, but I’m afraid 2 is not an odd number and hence lies outside our system. So there is no solution for x.
Where did we go wrong? Let’s try out our method and see.
1 + x = 3
(-1) + 1 + x = (-1) +3
x = (-1) +3
Well, (-1) + 3 does exist and equals 2, but just not in this system. So, (-1) + 3 is an illegal calculation, similar to 1/0. There isn’t a number in this system which could equal this.
There is a technical term for this idea in mathematics, which is called closure. It means, if we pick any a and b, a*b always exists within the system. The system is said to be closed under the operation＊.
Question to the reader: Is System 3 of real numbers under multiplication closed?
Unsuccessful, System 4: All odd numbers and 0, under addition
Reason: System is not closed
Taking stock, we’ve now seen that closure, the existence of identity elements and the existence of inverses are very important properties a system must have to satisfy our needs. But, is that all? Are we missing something which is really “obvious”? Unfortunately, there is one final thing we still need to clarify, but in order to do this let’s tackle the general problem head on.
The equation we are trying to solve is:
a * x = b
Here, a, x and b are elements of a particular system. We need to do the same steps we’ve done above with the specific examples of multiplication, symmetry and addition, but more carefully this time because we don’t really know how the operation behaves. As we go along, we will hopefully see our familiar puzzle pieces and discover the one we’ve missed out!
To start, we assume the bare minimum : Closure, Existence of Identity and Inverse elements.
Let us call the identity element e.
Your intuition might suggest taking a-1 (which we know exists) on both sides so that it cancels out the a, giving us just e * x = a-1 * b. So, it should now give us x=a-1 * b.
However, there is a small error in logic here. Let’s look at our steps carefully :
a-1 * (a * x) = a-1 * b
To write (a-1 * a) = e, we need a subtle change of brackets here:
(a-1 * a) * x = a-1 * b
And the rest follows smoothly:
e * x = a-1 * b
That is, x = a-1 * b and we have our solution.
So, the question is, what does the change of brackets actually mean? Consider the following: add the numbers 115, 42 and 10 on a calculator (or by hand). You might proceed by first adding 115 + 42 to get 157. Then you add it to 10 and get 157 + 10 = 167.
This is indeed correct, but what if you add the 42 and 10 first to get 52. Then you add 115 to get 167. You arrive at the same answer but in two different ways!
We use brackets to signify which two numbers are added first.
So, in our first method, since we start by adding 115 and 42, we write
(115 + 42) + 10 = 157 + 10 = 167
Whereas in our second method, we write
115 + (42 +10) = 115 + 52 = 167
So all we have done is change the brackets but we still get the same answer.
You might say this change of brackets is logical but remember, ＊ isn’t necessarily multiplication or addition. It can be anything. For example, consider ＊ as division :
4 / (2/2) = 4 / 1 = 4
But if you change the brackets,
(4/2) / 2 = 2 / 2 = 1
This would suggest that changing brackets is not always as straightforward as it might first seem. In fact, this is the final piece to our puzzle. The property of changing brackets is called Associativity, which says a * (b * c) = (a * b) * c
And that’s it! We are now finished. We obtain x = a-1＊b for a general operator * and we have only assumed a total of four properties:
- Closure: a＊b is an element in system G
- Identity: There is an element e in G, such that a ＊ e = a for any element a in G.
- Inverse: There is an element a-1 in G, such that a-1＊ a = e for any element a in G.
- Associativity: a＊(b＊c) = (a＊b)＊c for all elements a, b, c in G.
This gives us a really nice algebraic structure which we call a Group. We say the set G forms a group under operation＊. In a group you can always determine the solution of a＊x = b using the formula we have calculated above.
Groups are ubiquitous in maths and arise in other areas such as physics and chemistry. They have widespread applications, from counting all of the different arrangements of a Rubik’s Cube, to the discovery of quarks by studying symmetries of elementary particles. With just these 4 rules, we can deduce a lot of interesting properties. The symmetries of a rectangle which we looked at previously indeed form a group with just 4 elements. And in fact the set of symmetries of any object will form a group!
To fully appreciate the power of groups, we must dig a little deeper. A lot of times we can find deep connections between two algebraic structures, which make the two systems strikingly similar. These are called isomorphisms, but more on those later.
To conclude, let us try proving a property which will be very useful in our final goal of proving Fermat’s Little Theorem, called the Cancellation Property:
If c * a = c * b, then a = b
Armed with our new knowledge of a group, we can prove this in a few simple steps as follows:
Multiply by c-1 on both sides so that it cancels with c and we get:
e * a = e * b
Which just simplifies to,
a = b
In the next article, we will finally prove Fermat’s Little Theorem using the knowledge we have acquired throughout the course of this series of articles. Strap yourself in, it’s going to be a bumpy ride!
Article 1: Modular Arithmetic and calculating expenses
Article 2: The Importance of being Symmetric