Most people believe that the mathematics they learn in high school and in their early college years represents the essence of abstraction. What could be less tangible that the idealistic abstract notion of number? Mathematicians work with and delight in bizarre numbers such as i (the square root of -1) which to the uninitiated must seem like an irrational mix of mysticism and numerology. Nonetheless, the basic operations in arithmetic (addition, multiplication, etc.) apply remarkably well to many disparate real-life situations whether you are balancing your checkbook, keeping score in a game of cards, calculating the position of an object under the influence of gravity in physics, or measuring ingrediants for a recipe. In all these situations, the usual grade-school manner in which you add and multiply numbers models the situation perfectly.
However, there are times when the normal rules of arithmetic that you learn in grade school do not seem to apply. For example, suppose that you are compiling baseball statistics. To determine a player's batting average, you need to know both the number of hits that he has and the number of times he has been at-bat. If a player has 7 hits in 23 attempts, a natural way to record this information is using the fraction 7/23. (The way this is usually displayed when you are watching a game is the corresponding decimal, truncated to 3 places. In the case of 7 hits in 23 at-bats, it would be written as .304. Although the number .304 is more visually appealing and and conveys information more quickly (the batter gets a hit about 30.4% of the time), it is less precise because we are unable to determine whether the play has 7 hits in 23 at-bats, 14 hits in 46 at-bats, or 70 hits in 230 at-bats.)
Suppose that we have decided to record batting statistics in the above manner. On opening day, Joe has an excellent night, and has 3 hits in 4 at-bats. We record this as the fraction 3/4. The next night, Joe only has 1 hit in 5 at-bats, and we record this as the fraction 1/5. What fraction corresponds to Joe's batting average for the season after these 2 games? Well, Joe has a total of 4 hits and 9 at-bats, so we should use the fraction 4/9. Hence, in this situation, the natural way to "add" the two fractions 3/4 and 1/5 yields the result 4/9. This should be at least a little alarming. If you add the fractions 3/4 and 1/5 in the manner that you learned in grade school, you would obtain the result 19/20. What went wrong?
Here's another problem to consider. Normally when dealing with fractions, you consider 1/3 and 3/9 to be the same. However, in this case, we should not, because there is a definate difference between have 1 hit in 3 at-bats and 3 hits in 9 at-bats. Thus, when dealing with fractions in this context, we should never "reduce" them by dividing both the numerator and denominator by a common factor.
Probably the initial reaction among most people is that we should not use the word "add" in the above situation. After all, we were forced to sit through years and years of math classes to learn the rules of arithmetic, and we're not ready to throw it all out the window now just because I used the word "add" in a slightly different manner. Perhaps instead you find my use of fractions above a bit fishy. In either case, I'm not doing mathematics properly.
The bold step needed to reconcile this aberration is to change our definition of the word "addition" to encompass many more operations than the procedures that you learned in grade school. This may seem extreme. The key point, and the mantra of modern algebra, is that we should call an operation "addition" if it has similar properties to grade-school addition, even if the method of computing the "sum" of two objects differs radically. By shifting our attention from the procedures used to "add" objects, to the properties of "addition", we open our eyes to new and fundamental mathematical phenomena.
Enough abstract talk. What do I mean by the properties of addition?
Let's examine some of the properties of the usual addition of fractions.
By usual, I mean that we add two fractions x = a/b and y = c/d by setting
x+y = (ad+bc)/bd.
0) Given two fractions x and y, the sum x+y is also an fraction. (Closure)
We call this property closure because it tells us that we are unable to
leave the realm of the fractions using only addition.
1) x+(y+z) = (x+y)+z (Associative Law)
This property is called the associative law, because it says that, when
adding three fractions, we can "associate" or "pair off" the elements in
either way without affecting the outcome.
2) There is an element e such that x = e+x = x+e (Identity)
This property says that the element e (in the case of fractions, e = 0/1
works) acts as a neutral element. By adding 0/1 to any fraction x, you do not
change x.
3) For the identity element e in property 2, given any x, there exists y such
that x+y = y+x = e (Inverses)
In other words, given any integer x, we can always find a complementary
fraction which, when added to x, take us back to the neutral element e.
In the fraction case, the inverse of a/b is (-a)/b.
4) x+y = y+x (Commutative Law)
The law is named for the fact that we may commute, or exhange the order,
when we add two fractions.
Let's return to our original example of compiling baseball statistics. In that example, we wanted to add fractions in a different way. Given two fractions x = a/b and y = c/d, we wanted to define their sum to be (a+c)/(b+d). To avoid confusion, we will denote this operation by #, so we write x#y = (a+c)/(b+d). For example, we have 4/5 # 2/3 = (4+2)/(5+3) = 6/8. Notice also that we consider this fraction as different from 3/4, for reasons discussed above.
Does this # operation have the above properties of the usual + on fractions? One can check that x#y is a fraction whenever both x and y are fractions, that x#(y#z) = (x#y)#z, and that x#y = y#x. How about properties 2 and 3. Here it is not so clear. The element 0/1 was the identity or neutral element for +, but it does not work for # because we have 1/2 # 0/1 = 1/3, which is different from 1/2. However, the fraction 0/0 works as an identity. Hold on! What do I mean by the fraction 0/0? Aren't I dividing by 0, which is something we should never do? Remember that in this case, we are interpreting fractions differently that normal. It makes perfect sense, when compiling baseball statistics as described above, to use the fraction 0/0, because it represents the the statistics of a batter yet to have an at-bat. Once this initial psychological difficulty is overcome, one can show that 0/0 acts as an identity element for #, satisfying property 2. How about the inverses in property 3? This depends on whether you want to consider fractions such as -1/3 in the baseball context. If you feel like we should exclude these negative fractions because they do not correspond to actual at-bat statistics, then we do not have inverses. If you want to include them, then we do have inverses (notice that, in this context, the inverse of a/b is (-a)/(-b) because a/b # (-a)/(-b) = (a+(-a))/(b+(-b)) = 0/0).
Take some time to sit back and understand this example. We have both fundamentally changed how we think about fractions (by allowing the fraction 0/0, and not considering fractions like 1/2 and 2/4 to be equal), and have defined a new type of "addition" which we denoted by # on them. However, in the end, the operation of + on fractions in the old sense, and the operation of # on fractions in the new sense, share some very important properties. By taking a step back and considering the properties of + and #, instead of simply the way to compute + and #, we have uncovered a deep similarity.
Let's study another "addition" by examining Rubik's Cube. If you consider each valid transformation (which consists of a sequence of rotations of cross sections) an object, then we have a natural way of "adding" two objects, i.e. given two transformations, we first perform one, and then follow that by performing the other. One can check that this "addition" satisfies properties 0-3. For example, there is an identity transformation where you do nothing to the cube. Also, for any valid transformation, which consists of a sequence of rotations, we can undo the rotations one-by-one to arrive back at the identity transformation. However, this "addition" does not satisfy property 4, the commutative law. Given two transformation S and T, we may not have S+T = T+S. For example, if we rotate one vertical cross section up, then rotate a horizontal cross section right, we would get something different than the result of performing the two operations in the reverse order.
We are now ready to make our leap into abstraction. We call any collection of objects with an "addition" operation (that takes as input two objects and outputs a third) which satisfies properties 0,1,2, and 3 a group If, in addition, the "addition" satisfies property 4, we call it an abelian group (after the mathematician Niels Abel). We now open up a world of new questions. Are there new and bizarre groups that we have not yet seen? If so, how many? In what ways are they similar, and in what ways do they differ? Can we classify groups in some way? Many mathematicians of the past (and present) have worked (and continue to work) to find partial answers to all of these questions.
Abstract Algebra deals with many structures other than groups. What happens if we two operations called "addition" and "multiplication" that behave in ways simiar to the usual addition and multiplication of integers? We then have a structure that mathematicians call a ring. How about if the addition and multiplication behave like they do on fractions (so that we can divide, unlike in the integers)? This leads to a structure called a field. The investigation of these 3 types of structures (groups, rings, and fields) form the cornerstone of the field that mathematicians call Abstract Algebra (well, usually they just call it algebra, but that may sound funny at first).
Although at this point it may seem like the study of these new and strange objects is little more than an exercise in a mathematical fantasy world, the basic results and ideas of Abstract Algebra have permeated nearly every branch of mathematics. Furthermore, many cryptographic protocols and coding theory schemes are based on new groups and fields that have "additions" and/or "multiplications" which are quite different from the usual addition and multiplication.