Sometimes mathematics is moved forward by the discovery of new formulas and solutions to problems. However, sometimes mathematics grows by adding new kinds of numbers to the number system.
In the early days of mathematics, it was thought that whole numbers were the only kind that existed. Sure, there were fractions, but fractions are merely ratios of whole numbers. It was thought that every possible number could be written in terms of whole numbers. These numbers were called rational numbers because they could be written as a ratio.
There is a story about a Greek philosopher, Hippasus who discovered, roughly 2500 years ago, that certain numbers (specifically the square root of two) could not be written in terms of ratios at all. In other words, it was an irrational number. The story goes that he was on a ship with some Pythagoreans (some of the greatest mathematicians of the time). He showed them his proof and it so outraged them that they threw him overboard.
However, while Hippasus himself did not survive, the irrational numbers did. In fact, eventually we learned that the irrational numbers are actually far more numerous that the rational numbers! The rational numbers are tiny islands in a sea of irrationals. That is a story for another time.
Irrational numbers, if you think about them in decimal terms, are numbers that have an infinite number of numbers after the decimal that do not repeat. Many numbers do repeat. For instance, 1/13 has an infinite decimal expansion, 0.076923… , but those numbers keep repeating over and over again. However, the square root of two doesn’t have any such repetitive elements.
So then, if you combine the rational numbers and the irrational numbers, we have numbers that have every number of decimals, every repeating set of decimals, and even those with an infinite number of non-repeating decimal points. Today, these are known as the “real” numbers.
That sounds like all the numbers, doesn’t it? Well, as a matter of fact, no. It turns out that there are even more numbers that we need to solve even simple mathematical equations.
Take the equation x2 = -1. What is the answer? Does it have one? Well, since every positive number that is squared has a positive answer, and every negative number that is squared has a positive answer, it looks like this equation doesn’t have an answer. Or does it? This is where the complex numbers come in.
The complex numbers can be thought of as two dimensional numbers. Normally we think of a number as a point along a line the number line. It goes from left (negative) to the right (positive). What if, however, instead of all of the numbers being on the line, some of the numbers were above and below the line? That image gives you the complex numbers. Complex numbers have two components. The first part, known as the real part, tells you how far to the right or the left on the number line you should go to find the number. The imaginary part tells you how far up or down you should go to find the number.
So, it turns out that there are no numbers on the number line whose square is -1, but there is in fact a number that is off of the number line whose square is -1. This is i, the imaginary unit, whose multiples tells how far up and down off of the number line a number should go.
Beyond complex numbers, the concept of multiple numbers to represent a single value is generalized into matrices. A matrix is a set of numbers in a grid that represent a collection of values acting together.
That is usually as far as most undergraduate mathematics goes. However, there is one more useful extension to the concept of number that I think is extremely helpful, and also often overlooked: the hyperreal numbers.
Hyperreal numbers represent infinities and their reciprocals (known as infinitesimals). There are many different mathematical ways of dealing with infinities but the hyperreal have an advantage in that you can deal with the infinitely large and infinitely small almost precisely the way that you deal with ordinary numbers.
The problem that the real numbers have with infinities is that there is usually only one value for infinity: ∞
However, this symbol really just means “the value is outside of the scope that are considered for real numbers.” In other words, you can’t manipulate it because it doesn’t have a precise meaning.
Of course, with infinities, “precise” meanings are hard to come by. It’s hard to point to anything concrete as “infinity.” The hyperreal numbers, instead of trying to establish a precise meaning for an infinity symbol, simply assign a “landmark” value for an infinity and infinitesimal. I usually use the symbol to represent a landmark infinity (ω) and epsilon (ε) to represent the infinitesimal that is 1/ω. That is, I can’t tell you exactly how big ω is, but I can tell you that it is half as big as 2ω and just barely bigger than ω – 1.
Likewise, ε represents something infinitely small. Again, I can’t tell you exactly how small, but it is smaller than 2ε and bigger than ε/2.
So what’s the advantage of hyperreal numbers? The advantage is that, when using hyperreal numbers, many, many difficult subjects in mathematics turn into simple applications of algebra. Limits, calculus, infinite series (both convergent and divergent), and a host of other difficult problems wind up being rather trivial extensions of what we are already teaching students with algebra. What was difficult to express with real numbers becomes almost trivial once you expand your concept of numbers sufficiently.
I doubt that schools will add hyperreal to their curriculum anytime soon, but, if I were to guess, the next big change to high school and undergraduate curriculum will be the addition of hyperreal numbers.