Elementary school students spend most of their time working with counting numbers, that is, the non-negative integers. As students progress through secondary school, they work increasingly with non-integers, eventually entering the complex number plane. However, many of them maintain the desire to tie these back to their comfort zone in \(\mathbb{N}^0\). In this post, I’ll discuss the various number sets and how each one came about.
The Positive Integers
A long time ago, before there was mathematics in the modern sense, there were counting numbers. Indeed, originally, humans didn’t think in terms of a never-ending set of counting items. Even some modern tribal languages have number word sets along the lines of “one, two, meh… more than that”.
For the most part, the mathematics that people will encounter before college was originally developed for human need (as opposed to human curiosity, which drives much of college-level mathematics). Tribal people didn’t need large numbers. What did they need to count? People in the tribe. Available food. Animals in the herd. Plants in the field. So there wasn’t a need for numbers much above ten.
As societies grew more complex, we developed techniques for avoiding counting big numbers. Want to count the herd, but don’t want to bother with numbers? Here’s a trick: Make a pile of pebbles, one for each animal in your herd. Any time you want to count the herd, move one pebble for each animal. Easy. Arguably, this technique developed into the abacus, a Babylonian invention later adapted by the Romans; whether the Chinese suanpan, what we think of as an abacus, is a related or independent invention is not known.
Eventually, though, our society got complicated enough that we needed larger and larger numbers. This led to the development of the place value system we use today, which provides easy access to a virtually limitless set of integers; the later development of scientific notation allows for convenient reference to just about positive integer we might have need for in practical usage.
Zero
Because these early large numbers were used for things like inventory, positive integers satisfied that need. Zero wasn’t explicitly needed in writing because it didn’t make sense to report on zero items. If you’re out of goats, you simply fail to list them at all. Today, a manager might well ask, “Are we out of goats or did you forget to count them?”, but that was apparently not a significant concern back in the day. So zero took a surprisingly long time to develop, and was only developed independently by a few cultures.
Even these days, I have seen discussion in social media mathematics communities where people insist that zero is not a number, it’s some sort of ephemeral place-holder. Conceptually, I can see why some people would feel this way. Zero became solidified as a mathematical object not because it satisfied a practical need (Out of goats? Just don’t list them.) but because it satisfied a mathematical need. If \(3 – 2\) is meaningful, then \(3 – 3\) is likewise meaningful, and has a conceptually real answer. That answer is zero.
Nonetheless, zero creates mathematical awkwardness. It means that “the positive integers” and “the non-negative integers” are two ever-so-slightly different groups. Some people include zero in the natural numbers and other people don’t, leading to the need to explicitly state which set of natural numbers you’re referring to. There are many common functions that have singularities at zero. It is both a blessing and a nuisance.
When mathematicians are referring to the number sets, they use \(\mathbb{N}\), \(\mathbb{N}^*\), \(\mathbb{N}^+\), or \(\mathbb{N}_1\) for the positive integers and \(\mathbb{N}\), \(\mathbb{N}^0\), or \(\mathbb{N}_0\) for the non-negative integers. I prefer \(\mathbb{N}^*\) and \(\mathbb{N}^0\).
Rational Numbers
As with zero, the next two groups come about to fill mathematical need. Historically, ratios are more salient than negative numbers are. My son, at six, understands the notion of dividing a cookie up among multiple people, for instance.
In technical mathematical speak, we say that the natural numbers (both \(\mathbb{N}^*\) and \(\mathbb{N}^0\)) are closed under addition and multiplication. What that means is that, if you take any two positive integers and add them, you will get a positive integer. The same is true for adding any two non-negative integers, or multiplying any two positive integers, or multiplying any two non-negative integers.
This is not true for either division or subtraction. Neither \(1 \div 2\) nor \(1 – 2\) is a non-negative integer. To account for these numbers, mathematicians found the need to expand the number sets.
One of these expansions involves the rational numbers. A rational number is any number that can be expressed in terms of a division problem, where the divisor is a positive integer.
Note that most standard definitions of rational numbers refer to fractions where the denominator is not zero. This is effectively the same definition with a different focus, and this different focus is not trivial. Common Core’s expectations for elementary mathematics education emphasize that fractions ought to be taught primarily in terms of units. For instance, \(\frac{3}{7}\) represents breaking an abstract unit up into seven equal parts, and using three of those seven equal parts. I believe this is the appropriate approach when first teaching students about fractions. It’s reflected in the names: The divisor is called the denominator because it indicates the size of the unit to be considered, the same way that we call the values of our various money bills “denominations”. However, I think it’s also important in secondary school to transition to a recognition that fractions are unprocessed division. (I’ve discussed this elsewhere on this blog, such as here.)
Another reason for removing “fraction” from the formal definition of “rational number” is that not all fractions are in fact rational numbers. We can put anything we want in the numerator, and anything-but-zero in the denominator. And once we get to limits, we can even put zero in the denominator, just for fun. But a rational number must be expressible in terms of \(p \div q\) where \(p\) an integer and \(q\) is a positive integer.
Negative Numbers
We tend to introduce rational numbers before negative numbers in elementary school because it’s easier for a six-year-old to understand “break this cookie into four parts” and it is to understand negatives. Negatives don’t exist in inventories, and in exchanges of money between people we still speak in positive numbers: If I owe you ten dollars and buy you a twenty-dollar book, you now owe me ten dollars. We don’t normally say “I owe you negative ten dollars”.
Mathematically, though, the concept of rational numbers is not complete until we include negative numbers. So while rational numbers probably appeared earlier in our history (consistently, at least), negative numbers are more “basic”.
The need for rational numbers is driven by division. The need for negative numbers is driven by subtraction. When mathematicians refer to number sets, they often use \(\mathbb{Z}\) (from the German word Zahlen, “numbers”) for the complete set of integers and \(\mathbb{Q}\) (for “quotient”) for the complete set of rational numbers.
All positive integers are included in the non-negative integers, which are all included in the set of integers, which are all included in the set of rational numbers. The mathematical shorthand for this is \(\mathbb{N}^* \subset \mathbb{N}^0 \subset \mathbb{Z} \subset \mathbb{Q}\), where \(\subset\) means “is a subset of”.
Algebraic Numbers
At this point, we have a set of numbers that is closed for all four basic operations (with one exception). If you take any two rational numbers and apply addition, subtraction, multiplication, or division to them, you will get another rational number. The exception, which has already been noted, is that the divisor cannot be zero.
The story now gets more intricate. The next class of numbers comes about to satisfy a specific and ancient need for specific values, even though the class itself can be explained in general terms of the operations.
There were two numbers that particularly annoyed the ancients. The first was the length of the diagonal of a unit square; the second was the length of the perimeter of a circle. The first of these eventually lead to the development of the algebraic number system; the second will be discussed in the next section.
It was known to the Greeks that, given a right triangle, the lengths of the three sides had a predictable relationship. Geometry students know this as the Pythagorean Theorem: If you square the lengths of the two legs and add them, you will get the square of the length of the hypotenuse. Repeat along with me: \[a^2 + b^2 = c^2\]
The Pythagoreans, and most Greek mathematicians in general, were convinced that all numbers had to be rational. Thus, the length of the diagonal of a unit square had to be rational.
By modern standards, this position might seem irrational and intransigent. But we also have an understanding of the real number system that goes far beyond the rational numbers. The Greeks, who still worked largely within the practical realms of mathematics, saw the world in terms of numbers being “commensurable”. That’s a fancy word that means that any number can be expressed as an integer times a fixed measurement unit. “An eighth of a cookie” is a fixed measurement unit, so \(\frac{3}{8}\) was a commensurable number. They reasoned that since it’s possible to place a ruler alongside the diagonal of a unit square, it must be possible to give the length of that diagonal in meaningful units.
This is a reasonable conclusion. It just happens to be wrong. If a square has side lengths of one inch, for instance, its diagonals will both measure \(\sqrt{2}\) inches. That’s around 1.414 inches, but it’s not exactly 1.414 inches.
Legends abound of Pythagoras’s rage and refusal to accept this, but it is mathematical reality. Indeed, if we generalize it, we discover that the square root of a positive integer is either an integer or it’s not rational at all. There are absolutely no positive integers that have rational, non-integer roots.
This means that while the set of rational numbers is closed under the four basic operations we learn in elementary school, it is not closed under the additional operation of “square root” (and indeed, roots in general). This leads to another expansion and another caveat.
The expansion is the set of algebraic numbers, \(\mathbb{A}\). The precise definition gets rather complex, but basically, a number is algebraic if it can be expressed in terms of addition, multiplication, division, and rational exponents. The caveat is that we need another number set to account for, say \(\sqrt{-1}\), since there is no real number that, when multiplied by itself, is negative. Also, as with all number sets, division by zero creates a non-number.
In addition to the square roots, there is one well-known algebraic number that was named because it had to exist, not because it completed some set or other. This makes it similar to the named transcendental numbers, which I describe next. Specifically, mathematicians figured there was some number such that \(x – 1 = \frac{1}{x}\). This can also be written as \(\frac{a+b}{a} = \frac{a}{b}\). In words, we can describe the problem as: “A line segment is divided into two smaller segments. The ratio of the length of the entire segment to the longer sub-segment is the same as the ratio of the length of the longer sub-segment to the length of the shorter one. What is that ratio?”
This ratio is called the Golden Ratio, or \(\phi\) (“phi”), and \(\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618\). This value has some very interesting properties.
Transcendental Numbers
While there are more transcendental numbers than algebraic numbers, there are only a handful that have precise names or uses. You should know one (and possibly its friends) and may know a second. Unless you’ve completed college-level work in mathematics, that’s probably the extent of your exposure. As a result, people tend to think of the transcendental numbers (such as they’re familiar with them) as a very small set.
While the square root of two was troubling to the Greeks, the ratio of a circle’s perimeter to its diameter proved to be even more challenging. This is a case where, rather than measuring or calculating a specific number using core mathematical concepts, the first values were based on the conclusion that such a value had to exist, and therefore it existed. In other words, the transcendental values we have named are the result of theory requiring a specific value, and using methods to determine that value.
In the case of the number that is now known as \(\pi\) (“pi”), it has long been known that, however big a circle you have, the circumference (i.e., a circle’s perimeter) is a little more than three times the length of the diameter. If you have a wheel with a diameter of ten inches, it will travel a little over thirty inches in one rotation. If you have a rolling pin with a diameter of three centimeters, it will rotate once every ten centimeters or so. Given how useful wheels are to a variety of daily activities, it should be easy to see why it’s important to know the value of this ratio.
However, it turns out that there’s no way to write \(\pi\) in terms of finite algebraic operations, let alone as a rational value. It can be written as the limit of an infinite number of algebraic operations; indeed, while the first estimates of \(\pi\) were done through simple measuring, later estimates were done through calculating the perimeters of polygons with increasing numbers of sides. This was an early awareness that a circle is the limit of an n-gon as n goes to infinity, well before the concept of “limit” was mathematically formalized.
In recent years, \(\pi\) has acquired two frenemies: \(\tau\) (“tau”), which is \(2\pi\), and \(\eta\) (“eta”), which is \(\frac{\pi}{2}\). You can learn more at the links, and I personally wish that \(\tau\) were the standard, but \(\pi\) for now continues to be King of the Castle.
The second widely used transcendental number is e. Let’s say you had some money in the bank, and let’s say the interest was compounded on a very small cycle. What is the most interest you could possibly expect? To figure this out, mathematicians sought a number that could be used in a formula for interest in the case where the bank continuously compounds interest. This formula is \[A = Pe^{rt}\] where P is the amount you put in at the beginning, A is the amount the bank owes you at the end of t years at an interest rate of r per year.
The formula for compounded interest is \(A = P\left(1 + \frac{r}{n}\right)^n\). Based on this, a mathematician named Jacob Bernoulli calculated e as \[\lim_{n\to\inf} \left(1 + \frac{1}{n}\right)^n \approx 2.718\] There had been other needs for this number already, and there’s one more very significant detail of e: \(f(x) = e^x\) is the only function whose derivative is itself. So, again, e exists as a named value because it is mathematically useful to use it and hence to name it.
In summary, if a number is not algebraic, it is transcendental. There is no conventional symbol for the transcendental numbers. If a real number is not rational, it is called irrational.
Real and Complex Numbers
As mathematicians continued to explore, they realized that there were a limitless number of numbers that could be described. As the Greeks had learned, if we limited ourselves to rational numbers, there were numbers in between them (like the square roots). If we limited ourselves to algebraic numbers, there were still numbers in between them (like \(\pi\)). But we could talk about real numbers, as a set; there is no number that we could devise and write using an infinite number of digits that does not fall into the real numbers.
However, there were still two gaps: Dividing by zero and taking the even root of a negative number. Division by zero continues to be unaddressed by the number system: It is undefined if the numerator is not zero, and indeterminate if it is zero. This does not create significant enough problems for mathematicians that would be solved by creating it as some sort of number.
Even roots of negatives, though, did warrant an additional layer of numbers. In the days that mathematicians resisted considering such numbers, problems began piling up. As soon as mathematicians allowed themselves to use such numbers, they became useful in a variety of high-level mathematical contexts, and even have a place in practical engineering.
To simplify matters, mathematicians created a notation for a single value, called i for “imaginary”. \(i = \sqrt{-1}\). This simple definition allows a multitude of more numbers to be described and used. While the real number system (\(\mathbb{R}\)) was not closed under exponents, this new system, the complex number system (\(\mathbb{C}\)), is. If a number is complex, then its roots are all complex. There is no mathematical operation you can perform on a complex number and create a number that is not complex.
Complex numbers are written in the form \(a + bi\), where \(a\) and \(b\) are real numbers. If \(a = 0\), then the number is imaginary; if \(b = 0\), then the number is real.
There are number sets that are even more complicated (quaternions, octonions, and sedonions), but those have very limited use. Most career mathematicians, engineers, and physicists go their careers without using them in practice.
Summary
Our number set started out simple and has become increasingly complex over time, as different needs have emerged. We can break the current model that is used most widely down as follows:
- Complex \(\mathbb{C}\)
- Real \(\mathbb{R}\)
- Rational \(\mathbb{Q}\)
- Integer \(\mathbb{Z}\)
- Non-negative integer \(\mathbb{N}^0\)
- Positive integer \(\mathbb{N}^*\)
- Non-negative integer \(\mathbb{N}^0\)
- Integer \(\mathbb{Z}\)
- Irrational
- Rational \(\mathbb{Q}\)
- Imaginary
- Real \(\mathbb{R}\)
Algebraic and transcendental numbers can be real, but don’t have to be.