We know many kinds of numbers, . The simplest and earliest-discovered numbers (already known to early humans) are called the natural numbers: 1, 2, 3,…,to infinity, and this set is denoted by N. Then if you add to these numbers zero, you form what mathematicians call a group under addition, which means you can now define additive inverses, i.e., the negative integers. The enlarged set of numbers is called the integers and denoted by Z (from the German for numbers, Zahlen). Add another operation, multiplication, and you now also have multiplicative inverses (except for zero), which are all the fractions, meaning quotients of integers, and the enlarged set is now thefield of rational numbers, denoted by Q. When you add to this set all the irrational numbers (numbers that can’t be written as quotients of integers), you get the field R of all the real numbers (these are the numbers on the real number line, and we call them “real” to distinguish them from imaginary numbers; if you then also add to them all combinations of the imaginary numbers and real numbers you get the field C of complex numbers).
The German mathematician Georg Cantor proved in the 1800s that while all these sets of numbers are infinite, they are not of the same infinite size. Using ingenious methods, he showed that there are as many rational numbers as there are integers and positive integers. So N, Z, and Q have the same infinite size (or cardinality), while the real numbers, R, have a higher order of infinity (although we don’t know what it is – Cantor’s unprovable conjecture about it is called the continuum hypothesis). The “enlarged infinity” is because of all the irrational numbers – there are just too many of them! We say that N, Z, and Q are countable, whileR is uncountable (and so is C).
Algebraic Numbers
But the story gets complicated. Numbers that can be obtained as solutions of equations with rational-number coefficients are called algebraic. So algebraic numbers can be irrational, for example √2. This number is algebraic because it is the solution of the equation x2 – 2 = 0, whose coefficients are all rational (in fact, integer): 1 and -2.
One fascinating fact about algebraic numbers is that they are countable, i.e., they have the same order of infinity as N, Z, and Q – even though they are members of the (uncountable) higher set R. Thus their order of infinity is more pedestrian. The hard-core irrational numbers – those that are notalgebraic – are called transcendentalnumbers. These include π (pi) and e. There are “infinitely many more” such numbers than there are algebraic numbers, or integers, or rational numbers!
Squaring the CircleAn interesting fact is that it is because π is transcendental that it’s impossible to square the circle, as the ancients had tried so hard to do. This fact became known only in the nineteenth century, when algebraic numbers became well-understood. It so happens that to square the circle, meaning to construct with straightedge and compass a square whose area is the same as that of a given circle, is tantamount to solving an equation with rational coefficients and getting π as the solution. This is impossible because π is transcendental, and therefore not algebraic. There can never be such an equation that would yield π as a solution.