"How do I love thee? Let me count the ways." – Elizabeth Barrett Browning

For mathematicians, Ms. Browning may well have been referring to all types of numbers instead of just a person. For them, numbers are beautiful, mystical creatures that provoke wonder, defy explanation and even inspire love.

Every aspect of the modern world is entwined with the concept of numbers. From the simplest games like hopscotch to the most esoteric of twenty first century sciences, numbers form a backdrop that allows us to more easily and completely understand our world.

Numbers have been a powerful tool throughout the history of mankind and have been instrumental in every great invention. Indeed, their own creation and expansion have been some of the most fundamental and important creations ever. Today, numbers are integral to our most advanced technologies and figure in our greatest art.

We use numbers every day and they seem quite real but what are they really. It is a question best left to philosophers and mathematicians. Instead, as a partial explanation, here are the most common of all types of numbers and a little about their uses.

The Naturals

The natural numbers have been in constant use for over five millennia. These numbers and the associated concept of a “place system” provide a convenient framework for tracking and, more importantly, manipulating large sums of objects. The natural numbers are also known as the “counting numbers.”

The Wholes

Zero seems such a natural concept to the modern mind that it is, perhaps, surprising that mankind did not invent the concept of “zero” as an actual number and placeholder until sometime in the ninth century, A.D.  This is not to say that the ancients could not conceive of nothingness, just that they overlooked the utility of a placeholder digit to indicate a zero value. For instance, the ancient Babylonians used a space to indicate a zero value. This approach led to several different numbers having the same nomenclature with the context providing additional clues as to the actual number.

The Integers

The use of addition with the counting numbers allowed its users to combine varying amounts of objects with alacrity and ease. Subtraction, however, created a problem when the first number was smaller than the second. Hence, history saw the introduction of the concept of negative numbers and the development of integers which is the combination of the sets of whole and negative numbers. While there are written mentions of negative numbers as early as 200 B.C. in China and 600 A.D. in India, it wasn’t until the 16th century that Western mathematicians took up their study.

The Rationals

The rational numbers subsume all of the numbers described so far as well as any numbers that can be formed by their ratios; that is, create any fraction using an integer in the numerator and denominator and a rational number is created. In short, the rational numbers are the set of all fractions. Purists will note that division by zero is left undefined.

The “ratio” in “rational” is from the Latin for calculate and leads to the definition of these numbers as well as to our modern sense of the word. An interesting side note to the rational numbers is that they are countable even though they stretch to infinity in both directions and have an infinite number of elements between any two points. This fact was most ingeniously proven by Georg Cantor with his “diagonal” proof.

The Irrationals Credit: Yves BaeldeThe definition of the rational numbers also describes the set of irrationals. They are simply all the real numbers that cannot be written as a fraction. The introduction of the irrational numbers brings us to the first set of uncountable numbers. “Countable” in this sense means that the elements of the set cannot be ordered in such a way that the natural numbers form a one to one correspondence. If has been further proven that “almost all” of the real numbers are irrational. This definition is beyond reproach, easily verified and should be used to win bar bets on as many occasions as possible. Some of the more notable irrationals include √2, π, φ and e.

The Real

The reals  include all the rational and irrational numbers. For the purposes of the average user, the real numbers can be thought of as the entire set of numbers that occupy all the points on a “real” line. The nomenclature is purely mathematical by definition and this fact leads to the somewhat contradictory result that a “real” line is a purely hypothetical construct.

The Imaginary

Just as the operation of subtraction led to the development of negative numbers and the integers, the operation of finding the roots of numbers has led to concepts of imaginary and complex numbers. The most ubiquitous irrational number being the square root of -1 or “i”.  More formally, the imaginary numbers are that set of numbers whose squares are equal to or less than zero.

The term “imaginary” was originally intended as derogatory and was used by such mathematical luminaries as Descartes who considered the imaginaries to be worthless. Time, however, has shown the imaginary numbers to be invaluable tools and they have had, very real, practical applications in fields as diverse as electromagnetism & quantum mechanics, mapmaking, signal processing and vibration analysis.

The Complex

The complex numbers are the combination of the real and imaginary ones. While the complex numbers are a highly abstract construct and the real numbers are considered, well, real, they are often combined on a three dimensional graph with the complex numbers being denoted in the third dimension. As mentioned, complex numbers have numerous practical applications and are the heart of many proofs in higher mathematics.

Other Types of Numbers

Cardinal & Ordinal

Although originally conceived to convey the amounts and order, respectively, of actual objects, cardinality and ordinality have taken on new subtleties when denoting all types of numbers and  the various types of infinities.

Surreal & Hyperreal

The surreal numbers are an even larger set of numbers that includes all the complex numbers as well as the infinite and infinitesimal numbers. It is considered, though not yet proven, as the largest ordered field possible.

Algebraic & Transcendental

The algebraics are the set of numbers that can be derived from a non-zero, single variable polynomial with rational coefficients. Transcendental numbers do not satisfy this condition. Hope that helps. Seriously, there are far more transcendental numbers than algebraic ones, even though the algebraic numbers but not all types of numbers are countably infinite.