A comp sci joke claims that there are 10 kinds of people: those who understand binary numerals and those who don’t. Being a former computer science student myself, I find it tickles my funny bone.
Note that I wrote “binary numerals” and not “binary numbers.” There are no binary numbers, just like there are no decimal numbers. Numbers are numbers, not decimal or binary or hexadecimal.
Numbers and Numerals
The distinction between numbers and numerals is important. Numbers are abstract entities that exist independent of their representations. Representations of numbers are called numerals. The familiar numerals of the Indian decimal system are ten in number: 0 through 9. The numerals of the less familiar binary system are two in number: 0 and 1. The number two is represented in decimal (base ten) as 2 and in binary (base two) as 10.
There are an infinite number of integers (whole numbers), an infinite number of fractions, a greater infinite number of real numbers, and so on. But the number of symbols in the decimal system is only ten and with only the addition of just one more symbol — the decimal point — all numbers can be represented. That’s magic.
The question whether numbers exist has bothered philosophers for some time. We enumerate, count and do arithmetic using the cardinal numbers. But would numbers exist in a world without objects? Does that question even make sense? As a worldly philosopher, my guess is that numbers do exist but they don’t persist once the thought that considers them ends. Numerals, though, exist and persist even when no human mind is contemplating them.
Here endth the lesson.
Postscript: I am appending what I wrote in reply to a comment to this post —
Numbers and numerals differ in the same sense that an object and its name differs. Think of something concrete such as a cow. That thing has different representations in different languages — cow in English, gai in Hindi, guru in Bengali, vache in French, vaca in Spanish, etc. Different base systems are like different languages to express the same abstract object.
Another analogy we could use is that of a person and his name. The same person can be referred to by different people using different names: Daddy by the kids, Alexis by the wife, Alex by his coworkers, Mr Martin by his clients, and Al by his friends. So the number five is represented by the commonly used numeral 5 but as the “V” in the Roman numeral system. So the representation “xi” could refer to the number eleven or to the name of a Chinese person, depending on the context.
Anyway, I like words. They help us think. The words “numbers” and “numerals” refer to different classes of objects. Words matter. They help us comprehend the world as ideas. Numbers are ideas and people have discovered different ways of representing them. Some centuries ago, Indians invented the representation for zero and the decimal positional system.