Answers and Explanations

When talking about numbers, there are many **very different
contexts** that one could have in mind.

Here are the four most familiar ones:

*The Natural Numbers*. These are the counting numbers 1, 2, 3, ... that are possible answers to the question "how many?" They are abstract concepts that describe sizes of sets.*The Integers*. These are abstract concepts that describe, not sizes of sets, but the*relative sizes*of two sets. They are the possible answers to the question "how many more does A have than B has?" They include both positive numbers (meaning A has more than B) and negative numbers (meaning B has more than A).*The Rational Numbers*. These are abstract concepts that describe*ratios of sizes of sets*. They do not model sizes of sets the way that natural numbers do. If you say "I ate 3/4 of a pie", you are not saying that the set of things you ate had 3/4 elements. Instead, you are expressing a ratio of two integer quantities: 3, the number of pie-quarters that you ate, and 4, the number of pie-quarters that make up a whole pie.*The Real Numbers*. These are abstract concepts that describe*measurements of continuous quantities*, such as length, weight, quantity of fluid, etc. (Don't let the word "real" fool you; the real numbers are no more "real" in the ordinary English sense of the word than are any other kind of numbers.)

Concepts that exist in one of these contexts may not exist in another.
The question "does there exist a number between 1 and 2?" has the
answer **no** in the first two contexts (you cannot go to the
beach and pick up more than one but fewer than two pebbles), but
**yes** in the last two contexts (you could eat three cookie
halves, which is in between one whole cookie and two whole cookies).

Although in the first two contexts there does not exist a number between 1 and 2, most people are quite comfortable with the fact that such numbers do exist in other contexts. For instance, people don't usually have trouble accepting the existence of the fraction 3/2. Why then is it so hard to believe that the concept of "a number whose square is -1", though it does not exist in any of the four contexts mentioned above, might nonetheless exist in some other context?

*It is because we usually forget the fact that we already have
four quite different meanings for the word "number".* We have
become so familiar with each of the four contexts that we have jumbled
them together in our mind as if they were a single concept. When we
encounter a notion like "square root of -1" which does not exist in
any of these four contexts, we think that it cannot exist at all,
because we think the word "number" is a single concept that embodies
just these four contexts.

Instead, what we *should* be thinking is something like this:

Okay, I know about four different number systems: one in which "number" means a measurement of how many items are in a set, a second one in which "number" means a relative measurement of the sizes of two sets, a third one in which "number" means a ratio of sizes of two sets, and a fourth one in which "number" means a measurement of a continuous quantity.The answer to that final question is "yes, there is". It is called theIn neither of these four number systems does there exist a square root of -1.

Might there be a fifth context, a number system (where "number" means something different from any of the above four things) in which there

doesexist a square root of -1?

This page last updated: September 1, 1997

Original Web Site Creator / Mathematical Content Developer: Philip Spencer

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