How can one show that imaginary numbers really do exist?

One does it in exactly the same way one would show that fractions exist.

Let's look at a way to show that fractions exist. Of course, that's something you know already; you don't need a mathematical exposition to prove it to you. But the point of going through it is that exactly the same argument can be used to show that imaginary numbers exist. Having become convinced that the argument is a legitimate one by seeing it work in a familiar context, you should be more willing to accept it in the somewhat mysterious context of "imaginary" numbers.

Argument that Fractions Exist

Suppose the only things you knew about were the natural numbers (1, 2, 3, etc.), and you had to show that "three halves" exists. In other words, you need to show that there exists some number which, when doubled, gives you 3. You could argue as follows:

Granted, no such thing exists within the Natural Number System.

However, there is a different number system in which such a thing does exist: the Rational Number System. The "numbers" in this different number system will be fractions: totally different objects from the natural numbers (they won't represent sizes of sets; instead, they'll represent ratios of sizes), but that doesn't make them any less real.

Do fractions really exist? Yes. Do they really form a number system? Yes. Within this number system, is there a number which, when doubled, gives 3? Yes. Therefore, "three halves" exists.

Validity of the Argument

To see that the three key answers (in the last paragraph of the argument) really are "yes", let's look at the questions one by one.

Do fractions really exist? Yes; they're just pairs of natural numbers. (Let's just talk about positive fractions here, to make the discussion as simple as possible and avoid having to worry about things like the denominator being zero). Pairs of natural numbers certainly exist, so fractions exist. We write such a pair by writing the first number over the second number, e.g. a/b.

Do fractions really form a number system? Yes. A number system is just a collection of objects for which

•there's a definition of what it means for two objects to be equal, •there is a rule for how to add two objects together and a rule for how to multiply two objects together (subtraction and division can be deduced from these, provided that all objects have corresponding negatives and some objects have corresponding reciprocals), and •these rules for addition and multiplication satisfy the familiar properties of arithmetic, such as commutativity (order doesn't matter), associativity (in a sum of three of more terms, it doesn't matter which two you add first, and likewise for products), and distributivity (a(b+c) = ab + ac).

Roughly speaking, any collection of objects that satisfies these properties is, by definition, a number system. (Strictly speaking, some of these properties need to be stated a little more precisely, but the rough statement is quite enough for our purposes!)

These properties are all satisfied by fractions. We have a definition of when two fractions are to be considered equal:

a/b = c/d if and only if ad = bc.

We have a rule for adding two fractions:

a/b + c/d = (ad+bc)/(bd)

and a rule for multiplying two fractions:

(a/b)(c/d) = (ac)/(bd).

One can check that these rules do indeed satisfy the familiar properties of arithmetic.

Therefore, fractions form a number system.

Within this number system, is there an object which, when doubled, gives 3? Yes. It is the fraction 3/2 . When you double it, you get the fraction 3/1.

  Strictly speaking, 3/1 is something different from the natural number 3. After all, it's a pair of natural numbers, 3 and 1 (representing the ratio "3 to 1"), not a single natural number.

However, fractions of the form a/1 behave identically to the way ordinary natural numbers a behave. They add and multiply in exactly the same way that ordinary natural numbers do:

a/1 + b/1 = (a+b)/1

(a/1)(b/1) = (ab)/1.

The "/1" just "comes along for the ride".

Since numbers are just abstract concepts anyway, and since natural numbers a and fractions of the form a/1 are completely identical as far as their arithmetic behaviour is concerned, it is perfectly legitimate to view them as just two different representations of the same underlying concept.

With this in mind, we can consider the fraction 3/1 (the ratio "3 to 1") and the natural number "3" to be the same thing. This enables us to say that 3/2, when doubled, gives 3.

This completes the argument that "three halves" exists. Of course, that's something you knew already; it's obvious that fractions exist. But even though you already knew that fractions exist, and didn't need this long argument proving it, the point of going through the details of the argument is that exactly the same argument can be used to show that imaginary numbers exist.

The argument that "imaginary" numbers exist is almost word-for-word identical to the above argument. So, being convinced that the above argument is a valid one, you should be better able to accept the argument that imaginary numbers exist.

Argument that Imaginary Numbers Exist

This argument is patterned after the above argument that fractions exist; you'll probably find it helpful to open another window on your web browser and view the two of them side by side.

The issue is the existence of the mysterious quantity "i", since imaginary numbers are just multiples of i. In other words, we want to see that there exists some number which, when squared, gives you -1. Here is such an argument:

Granted, no such thing exists within any of the four familiar number systems (the Natural Number System, the Integers, the Rational Number System, or the Real Number System).

However, there is a different number system in which such a thing does exist: the Complex Number System. The "numbers" in this different number system will be totally different objects from the familiar real numbers (they will in fact be pairs of real numbers), but that doesn't make them any less real.

Do complex numbers really exist? Yes. Do they really form a number system? Yes. Within this number system, is there a number which, when squared, gives -1? Yes. Therefore, i exists.

Validity of the Argument

To see that the three key answers (in the last paragraph of the argument) really are "yes", let's look at the questions one by one.

Do complex numbers really exist? Yes; we just define a complex number to be a pair of real numbers. Real numbers certainly exist, so pairs of them exist.

Do complex numbers really form a number system? Yes. Remember that any collection of objects for which

•there is a definition of what the objects are and when two objects are equal, •there is a rule for how to add two objects, •there is a rule for how to multiply two objects, and •these rules obey familiar arithmetic laws like commutativity, associativity, and distributivity,

is, by definition, a number system.

These properties are all satisfied by complex numbers.

We have a definition of when two complex numbers are to be considered equal: they are equal if and only if they are the same pair of real numbers.

We have a rule for adding two complex numbers (which, remember, are nothing more than pairs of real numbers):

(a,b) + (c,d) = (a+c, b+d)

and a rule for multiplying two complex numbers:

(a,b)(c,d) = (ac-bd, ad+bc)

The rule for multiplication may look very strange, but there's nothing wrong with that; one can still verify that these rules do indeed satisfy the familiar properties of arithmetic.

Therefore, complex numbers form a number system.

Within this number system, is there an object which, when squared, gives -1? Yes. It is the pair (0,1). When you square it using the above rule of multiplication, you get

(0,1)(0,1) = ( (0)(0) - (1)(1), (0)(1)+(1)(0) ) = (-1,0).

Strictly speaking, the complex number (-1,0) is something different from the real number -1. After all, it's a pair of real numbers, -1 and 0, not a single real number.

However, complex numbers of the form (a,0) behave identically to the way ordinary real numbers a behave. They add and multiply in exactly the same way that ordinary real numbers do:

(a,0) + (b,0) = (a+b,0)

(a,0)(b,0) = (ab,0).

The ",0" just "comes along for the ride".

Since numbers are just abstract concepts anyway, and since real numbers a and complex numbers of the form (a,0) are completely identical as far as their arithmetic behaviour is concerned, it is perfectly legitimate to view them as just two different representations of the same underlying concept.

With this in mind, we can consider the complex number (-1,0) and the real number -1 to be the same thing (this may seem a little hard to swallow, but remember it is no different from saying that the fraction 3/1 and the natural number 3 are the same thing, something that we do all the time; it may be helpful to re-read the corresponding paragraph for fractions to see just how similar the two cases are).

This enables us to say that (0,1), when squared, gives -1. Therefore, i exists; it is merely the pair of numbers (0,1) under the above rules for adding and multiplying.