Answers and Explanations

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.

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.

**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).

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

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

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*.

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.

**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,

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

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

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

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.

This page last updated: September 1, 1997

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