Math Project #2: How does the consistency requirement of the number system lead to the real numbers?

George E. Hrabovsky, President, MAST

The first column in this series of math projects was "A New Year," where I introduced the format change and presented the development of the number system from natural numbers to the rationals.

Background

In order to understand the question we need to know two things: What do we mean by the number system? What do we mean by self-consistent?

In the previous column we explored the ideas of consistency and number system. We concluded that neither the natural numbers, the integers, nor the rationals were self-consistent. In fact, we determined that the integers are a generalization of the natural numbers allowing subtraction to be carried out for any pair of numbers. The rational numbers are, in turn, a generalization of integers allowing division to be carried out.

The question then becomes, are the rational numbers self-consistent?

Necessary Principles

The notion of a set as a collection of elements, each having a specific property that allows their inclusion in the set.

Definition 1: is the set of natural numbers, listed as .

The seven binary arithmetic operations. Recall that a binary operation on a set is any operation that combines any two elements of the set.

Definition 2: + is the symbol for the binary operation of addition. I assume the reader is familiar with addition.

Definition 3: I assume the reader is familiar with multiplication.

Definition 4: is the symbol for binary operation of taking the th power of . I assume the reader is familiar with exponents and powers.

Definition 5: - is the symbol for the binary operation of subtraction. I assume the reader is familiar with subtraction.

Definition 6: is the symbol for the binary operation of division. I assume the reader is familiar with division.

Definition 7: is the symbol for the binary operation of extracting the th root of . I assume the reader is familiar with root extraction.

Definition 8: is the symbol for the binary operation of extracting the th-base logarithm of . I assume the reader is familiar with logarithms

Definition 9: Closure is the property that a set has when the result of the application of a particular binary operation between any two elements of set is itself an element of the set. We can write this as if , and we apply a binary operation symbolized by , then .

Theorem 1: Principle of Induction. Given a set of elements, a property that is true first for , and then if we assume the property is true for an arbitrary that it must also be true for is true for every element of the set. This can be found here.

Definition 10: is the set of integers, listed as {..., -2, -1, 0, 1, 2, ...}.

Definition 11: is the set of rational numbers, .

The Problem of Certain Roots

At the end of the last column we were left with the question of whether was closed under the extraction of roots or logarithms? It can be shown (in many different ways, see Bogomolny (2006) below) that there is no rational number equal to . As such, we can see that the set of rationals is not closed under root extraction.

The Answer

In order to create a system of numbers that allows for the extraction of any rational root we must realize that if we look at the set of rationals as points on a line, no matter how small we make each rational number there will always be holes separating these points. One such hole is , as there is no rational number that corresponds to this number. Another example is , and there are an infinity of others, too. We can see that between any two points on a line, there is always another point between them.

Any time that we have a number, such as , we can consider the set of all numbers as split into two other parts, all numbers that preceded and all numbers that follow . In this way we are said to have cut the number system. This method of describing a set of numbers is due to the famous mathematician Richard Dedekind (see Eric W. Weistein (1999)).  The set of all irrationals can be described this way (indeed, so can the rationals). The set of both the rational numbers and the irrational numbers is written and is called the set of real numbers.

The Problem of Certain Other Roots

We now have a number system that is self-consistent for all irrational roots, right? Not so fast. What happens is we have the situation , then we have . No real number allows us to calculate this root.

The Final Answer

If we define a new symbol, , and call it the imaginary number we can then extend the real numbers into a new set that is completely self-consistent. We can write . This defines the set of complex numbers and completes the number system.

References

Created by Mathematica  (August 1, 2005)