The Irrational Numbers

by George E. Hrabovsky, President, MAST

Where We Have Been

We have been continuing our focus on the foundations of mathematics. We have discussed the basic ideas of set theory and logic, proof techniques, quantifiers, and have begun to develop the number system. Last column we discussed the properties of rational numbers.

Where Will We Go in This Column

In this column we will continue our exploration of preimage problems to include the problem of powers and roots.

Common Factors

Before getting to the meat of this column I need to mention something about multiplication. If two integers, say p and q are even then they both are multiples of two. Thus you can divide both by 2. For these numbers 2 is a common factor. For a rational number

if p and q have common factors they must be divided out so the rational number will be in simplest form.

Powers

If we multiply a number x by itself n times, then we are raising x to the power of n. We call this exponentiation and we call x the base, and n the power, and the result of the exponentiation is called the exponent. We can write this,

Properties of Powers

Bases that have similar powers may be added directly,

while those with different powers may not, for example the expression

is as simple as it can be stated.
Powers themselves have properties, we can define

Here are the properties of powers,

$FormBox[RowBox[{x^a x^b, , =, , RowBox[{x^(a + b), Cell[ ... nbsp; ]}], , (5)}]}], TraditionalForm]$

Here are the two most common powers,

Roots

If we define

then the nth root of y is x. We can write this,

This is an inverse operation of exponentiation in the same way that subtraction is the inverse of addition, or division is the inverse of multiplication.
Over the years some special common roots have been discovered that correspond to the common powers,

Roots as Rational Powers

We can also define roots as rational powers. We can equate,

We can write the special roots,

Square Root : y^1/2 Cube Root : y^1/3 .

Properties of Roots

Roots have properties just like powers,

y^m^(1/n) = y^m/n x/y^(1/n) = x^(1/n)/y^(1/n) .

Square Root of Two

For centuries it was thought that the exact rational numbers were the only possible numbers. They were perfect and it was thought that only perfect numbers existed. There was a problem though; what happens if ? If the answer was a rational number it would have the form,

where p and q are both integers and that they have no common factors (they are in simplest form). We have

this implies

p^2/q^2 = 2 p^2 = 2 q^2 .

This implies that p is even since it is a multiple of 2. Since p is even is divisible by 4. In this way is also divisible by 4. This means that q is also even. This implies that p and q have a common factor of 2. Thus is not rational because it can never be in simplest form. Thus is not a rational number.

Irrational Numbers

So, we need to expand our number system again. This time in a vague way. We need to include numbers that are not rational. This leads us to an uneasy set of numbers called the set of irrational numbers. If we look at a number line and place points at all rational numbers we would find an even distribution of these points with spaces between them. One way of thinking about irrational numbers is that they fill in the spaces on the number line. We will explore this more carefully next time.