TheMathPage

The Evolution of the

R E A L  N U M B E R S

Table of Contents | Home

13

THE EXISTENCE
OF
IRRATIONAL NUMBERS


This continues the previous Lesson.

NO ONE HAS EVER BEHELD the actual value of Square root of 2. Or π. Or any irrational number.  All we ever see are rational approximations -- even to hundreds of thousands of decimal places. In what sense, then, are these irrationals numbers?  We may not assume that just because something has been defined and given name and a symbol, that it is something that actually exists.  (A dragon has been defined. Do dragons exist?)

The Greeks did not admit irrational numbers. They were quite content to say that incommensurable magnitudes simply are not in the same ratio as two natural numbers.  That did not prevent them from approximating Square root of 2, just as we must if we want to know it and use it.

Modern mathematics, on the other hand, is virtually based on calculating with symbols for the irrationals.  For with those symbols -- Square root of 2, π, and so on -- scientists can perform written calculations that previously were impossible.  No man would have landed on the Moon had it not been for those calculations. They are the essential link between arithmetic and geometry; between written computations and space-time.  As far as the mathematics of measurement is concerned, calculation with symbols for irrationals has been the main logical development since the 3rd century, B.C.

Knowledge of irrational numbers

Arithmetic, if it is anything, is knowledge of numbers.  Knowledge not only of how to calculate with numbers, but knowledge of their properties. Numbers have not only their properties as individuals, such as being even, or odd, or prime. They have their properties in the society of numbers, which is how they relate to others. One way a number relates is by its position in the sequence.  5 comes before 6 and after 4.  6.184 is less than 6.185, and more than 6.183.  Having a relative position is the essence of being a number.  For an irrational number to exist, then, must mean that we can place it with respect to order relative to any rational number.  For the rational numbers are what we definitely know.

Is Square root of 2, then, less than or greater than 1.41?

We can compare them by squaring.  (Square root of 2)2 = 2.   (1.41)2 = 1.9881.  Therefore, Square root of 2 is greater than 1.41.

Is Square root of 2 less than or greater than 1.42?

(1.42)2 = 2.0164.  Therefore, Square root of 2 is less than 1.42.

We have found, then, that Square root of 2 falls between 1.41 and 1.42 --

1.41 < Square root of 2 < 1.42.

Continuing to the third decimal digit, we would find

1.414 < Square root of 2 < 1.415.

In this way, we could place Square root of 2 with respect to order relative to any rational number.  We may say then that Square root of 2 is, in fact, a number.

Thus if the symbol for an irrational number is to refer to something that actually exists, then there must be a method to compute its decimal approximation to as many decimal digits as we please; for we could then place it with respect to order relative to any rational number.  That is true of π, for example.  π is approximately

3.1415926535897932384626433.

Real numbers

We have two ideas of number.

1)  Number as discrete units.  These are the numbers whose names we use for counting.  They are the natural numbers.

2)  Number as measurement.  These are the numbers whose names we need for measuring.  They are the rationals and irrationals.

The technical term for number as measurement is real number.

Here are 6 discrete units,

//////

which, we must be admit, is our fundamental idea of 6.

A length of 6 units

The line AB is a picture of the real number 6, in the sense that if AE is the unit, then, proportionally,

AB : AE = 6 : 1.

We require a real number, then, to name the distance from 0 of a

A number line

point P on the number line.  We have seen that the rational numbers are not sufficient for that task, because lengths can be incommensurable. Irrational numbers therefore were invented.

Problem 1.   In terms of parts, what is the difference between the natural number 10 and the real number 10?

The natural number 10 has only half, a fifth part, and a tenth part. The real number 10 could be divided into any parts.

Problem 2.   We have classified numbers as rational, irrational, and real.  Name all the categories to which each of the following belongs.

  a)   2   Real, rational.   b)    3
5
   Real, rational.
  c)   11 1
9
   Real, rational.     d)   Square root of 2   Real, irrational.

e)   Cube root of 8   = 2.  Real, rational.

f)   Cube root of 9.  Real, irrational.

g)   3.1415926535897932384626433    Real, rational. Every decimal is
g) rational. This decimal is an approximation to pi.

  h)   Square root of 36   Real, rational.     i)   Square root of 37   Real, irrational.

j)   Fourth root of 10.   Real, irrational.

*

So.  We can now return to the question we posed at the beginning of this inquiry:

Two lengths

If AB, CD are lengths, will there always be a number n, rational or irrational, such that, proportionally,

AB is to CD  as  1 is to n?

Can we always name the "ratio of two lengths"?  For, numbers have names. That is their usefullness:  1,  9.6,  Square root of 2,  Pi.  But names are discrete, while lengths are continuous.

Any claim, then, that there is a number to name every length will require that the names themselves be continuous.  A continuum of lengths make sense.  But a continuum of names is an absurdity.

Or are there "numbers" with no names?  If so, then they will be "numbers" that are not the measures of anything.  They will be "numbers" of which we have no knowledge -- which we could never place with respect to order. In other words, they would not be numbers.  Or rather, the word number would then have a very unusual and unaccustomed meaning.

Just as all the words in a dictionary are no match for all of reality, so the names of numbers can never exhaust the lengths of lines.  That is the separation of arithmetic and geometry:  It is not possible to name every ratio of magnitudes.

(See Are the real numbers really numbers?)


Table of Contents | Home


Please make a donation to keep TheMathPage online.
Even $1 will help.


Copyright © 2014 Lawrence Spector

Questions or comments?

E-mail:  themathpage@nyc.rr.com