The myth of irrational numbers - TPR World

The myth of irrational numbers
January 2008
James J. Asher
Prize-winning math instructor
tprworld@aol.com
www.tpr-world.com
If you think about it, mathematics is all about a straight line. That's all it
is. That straight line is called "the number line" which I will refer to as the
"line."
If a number can be located on a point on the line, that number is called a
"real number." All real numbers are points on the line. According to books
on the mathematics of numbers, all "rational" and "irrational" numbers
are points on the line. Therefore, the set of all real numbers consists of
rational and irrational numbers.
What are rational and irrational numbers?
All positive integers such as 1, 2 and 3 are rational numbers.
All positive integers with an exponent are rational numbers such as:
1(sq) = 1, 2(sq) = 4, and 3(sq) = 9.
Also, for instance:
1(cube) = 1, 2(cube) = 8, and 3(cube) = 27.
So far so good! Now let's look at fractions and roots
Some fractions are rational numbers such as:
1/ 2 = .50, 1 / 4 = .25, and 1/5 = .20.

Some roots are rational such as:
(sq rt) 4 = 2, (sq rt) 9 = 3, and (sq rt) 16 = 4.
It is easy to find these rationals on the line
Now for the fun part of the puzzle: The irrationals
Any number that ends in a decimal that continues forever is called
"irrational." The mathematics books on numbers claim that irrationals can
also be located on the line and hence are in the set of real numbers. I
don't think this is true. I will tell you why.
But first, here are come examples of irrationals:
Some fractions are irrational. Examples would be:
1/3 = .333..., 1/6 = .1666..., 1/7 = .1428571...
Some roots are irrational. Examples would be:
(sq rt) 2 = 1.4142135..., (sq rt) 3 = 1.7320508..., and (sq rt) 5 =
2.2360679...
I don't think irrational are on the line because
irrational are not numbers
I know this sounds like heresy. The mathematics books refer to irrationals
as "irrational numbers." I think that is the myth. I think that irrationals are
non-numbers. Irrationals are becoming numbers but they will never, for all
of eternity, be numbers. Think about this: (s rt) 2 = 1.4142135... and the
integers continue on forever approaching some value but never ever
reaching it. Since (sq rt) 2 will never be a number because it is in
continual motion moving towards some number, it is a non-number. Since

it is a non-number, it cannot be located on the number line. I conclude
then that only rational numbers have a location on the line and hence only
rational numbers are real numbers.
Here is the puzzle
Consider this: Some fractions are irrational. Examples would be:
1/3 = .333..., 1/6 = .1666..., 1/7 = .1428571...
Since .333..., .1666..., and .1428571... continue forever without
stopping, they are non-numbers which means they are not in the set of
real numbers, and hence cannot be located on the line. But there is a
contradiction since 1/3, 1/6. And 1/7 can indeed be located on the line
like this:
If 1/3 means to divide the line into three parts, then 1/3 represents
three points on the line:
The puzzle is this: If we apply long division, 1/3 is irrational since it
results in .333...and hence is a non-number and therefore cannot be
located with a single point on the line. However, if 1/3 represents
cutting the line or any unit on the line into three pieces or parts, then 1/3
is indeed located on the line. This suggests that 1/3 has two different
meanings, but what is it?
Another famous example may help clarify
The Pythagorean theorem allows us to find the diagonal of any rectangle
(and remember that a square is a rectangle) with this famous equation:
a(sq) + b(sq) = c(sq).
a and b are the length and width of the rectangle and the square root of
c is the length of the diagonal. Obviously, the diagonal is a finite
measurement like this:

Applying the Pythagorean theorem, 1(sq) + 1(sq) = 2 and the square
root of 2 is an irrational of 1.4142135... which is a non-number. But, how
can this be since we can measure the diagonal's length with a ruler and
get a definite answer? The length of the diagonal is not becoming a
number; it has an exact beginning and an exact ending point on the ruler.
The mystery can be solved
by looking at the nature of long division
Most of us acquired long division by rote, that is, by following a step-bystep
recipe until the procedure was automatic. We assume it is a valid
procedure, but is it?
Let's set up 1/3 with long division:
The set up tells us to divide 3 into 1, but we are told that we cannot do
this. Why not? Because 3 does not go into 1. But if I have one inch on a
ruler, for example, I can divide it into three parts. So 3 does go into one if
one means the length of a unit. I cannot divide 3 into if one means a point
on the line because a point is the smallest indivisible thing on earth. A
point cannot be divided.
So far, it all makes some sense. Since I cannot divide 3 into 1, which is a
point on the line, I am told to put a period after the 1 and add a zero like
this:
Now instead of trying to divide a point labeled 1 on the line, I have a
length of 10 to work with. The problem here, as I see it, is we have
switched from a scale consisting of points on the line to a scale with
lengths on the line. This may be the same flaw as mixing apples and
oranges in arithmetic. For example, 2 apples + 2 oranges = ? We might
say 4 pieces of fruit but this answer is wildly ambiguous since we could
have 1 apple and 3 oranges, 3 apples and 1 orange or 4 pieces of
pineapple, or any combination of fruit.

The length on the line is now 10 and we sort the 10 units into three sets
of three with one unit remaining like this:
Now, what are we going to do with the remainder? Well, the recipe calls
for adding another zero like this:
So the length on the line is 10, like this:
This continues forever because there will always be another remainder of
1. The fallacy is that we started with the task of dividing 3 into a point on
the line labeled 1. The point could have been labeled anything. We could
have called it James or Mary. It is still a point and a point is indivisible.
Hence, the instructor is exactly right is saying, "Under these
circumstances, we cannot divide 3 into 1." If we conceptualized 1 as a
length rather than a point, then of course we can divide 3 into 1 to get
three parts.
The first step in long division is fine (i.e., 3/1 =?). The next step however,
is questionable. That is, changing the scale from points on the line to
lengths on the line. I think that this is where irrationals are manufactured
creating the myth that the irrationals are actually numbers. They are not.
They are artifacts of a flaw in long division. They are a "red herring" to
use a metaphor from Sherlock Homes. They led us on a merry chase
through volumes of math books. The irrationals are not in the set of real
numbers.
Conclusion
I think Pythagoras, Euclid, Descartes, Newton, and Einstein were on target
with their concept of a universe with simple, elegant, symmetrical
patterns of rational numbers that decode the blueprint of nature.
European mathematicians of the 17-century were attracted to intricate
models inspired by the mechanical age of more and more complex

machines. Hence, they were smitten with mathematical models that were
more and more complicated. The more complex, the closer they felt they
were to reality.
R E F E R E N C E S
To order James Asher’s books online, go to www.tpr-world.com
Asher, James J. Learning Another Language Through Actions (6th
edition).
Los Gatos, CA., Sky Oaks Productions, Inc.
Asher, James J. Brainswitching: Learning on the right side of the brain.
Los Gatos, CA., Sky Oaks Productions, Inc.
Asher, James J. The Super School: Teaching on the right side of the brain.
Los Gatos, CA., Sky Oaks Productions, Inc.
Asher, James J. The Weird and Wonderful World of Mathematical
Mysteries: Conversations with famous scientists and mathematicians.
Los Gatos, CA., Sky Oaks Productions, Inc.
Asher, James J. A Simplified Guide to Statistics for Non-Mathematicians.
Los Gatos, CA., Sky Oaks Productions, Inc.
Asher, James J. Prize-winning TPR Research (Booklet and CD).
Los Gatos, CA., Sky Oaks