Pi and Finite Measurements
The value pi is an irrational number and never
ending. If circles have areas and perimeters that are multiples
of pi or just simply involve the value pi, doesn't a circle have
a never ending length?
There are two points here: Not only is pi irrational (its value
cannot be expressed exactly as the ratio of two integers), it is
also transcendental (it is not a solution to any algebraic equation
that involves the arithmetic operations of addition, subtraction,
multiplication, division, and taking "roots" of numbers). That
doesn't mean that it is not finite, rather the ration of the
circumference of a circle and its radius cannot be expressed as a
number meeting the two negative stipulations above -- irrationality
Sounds a little weird, but numbers are a lot weirder than they
appear at first glance.
Let us think about Pi, an irrational number, and the circle.
An irrational number is a number that can not be written as one whole
number divided by another whole number. A real number that is not a rational
number is called an irrational number. The decimal expansion of an
irrational number never repeats or terminates, unlike a rational number.
You might think that if Pi is the ratio of the circumference
to its diameter, then it must be able to be expressed as the ratio of two
whole numbers; but it can not despite the fact that the circumference is of
finite length. If you measure the diameter and express it as a whole number
the circumference can not be a whole number. Therefore the ratio is an
irrational number. If you do get a whole number, as you would for the
perimeter of a polygon that approximates a circle, you are not dealing with
Now consider this:
The length of the perimeter of a polygon of N sides can be written as
Where R is the distance from the center of the polygon to one corner. As N
gets large, and approximates a circle, the Perimeter approaches
2*R*Pi, the equation for the circumference of a circle of radius R.
For a 100 sided polygon
The perimeter is 2R Sin (180/100)*100=2R*3.141
A circle is just a polygon with an infinite number of sides.
Yes, pi is never ending. But extra digits are added after the decimal
place. Every added place means we get closer to the real value of the
ratio between a circle's radius and circumference. They do not affect
the length as such. In other words, places added to pi make our answers
that much closer to the real world numbers.
At over a billion decimal places for pi, the difference between
calculation and the real values is getting very small.
Almost every measurement in the world is really a never ending number.
Measure a stick. Your ruler may produce a length of 4.25cm. The stick
might really be 4.249999678523987cm. You will never really know, but
you can often see a certain "uncertainty" with your measurement. Was it
right on the EXACT CENTER of the black line, or maybe just a little bit
off? With digital devices, this uncertainty can be even harder to
notice but is still there. A stopwatch is on 24.57 seconds. The watch
reading indicates the watch was running for 24.57s, but it might have
been 24.571234231199560782 seconds. The digital stopwatch will stay on
24.57 seconds until 24.58 seconds have passed.
What this means is that we can never truly know exactly how big or small
something is. We can be extremely close, but we will never know whether
we are exact.
Dr. Ken Mellendorf
Illinois Central College
I am not sure if you are confusing an irrational number being 'never
ending' with being infinite. Infinite would be a circle with a never
ending length. But any circle you can draw or describe obviously HAS
a limit to it's size, although trying to get a very precise measurement
of that size might prove to be a little bit of a problem.
That's where Pi being an irrational number comes in. Although the exact
ratio of the circumference of a circle to it's diameter may be hard to
define, it is always a constant value, slightly higher than 3.14. Okay,
slightly higher than 3.14159. Make that a little higher than 3.14159265...
By Irrational, it is simply meant that the precise value extends out an
apparently infinite number of digits, but the value itself is by no
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Update: June 2012