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Pi
The Most Important Number in the Universe?
TRANSCRIPT
FROM A LECTURE BY PROFESSOR EDWARD B.
BURGER, PH.D.
One of the most
important numbers in our universe is the number Pi or π.
While the origins of
π are not known for certain, we know that the Babylonians approximated π in
base 60 around 1800 B.C.E.
A Definition
The definition of π centers around circles. In
fact, it’s the ratio of the circumference of a circle to its diameter — a
number just a little bit bigger than 3.
We’ll explore humankind’s odyssey to compute,
approximate, and understand this enigmatic number, π. These attempts throughout the
ages truly transcend cultures.
The constant π helps us understand our universe with greater
clarity. In fact, the definition of π inspired
a new notion of measurement of angles, a new unit of measurement.
This important angle measure is known as “radian
measure” and gave rise to many important insights in our physical world.
As for π itself, Johann Lambert showed in 1761
that π is an irrational number,
and later, in 1882, Ferdinand von Lindemann proved that π is not a solution to any polynomial equation
with integers.
However, many questions about π remain unanswered.
Experimenting with Pi
Any discussion of the origins of pi must begin with an
interesting experiment involving circles that we can all try.
Take any circle at all and take the length of the
circumference — which is the length around — and measure it in terms of the
diameter, which is the length across.
You will end up with three diameters and just a little
bit more, and if you look really closely, it’s actually a little bit more than
1/10 of the way extra.
So, this little experiment shows us that that ratio of
the circumference to the diameter is going to be a number that’s around, or a
little bit bigger than, 3.1.
No matter what the size of the circle is, the
circumference is slightly greater than three times its diameter.
We’ve given this fixed, constant value a name, and we
call it π.
So, let’s say this more precisely. The number π is defined to equal the ratio of the circumference
of any circle to its diameter across.
This ratio is constant. No matter what size of circle we
try this with, that number will be always the same. It begins 3.141592653589,
and it keeps going.
We’ll first take a look at the early history
of π and the ancient struggle
to pin down its exact value — first, a word about the symbol π.
We use the Greek letter π for this number, because the Greek word for
“periphery” begins with the Greek letter π.
Now, the periphery of a circle was the precursor
to the perimeter of a circle, which today we call circumference.
The symbol π first appears in William Jones’s 1709
text A New Introduction to Mathematics.
The symbol was later made popular by the great 18th-century Swiss mathematician Leonhard Euler around
1737.
From Babylon to the Bible
Moving from its name to its value, we have evidence that
the Babylonians approximated π in base 60 around 1800 B.C.E.
In fact, they believed that π = 25/8, or 3.125 — an amazing
approximation for so early in human history.
The ancient Egyptian scribe Ahmes, who is associated
with the famous Rhind Papyrus, offered the approximation 256/81, which works
out to be 3.16049.
Again, we see very impressive approximation to this
constant.
There’s even an implicit value of π given in the Bible. In 1 Kings 7:23, a round basin
is said to have 30-cubit circumference and 10-cubit diameter.
Thus, in the Bible, implicitly it states that π equals 3 (30/10).
Not surprisingly, as humankind’s understanding of number
evolved, so did its ability to better understand and thus estimate π itself.
In the year 263, the Chinese mathematician Liu Hui
believed that π =
3.141014.
Approximately 200 years later, the Indian mathematician
and astronomer Aryabhata approximated π with the fraction 62,832/20,000, which is
3.1416—a truly amazing estimate.
Around 1400, the Persian astronomer Kashani
computed πcorrectly to 16
digits.
How to Measure Angles with Pi
Let’s break away from this historical hunt for the
digits of π for a
moment, and consider π as an
important number in our universe.
Given π’s connection
with measuring circumferences of circles, scholars were inspired to use it as a
measure of angle distance.
Now, let’s consider a circle having radius 1. Radius is
just the measure from the center out to the side. It’s half the diameter.
The traditional units for measures of angles are, of
course, degrees.
With degrees, one complete rotation around the circle
has a measure of 360 degrees, which, by the way, happens to approximately equal
the number of days in one complete year and which might explain why we think of
once around as 360.
Instead of the arbitrary measure of 360 to mean
once around the circle, let’s figure out the actual length of traveling around
this particular circle, a circle of radius 1, once around.
So what’s the length? What’s the circumference of
that? Well, let’s see.
If we have a radius of 1, then our diameter is
twice that, 2, and so we know that the once-around will be 2 times π, because the circumference is πtimes the diameter.
Once around will be 2π. One full rotation around, which is an angle of
360 degrees, would be swept out with circumference length of 2π in this particular circle.
In fact, what would be halfway around? Well, that would
be 180 degrees, and we would sweep out half of the circumference, which, in
this case, would be π.
Ninety degrees would sweep out a quarter of the circle,
and for this particular circle, that would have length π/2, or one-half π.
We’re beginning to see that every angle corresponds to a
distance measured partway or all the way around this particular circle of
radius 1.
In other words, for any angle, we can measure the length
of the arc of this circle swept out by that angle.
This arc length provides a new way of representing the measure
of an angle, and we call this measure of angles “radian measure.”
So, for example, 360 degrees = 2π radians, those are the units; 180 degrees
equals π radians,
and 90 degrees would equal π/2 radians.
Remember, all these measures are always based on a
special circle that has radius 1.
Radian Measures and the Power of Pi
It turns out that this radian measure is much more
useful in measuring angles for mathematics and physics than the more familiar
degree measure.
This fact is not too surprising, since radian measure is
naturally connected through the circumference length with the angle, rather
than the more arbitrary degree measure that has no mathematical underpinnings,
but just represents an approximation through a complete year.
The term radian first appeared in print in the 1870s,
but by that time, great mathematicians, including the great mathematician
Leonhard Euler, had been using angles measured in radians for over a hundred
years.
Well, beyond angle measures, π is central in our understanding of our
universe.
In fact, the number π appears in countless important formulas and
theories, including the Heisenberg uncertainty principle and Einstein’s field
equation from general relativity.
So it’s a very, very important formula, a very important
number indeed.
From the
lecture series Zero to
Infinity: A History of Numbers
Taught by Professor Edward B. Burger, Southwestern University
Taught by Professor Edward B. Burger, Southwestern University
Dr. Edward B. Burger is
President of Southwestern University in Georgetown, Texas. Previously, he was
Francis Christopher Oakley Third Century Professor of Mathematics at Williams
College. He graduated summa cum laude from Connecticut College, where he earned
a B.A. with distinction in Mathematics. He earned his Ph.D. in Mathematics from
The University of Texas at Austin. Professor Burger is the recipient of many
teaching awards and accolades. He was named by Baylor University as the 2010
recipient of the Robert Foster Cherry Award for Great Teaching for his proven
record as an extraordinary teacher and distinguished scholar. Baylor University
lauded Dr. Burger as truly one of our nation's most outstanding, passionate,
and creative mathematics professors.
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Johann Lambert, the
Swiss mathematician was the first to prove that pi was an irrational number.
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William Jones, the Welsh mathematician
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