Happy Pi Day and The Hierarchy of Numbers

Young Astronomers Blog, Volume 28, Number 6.

It’s March 14, 2020. So, HAPPY PI DAY! Yes, Pi = 3.14, get it?

Anyway, Pi is a number. It’s one of many numbers. Pi is actually a calculated number. It is the ratio of the circumference of a circle to its diameter. Check it out – it always works (sort of). It is also an irrational number. It can’t be expressed as the ratio of two integers and the number of decimals goes on and on, never repeating.

You would think a number is a simple thing; 1, 2, 3 and so on. But there is an entire hierarchy to our numbering system.

Start with the numbers 1, 2, 3, and so on. These are the natural numbers. Then add zero (0) and you have the whole numbers. Numbers can be negative, -1, -2, -3, and so on. These along with the whole numbers are the integers. Take the ratio of any two integers and you have the rational numbers, 1/2, 1/3, 1/4, and so on. Yes 1/3 is a rational number although in decimal form it is .33333333333 on and on. Because the decimals repeat it is rational. Numbers, such as Pi, cannot be expressed as the ratio of two integers and are called irrational numbers. The rational and irrational numbers are grouped together into the real numbers. If we take the square root of a positive number, such as 4, we have 2 or -2 (2 x 2 = 4 and -2 x -2 = 4). But what if we take the square root of a negative number? No number can be multiplied by itself to get a negative number. We call these numbers imaginary numbers. Imaginary numbers are usually written in terms of i (the square root of -1). A real number plus an imaginary number is called a complex number.

Of course, all these sets of numbers are based on ten symbols (base 10 system). We can also define numbers based on a different number of symbols, such as two (binary system) or 16 (hexadecimal system). We’ll leave this for later.

Now, how many numbers are there? I bet you would say an infinite number and you would be correct. But are all infinities the same? No! All the numbers up through the rational numbers are countable. That is, you can line each of these sets of numbers up and count them, 1, 2, 3, 4, and so on or 1/2, 1/3, 1/4, 1/5 and so on. Each of these sets contains a countable infinite number of elements denoted Aleph-null. The set of all irrational numbers is also infinite. If I take any two irrational numbers, I can always find more irrational numbers between them. I can’t line them up and count them. Therefore, the set of irrational numbers is infinite, but not countable and is denoted Aleph-one. So, there are more infinite irrational numbers than infinite rational numbers.

Going back to Pi, remember it is the ratio of a circle’s circumference to its diameter, but only if our geometry is flat! On a flat geometry, there is a zero curvature and the sum of the angles in a triangle equals 180^o. But we live on a curved surface (that is the Earth) and the Earth’s geometry is not flat, it is elliptic (spherical) with a positive curvature. So, if we mapped out a huge triangle, we would discover that the sum of the angles would be greater than 180^o. And, if we draw a huge circle, the ratio of the circumference to the diameter depends on the size of the circle. There are other geometries, such as one that looks like a saddle (hyperbolic) with a negative curvature. Here the sum of the angles in a triangle is less than 180^o and the ratio of a circle’s circumference to its diameter also varies.

Flat geometry is called Euclidean and other geometries Non-Euclidean. These different geometries are usually discussed using parallel lines.

• In Flat geometry only one parallel line can be drawn through a point next to a line.
• In Elliptic (spherical) geometry no parallel lines can be drawn through a point next to a line.
• In Hyperbolic geometry an infinite number of parallel lines can be drawn through a point next to a line.

Again, the surface of the Earth is spherical, so airplanes travel “straight lines” above this curved surface. We call these lines geodesics, which are segments of what are called great circles. To fly from New York to London, a plane will appear to travel a curved path to the north. However, this is the shortest distance between the two cities: a geodesic on a curved surface.

Einstein in his General Theory of Relativity used geometry to explain gravity. Without mass, space-time is flat. But if an object with mass appears, its mass curves space-time, and there is a different geometry. Other objects around it follow lines (geodesics) in curved space-time and this is what we see as gravity.

What does this have to do with astronomy you ask? Well, astronomers would like to know what the geometry of the universe is. It appears to be flat, but this might be because it is so large, that the local geometry appears flat, just as the local surface of the Earth appears flat. But, out beyond our horizon, it could have a different shape.

Anyway, happy Pi day. Remember, it is just a number.