Irrational Behavior

Here’s some random stuff I learned this week:

File this one under “duh”: I’ve always assumed that the terms ‘rational’ and ‘irrational’ when relating to numbers was an indication of their behavior. 1/4, for instance, is a rational number, because it equates to 0.25 – it behaves in a rational way.

1/3, on the other hand, is 0.333333…, never ending – it behaves in an irrational manner.

Not only is the above NOT true mathematically, that is NOT where the terms come from.

Rational numbers are numbers that can be expressed as the ratio of two integers – i.e., a fraction:

1/4 = 0.25

1/3 = 0.33333…

98/100 = 0.98

On the other hand, numbers that CANNOT be expressed as a ratio between two integers – for example, π, e, or the square root of 2 – are all irrational numbers:

π = 3.1415926535…

e = 2.7182818…

√2 = 1.4142135…

The Golden Ratio (1.6180339…) is another example of an irrational number. The difference between the numbers in the Fibonacci Sequence, to use one example, follow the Golden Ratio as the sequence increases.

And speaking of cool numbers, if you divide 987654321 by 123456789 on your phone calculator, you may get an answer of 8. I’ve always enjoyed this, as I’ve always considered 8 to be my lucky number, based solely upon a fortune cookie I got when I was a kid. (This is also why I say my favorite color is red – the fortune cookie told me so – when it’s actually blue.)

However, you’re more likely to get an answer like 8.00000007, which seems much more accurate. But still close to 8, right?

But turn your phone calculator on its side to engage the scientific calculator functions, and the answer becomes 8.000000072900001. Much more accurate still!

But it doesn’t end there. The full remainder of this operation is another example of digits that repeat to infinity:

So, how many irrational numbers are there? That’s an interesting question, and involves countable versus uncountable infinity.

We already know that there are an infinite number of counting (or whole) numbers. “Countable” in this context doesn’t necessarily mean we can count them, just that it is possible to count them in theory. We can say n=1, for instance, and then apply the formula n=(n+1) to generate the series of natural numbers that increment by one.

Real numbers, on the other hand, are uncountable. In fact, there are more real numbers between 0 and 1 then there are natural numbers between 0 and infinity.

Think of it this way: where would you start counting the real numbers between 0 and 1? You can start with, say, 0.1. Well, there’s at least one number between 0 and 0.1 – it’s 0.01. And there’s at least one number between 0 and 0.01 – 0.001. And you can do this literally forever, and still not determine where to start counting the real numbers between 0 and 1.

Irrational numbers are similar – they are part of the set of uncountable infinite numbers. So in a sense, even though both the sets of rational numbers and irrational numbers are infinite, there are more irrational numbers than there are rational numbers, just as there are more real numbers than there are natural numbers, even though both sets are infinite.

My brain is now a smoldering heap, so that will be it for today. But tomorrow I have a nifty little thing to share regarding an interesting characteristic of numbers that are a power of 2, so until then – thanks for reading!

Further reading:

Math Is Fun Irrational Number Page: https://www.mathsisfun.com/irrational-numbers.html

Quora discussion on 987654321/123456789: https://www.quora.com/With-these-numbers-as-mirrors-why-does-987-654-321-divided-by-123-456-789-equal-8-and-is-this-a-coincidence-or-is-there-more-to-it

Maths Doctor UK rational vs. irrational discussion: https://mathsdoctor.co.uk/revision-help/gcse/number/rational-and-irrational-number/

Irrational Pi header courtesy of this Mathologer video.

Is your name in π?

Mmmm…pi pie.
Image courtesy of pillsbury.com

I preface this by saying zero original research went into this other than my interest in the subject and the God-given ability to Google stuff.

You probably already know that we use a base-10 number system to count stuff – 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. But there are other number bases – some of which you already know about, even if you didn’t realize it.

The language of computers, binary, is nothing more than a base-2 system, where every digit can be represented by zeroes and ones. Thus, when converting from base-2 to base-10, you’d count 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010…and so on. Here is what these base-2 numbers equate to:

0 = 0

1 = 1

10 = 3

11 = 4

100 = 5

101 = 6

110 = 7

and so on, so that (for example):

111001 = 57

(Links are at the bottom of the page if you want to explore this, as well as the other things discussed below)

But there are other bases – an infinite number of bases, actually. The Babylonians used a base-60(!) number system, and Mayans used a base-20 system. Hexadecimal (base-16) systems are relatively common, used in everything from computer programming (as a way to condense binary) to our everyday usage of pounds and ounces.

In fact, you use an octovigesimal system every day. Better known as base-28, this is what our current Gregorian calendar is (loosely) based upon.

So, what if I wanted to see if my name is represented in the digits of π? How would this even be possible? What would be the best way to convert the base-10 digits of π to letters?

Well, we’d want to use base-27 for this. Base-27 is represented thus:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J, K, L, M, N, O and P

To convert this to alpha, we’d assign 1=A, 2=B, 3=C and so on, letting zero represent a space, i.e. 0=””.

Doing this let’s us then convert π in base-27 from 3.3MQ53… to C.CVEZC. And we’re off and running!

(“Why not use base-26?” I hear you thinking. Well, we want to be able to represent each letter of the alphabet with a letter so that A=1, B=2, etc., and we need a 27th place to capture our zero value, so base-27 it is!)

There is a conversion tool online (also linked below) that allows you to enter your name (or any word, really) to see if it is represented in this converted base-27 π. My name, STEVE, appears starting with the 8,857,158th digit of π:

Image courtesy of Dr Mike’s Math Games

Some fun facts about the occurrence of words in (this version of) π-27:

The first spelled-out number appears at the 4,259th position: SIX

The first ordinal number appears at the 222,386th position, and of course it’s FIRST

JEDI appears at position 1,126,698, but the SITH overpowers them at the earlier position of 804,693.

Within the first 30,000,000 digits of our converted π-27, the word MATHEMATICS does not appear. Hopefully it will show up in the second group of thirty million characters…

And where does PI appear? Not surprisingly (given that it’s only two letters long), it pops up for the first time starting at position 18.

So, click on the link below (or right here if you’re impatient) and see where your name first appears!

One caveat, though – this database only includes the first ~30 million converted digits of π, and your chances of finding your name decreases substantially as the length of your name grows longer. If your name is three (or fewer) letters long, you have virtually a 100% chance of finding it. Five letter names (like mine) carry a 56% chance of success, but seven letter names have virtually zero chance of being represented.

Good luck, and let me know in the comments if you’ve found your name in the first thirty million digits of π!

Here are the links I promised:

Check π for your name here: http://www.dr-mikes-math-games-for-kids.com/your-name-in-pi.html

How π in base-27 works: https://www.atractor.pt/mat/fromPI/pib27-_en.html

How counting in different bases works: http://mae.engr.ucdavis.edu/dsouza/Classes/ECS15-W13/counting.pdf

Math.com’s π facts page: http://www.math.com/tables/constants/pi.htm

More stuff about π-27: http://www.dr-mikes-math-games-for-kids.com/your-name-in-pi-notes.html