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.

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