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Irrational Numbers – More than Infinite

Irrational numbers are numerous and yet, you can’t think of any of them. “But I can think of pi!” you might say to yourself. Well, ok. How does it go, then? Is it a fraction of some number we are already familiar with? No it isn’t.

Today, we have many things that are absolutely too complicated to understand completely, such as quantum physics. Unless you spend your whole life researching it, of course. But there is something much more familiar yet still surprisingly complicated: the irrational numbers.

Let’s think of it this way: You start in the morning with a notebook and start writing random decimals to a number. For example, you start with 0.628465963 and add random digits until you have five pages full at lunchtime. What you now have in your hands, it’s a rational number, not irrational. That’s because you can write it as a fraction. The numerator will be all the digits after the decimal point and the denominator is a power of ten; it has one and as many zeros as your number has decimals.

So, any number you can create decimal by decimal will be a rational number. Irrational numbers will look like special cases, square roots etc. And you would probably think: “Well, there must be only a limited amount of irrational numbers, then.” 

But that is not the case. There are a lot more irrational numbers than rational numbers. That’s where the infinity really comes into play. If you think of a decimal number as an endless journey to infinity, digit by digit, you realize that every time we take the next step, there will be ten directions we can go. And at the next step, again ten different directions. So numbers we get this way is countless. And only rare cases are rationals, like those that happen to have repeating decimals. If you want a rounded estimate, 0% of all real numbers are rational, the rest about 100% are irrational. 

Whoa! That was something. If you want to familiarize with the subject more, I made a free pdf material that you can download from my tpt shop. It has a concept understanding crossword and a story about how we even know that irrationals really exist.

If you need to teach this subject to let’s say, eight graders, there is presentation & worksheet bundles in my shop: 

Version for Google Slides™

Version for PowerPoint

These are essentially the same, but one has presentations for PowerPoint and the other for Google Slides™.

Hopefully you enjoyed this article! Irrationals are so intriguing, because they are easy and very complicated at the same time. If you want to challenge something, please leave a comment. 

The writer used to be a math teacher in Finland and has a master’s degree in math.