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At a Glance - Irrational Numbers

Have your parents ever accused you of being irrational? Probably when you were doing something like throwing a hissy-fit in Toys R Us just because they wouldn't buy you a new Thomas the Tank Engine train, right? That was just last week.

Well, irrational numbers can be just as big a pain to deal with. They're a little weird, in that they can't be written as fractions. How in the world did anyone ever find a need for a number that can't be written as a fraction, you ask? You ask a lot of good questions.

Let's take a look at how and why mathematicians stumbled upon irrational numbers.

Solving Equations

What number times itself equals 1? That's easy: 1 × 1 = 1. What number times itself equals 9? Boom: 3 × 3 = 9. What number times itself equals 2? Ahhh...now you see where we're going with this. There is no rational number that, when multiplied by itself, gives us an answer of 2. For this reason, mathematicians made up just such a number, and called it √2. Mathematicians don't generally have a ton of friends, so they have to resort to naming everything around them so they won't feel so alone. Sometimes they'll even take √2 out on walks.

Why do we need √2 so badly? Here's an example of when it may come in handy.

The Pythagorean Theorem states that a2 + b2 = c2. In other words, one leg of a right triangle multiplied by itself plus the other leg multiplied by itself equals the hypotenuse multiplied by itself. In a very basic right triangle where each of the legs has a measure of 1, the hypotenuse would be exactly √2,  since 12 + 12 = (√2)2.

So say you work at Subway and you're in charge of cutting those perfect right triangle cheese slices. How in the world would you ever be able to do your job if you couldn't figure out the length of the cheese's hypotenuse?


Many sequences of rational numbers come very close to another number, without ever quite getting there.

Sample Problem

The sequence 1/2, 2/3, 3/4, 4/5, 5/6,... gets closer and closer to 1, but as long as you keep following this pattern, you're never going to hit it precisely.

The sequence 1, 3/2, 7/5, 17/12, 41/29,... approaches √2, but √2 is not rational. This means that our set of rational numbers has a "hole" in it! If we try to fill in the hole, we're filling it with something that's not a rational number. And if we try to patch up the hole, we're probably going to run out of fabric. And we really need the rest of that in order to make those doilies.

Here are some of the holes in our number line:

By taking a look at different sequences we can find lots of other "holes" in the rational numbers, and each time we fill them in with something, we create an irrational number. Whenever possible, we should try filling it in with Crack-B-Gone liquid cement, because that stuff is amazing.

This is going to sound really crazy (like, crazier than filling in a sequence of numbers with liquid cement), but it turns out that there are a lot more irrational numbers than rational numbers. The reason this seems a little crazy is that we know there are infinitely many rational numbers, so how can there be more of anything else?

The thing is, you can systematically count rational numbers, but there are so many irrational numbers in between each pair of rational ones that there are just way too many to even systematically count them. For this reason, we say that even though both the rationals and the irrationals are infinite, the irrationals are a much bigger infinity than the rationals. How's that for brain stretching? Just when you thought you had this whole "infinity" thing figured out, right?

A couple other popular irrational numbers are π (approximately 3.14159...), which is the ratio of the perimeter of a circle to its diameter, and √5. What we're getting at is that irrational numbers are pretty abstract. 

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