Learn what irrational numbers are, where they fall along the number line, and how they relate to the rest of the world of numbers.
In the last article, we began to put names with some of the faces of the numbers that we know and love. In particular, we learned that all the integers and all the fractions made from the ratio of two integers together are called rational numbers. But the fact that we bothered to do that at all hints at the idea that there are other numbers that are not rational. And it’s these non-rational numbers—headlined by what are known as irrational numbers—that we’ll be talking about today.
Recap: Rational Numbers
Before we jump into the deep end of the pool and start talking about irrational numbers, let’s first recap what makes some numbers rational. The word “rational” here comes from the word “ratio” which tells us that rational numbers are all the numbers that can be represented as the ratio of two integers. In other words, they all can all be written as:
“any rational number” = “integer #1” / “integer #2”
(as long as “integer #2” here isn’t zero!). So what numbers fit this bill? Well, as we talked about last time, the rational numbers include all the integers (since we can write an integer like 5 as the ratio 5/1) and all the fractions that can be written as the ratio of two integers too.
How to Picture Rational Numbers
Let’s now take a few minutes and think about how to picture the meaning of rational and eventually irrational numbers. Imagine you’ve been given a meter stick and then asked to measure the lengths of a bunch of lines drawn on the ground. Your goal with each of these lines is to figure out whether or not they are a rational number of meters long.
You find that the first two lines are exactly 1 and 2 meter sticks long, which means that they both have lengths that are rational numbers—they’re 1/1 and 2/1 meters long. When you go to measure the length of the next line on the ground though, you find that it’s somewhere between 1 and 2 meter sticks long. So you decide to cut your meter stick in half and try again. This time you find that the line on the ground is exactly three half-meter sticks long. So the length of the line is 3/2 of a meter long, which as a ratio of two integers, is a rational number.
How to Picture Irrational Numbers
But then you come to a totally different type of drawing. This drawing isn’t just a single line but is instead made of three lines connected together to form a special type of triangle called a right triangle (we’ll talk more about exactly what that means in a future article). You measure the lengths of two of the sides of this triangle and find that they’re each exactly one meter long. Since the goal here is to figure out whether or not all the lines on the ground are some rational number of meters long, you now need to measure the length of the third side of the triangle.
Right away you see that the third side of the triangle is longer than your meter stick. Instead of breaking up your stick and trying to measure the total length of the entire line as you’ve done so far, you decide to start by erasing the one meter of the line that you’ve measured so that you only have to measure the length of the leftover part. After doing that, you break up your meter stick into 10 equal pieces and find that the leftover length is a little longer than four of these 1/10 of a meter long pieces. In other words, the total length of the line so far is 1 + 4/10 = 1.4 meters long.
You next erase the 0.4 meters of the line you just measured, and then you chop one of those 1/10 of a meter long pieces of your meter stick into 10 more even pieces to see if they’ll fit evenly into what’s left of the line. But much to your chagrin, you find that the line is still a little longer than one of these now 1/100 of a meter long pieces.
So you keep on breaking up your meter stick into tinier and tinier pieces to measure smaller and smaller lengths of the third side of the triangle. But no matter how tiny those pieces get you can never get an even number to fit exactly. In fact, you can do this literally forever and you’ll never finish. Why? Because the length of the third side of that particular triangle is the square root of 2—an irrational number!
What are Irrational Numbers?
So what exactly are irrational numbers? Well, irrational numbers are numbers that cannot be written as the ratio of two integers. And it turns out that when a number can’t be written as the ratio of two integers, it also can’t be written using a finite number of decimal digits. For example, the square root of 2, which I just told you is irrational, is equal to 1.414213562… and on and on forever. In other words, irrational numbers require an infinite number of decimal digits to write—and these digits never form patterns that allow you to predict what the next one will be. Which means that the only way to find the next digit is to calculate it.
How to Write Irrational Numbers as Decimals
You can check that this is true by using your calculator to calculate the decimal form of some rational and irrational numbers. For example, here are the decimal representations of some rational numbers:
3/4 = 0.75
23/16 = 1.4375
1/3 = 0.3333…
[[AdMiddle]Hmm, the first two work as expected since they show that these two rational numbers can be represented by a finite number of decimal digits. But what about 1/3 = 0.3333…? Well, it’s true that this decimal goes on forever. However, the infinite number of decimal digits in 1/3 simply repeat over and over again. In other words, we can tell what the next digit is going to be without actually calculating it—once you know the pattern, you see that it’s always 3. So the decimal representation of some rational numbers can go on forever, but those digits will always form a predictable repeating pattern. We’ll talk more about this next week.
So those are rational numbers, now let’s look at some examples of irrational numbers:
the square root of 2 = 1.41421… (and on and on in a never repeating pattern)
the square root of 3 = 1.73205…
π = 3.14159…
ø = 1.61803… (this is the golden ratio that we talked about before)
How to to Check if a Number is Rational or Irrational
We can summarize all of this by saying that:
Rational numbers can be written with a finite number of (possibly repeating) decimal digits.
Irrational numbers require an infinite number of decimal digits to write.
So the quick and dirty tip for checking whether a number is rational or irrational is to write it in decimal form. If the decimal goes on and on forever and never stops or begins to repeat predictably, it’s irrational. If the decimal stops after a finite number of digits or begins to repeat predictably, it’s rational.
Okay, that’s all the math we have time for today. Please email your math questions and comments to firstname.lastname@example.org. You can get updates about the Math Dude podcast, the “Video Extra!” episodes on YouTube, and all my other musings about math, science, and life in general by following me on Twitter. And don’t forget to join our great community of social networking math fans by becoming a fan of the Math Dude on Facebook.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading math fans!