What are irrational numbers? Good question, but don’t worry! Irrational numbers aren’t unreasonable, though they may seem a bit mysterious. After all, one of my favorite numbers, pi (π = 3.14159…), is irrational! Irrational numbers are just particular kinds of real numbers — specifically, those numbers that are not rational.
In this article, we’ll review rational and irrational numbers, focusing on the unique properties of the irrationals. We’ll learn how important and ubiquitous these fascinating numbers are. And we’ll even talk about a famous murder allegedly caused by an irrational number!
Real, Rational, and Irrational Numbers
By definition, a real number is irrational if it is not rational.
But what exactly is a real number? And what are the rational numbers?
We’d better start at the beginning!
The Ever Expanding Number System
People have been using numbers since as long as we have records and artifacts to prove it.
At first, people only used the counting (or natural) numbers, 1, 2, 3, 4, 5, and so on.
So for most of human history, the only “real” numbers were those that you could count things with.
Image by Field Museum of Natural History
Eventually, as civilizations grew it became necessary to record other kinds of quantities. For example, how do you “count” the amount of water in a jar? When the jar is full, you can call it 1 jar. But take some water out, and now you have less than “1 water.” We needed numbers to record concepts like “half” and “three-quarters” etc.
Fractions to the Rescue!
Fractions are ratios of counting numbers, a/b. We’re not going to worry about whether the fraction is proper or improper; all kinds of fractions are welcome here! 1/2, 3/7, 18/11, 9393827/1000001 — these are all fractions.
Fractions, together with the counting numbers, constitute half of what we call rational numbers. What are we missing, you ask? Well, so far we haven’t said anything about negatives or zero. That’s because those concepts didn’t become widely accepted by us humans until further along in history!
Don’t worry — we’ll get there!
Pythagoras and the Cult of Number
Remember old Pythagoras?
Yeah, the one with the famous theorem… you know, a2 + b2 = c2 for a right triangle. Well, he was also a cult leader and possibly also a murderer!
You see, the Pythagoreans were a Cult of Number. They believed that “All is Number.” However, for the Pythagoreans, the only numbers that they recognized were natural numbers and ratios of natural numbers (in other words, the positive rational numbers).
Ironically, the theorem that bears his name caused a great disturbance in Pythagoras’ cult. As the story goes, while out in a ship on the sea, one of the Pythagoreans made the mistake of proving that there were other numbers that were not rational.
Consider the simplest right triangle possible, one that has base and height both equal to 1 unit. What length must the hypotenuse be?
Applying the Pythagorean Theorem with a = 1 and b = 1, we find that c2 = 2. In other words, c is a number whose square is equal to 2, what we would call the square root of 2 nowadays (√2).
Well this wouldn’t be so bad, except that there is no natural nor fractional number whose square is equal to 2. In other words, √2 is not a rational number!
There are some good approximations to √2 by fractional numbers: 7/5, 141/100, 141421356237/100000000000, but none of these fractions is exactly right.
This revelation was apparently too much for Pythagoras, who had the unfortunate fellow thrown overboard and drowned.
The Complete Real Number Line
It took hundreds of years before the real and rational number systems were rigorously formalized by mathematicians. Along the way, we adopted zero and the negative numbers, expanding the natural numbers to the integers, and the fractional numbers to the full set of rationals. (For more about integers, check out: What Are Integers?)
Modern definitions of real numbers essentially boil down to this: A real number is any length that can be measured on a number line, beginning at a special point called the origin (0).
So, every counting number, fraction, their negatives, zero, and even that troublesome √2 are real numbers.
Spotting Rationals and Irrationals
One nice thing about real numbers is that they all have decimal expansions.
All integers have decimal expansions with no fractional part: -9 = -9.0.
Sometimes the expansion of a number terminates, as in 2.25. This includes the integers of course.
Sometimes it doesn’t terminate but instead has a block of numbers that repeat forever. For example, -2.836 = -2.836363636… (The bar on top of the digits 36 mean to repeat those digits forever.)
It’s a general fact that if a decimal either terminates or has a repeating block, then it must be rational. In other words, there is some fraction that represents the number exactly. Even integers can be expressed as fractions.
- -9 = -9/1
- 2.25 = 9/4
- -2.836 = -156/55
Irrational numbers, on the other hand, can be identified as having decimal expansions that do not terminate and have no block of digits that repeat forever.
There are a few basic cases for Irrational Numbers:
- The digits go on forever in more or less an arbitrary way (random). This is the case with most irrational numbers, including any square root of a number that’s not a perfect square (√2, √3, √5, etc.) and special constants such as pi (π = 3.1415926…) and e (= 2.718281828459…).
- The digits go on forever in some kind of pattern, but it’s not a repeating block of the same length. For example, 0.1010010001000010000010000001…, is irrational, even though there is an obvious pattern in its digits. The problem is that the block of digits grows each time, rather than simply repeating.
See if you can identify which of the following numbers are rational and which are irrational. Be careful; some of them are tricky!
-5.6 8/3 π/2 √49 -0.123123123… 0 3.14 1.112123123412345…
- -5.6 is a terminating decimal. Therefore, it’s rational. If you want to know what fraction represents it, first write -5.6 = -56/10, and then reduce: -28/5.
- 8/3 is clearly a fraction of two whole numbers — thus rational.
- π/2 looks like a fraction. However, the numerator is π (pi). So, π/2 cannot be rational, because that would imply π itself would be rational, contradicting what we know about the irrational number π. Therefore π/2 is irrational.
- √49 = 7, and so it’s a rational number. In fact, this number is an integer. (Not every radical is irrational.)
- -0.123123123… is rational because of the repeating block of digits 123.
- 0 is an integer, so it’s also rational.
- 3.14 is just an approximation for π While π is definitely irrational, this approximation, 3.14, is a terminating decimal, which makes it rational.
- 1.112123123412345… does exhibit a pattern, but it’s not a strictly repeating pattern of the same block of numbers forever. If it’s not rational, then it must be irrational.
Irrational Numbers in Geometry and Other Areas of Math
We’ll end with a topic that shows why the irrational numbers aren’t so strange. In fact, they seem to pop up everywhere in mathematics.
The irrational numbers spring up naturally in Geometry.
- As we saw above, the Pythagorean Theorem for right triangles implied that the hypotenuse of a certain triangle is √2. Other square roots arise as hypotenuse lengths of various right triangles.
- The ratio of the circumference of any circle to its diameter is equal to the irrational number π. The area formula for a circle of radius r is: A = πr2. So anytime you compute the circumference or area of a circle, you’re working with irrational numbers.
- There is a special rectangle called the Golden Rectangle whose sides are in the ratio φ = 1.6180339887…, an irrational number called the Golden Ratio. Golden Rectangles often show up in classical Greek and Roman architecture and art, because their proportions are supposed to be the most “perfect” and pleasing to the eyes. (I’ll reserve judgement on what is pleasing or not in art.)
Anyway, the Golden Rectangle has one amazing property that sets it apart from all other rectangles. Starting with a Golden Rectangle with long side a and short side b, if you attach a square of side length a to one of the long sides, the resulting rectangle is also Golden!
The Golden Ratio also shows up in nature as the Golden Spiral that some creatures seem to use as a blueprint for their shells.
In other areas of mathematics, including financial math, analysis, and dynamics, there’s a very famous irrational number called Euler’s Number (e = 2.718281828459045…) that seems to appear out of nowhere to solve all the world’s problems (well, not all of them).
Irrational numbers aren’t so crazy after all! But they’re definitely mysterious and fascinating.