When you hear the words "rational" and "irrational," it might bring to mind the difference between, say, the cool, relentlessly analytical Mr. Spock and the hardheaded, emotionally volatile Dr. "Bones" McCoy in the "Star Trek" television and movie universe. Unless you're a mathematician, though, you probably don't think of ratios between integers versus square roots, the sort of stuff that makes the non-mathematicians among us feel as confused as we do hearing Queen's "Bohemian Rhapsody" sung in Klingon.
But in the realm of mathematics, where words sometimes have specific meanings that are very different from everyday usage, the difference between rational and irrational numbers doesn't have anything to do with reasoning and logic versus raw emotional urges.
Remember the Word 'Ratio'
"In remembering the difference between rational and irrational numbers, think one word: ratio," explains Eric D. Kolaczyk. He's a professor in the department of mathematics and statistics at Boston University and the director of the university's Rafik B. Hariri Institute for Computing and Computational Science & Engineering.
"If you can write a number as a ratio of two integers (e.g., 1 over 10, -5 over 23, 1,543 over 10, etc.) then we put it in the category of rational numbers," Kolaczyk explains in an email. "Otherwise, we say it is irrational."
You can express either a whole number or a fraction — parts of whole numbers — as a ratio, with an integer called a numerator on top of another integer called a denominator. You divide the denominator into the numerator. That can give you a number such as 1/4 or 500/10 (otherwise known as 50).
Irrational numbers, in contrast to rational numbers, are pretty complicated. As Wolfram MathWorld explains, they can't be expressed by fractions, and when you try to write them as a number with a decimal point, the digits just keep going on and on, without ever stopping or repeating a pattern.
So what sort of numbers behave in such a crazy fashion? Basically, ones that describe complicated things. Perhaps the most famous irrational number is pi — sometimes written as π, the Greek letter for p — which expresses the ratio of the circumference of a circle to the diameter of that circle. As mathematician Steven Bogart explained in this 1999 Scientific American article that ratio will always equal pi, regardless of the size of the circle. Since the earliest attempts to calculate pi were performed by Babylonian mathematicians nearly 4,000 years ago, successive generations of mathematicians have kept plugging away, and coming up with longer and longer strings of decimals with non-repeating patterns. In 2019, Google researcher Hakura Iwao managed to extend pi to 31,415,926,535,897 digits, as this Cnet article details.
Sometimes, a square root — that is, a factor of a number that, when multiplied by itself, produces the number that you started with — is irrational number, unless it's a perfect square that's a whole number, such as 4, the square root of 16. One of the most conspicuous examples is the square root of 2, which works out to 1.414 plus an endless string of nonrepeating digits. That value corresponds to the length of the diagonal within a square, as first described by the ancient Greeks in the Pythagorean theorem.
Why Do We Use the Words 'Rational' and 'Irrational'?
Why do we call them rational and irrational? That seems to be a little murky. "We do indeed typically use 'rational' to mean something more like based on reason or similar," Kolaczyk says. "Its use in mathematics seems to have cropped up as early as the 1200s in British sources (per the Oxford English Dictionary). If you trace both 'rational' and 'ratio' back to their Latin roots, you find that in both cases the root is about 'reasoning,' broadly speaking."
What's clearer is that both rational and irrational numbers have played important roles in the advance of civilization. While language probably dates back to around the origin of the human species, numbers came along much later, explains Mark Zegarelli, a math tutor and author who has written 10 books in the "For Dummies" series. Hunter-gatherers, he says, probably didn't need much numerical precision, other than the ability to roughly estimate and compare quantities.
"They needed concepts like, 'we have no more apples,'" Zegarelli says. "They didn't need to know, 'we have exactly 152 apples.'"
But as humans began to carve out plots of land to create farms, erect cities and manufacture and trade goods, traveling farther away from their homes, they needed a more complex math.
"Suppose you build a house with a roof for which the rise is the same length as the run from the base at its highest point," Kolaczyk says. "How long is the stretch of roof surface itself from top to outer edge? Always a factor of the square root of 2 of the rise (run). And that's an irrational number as well."
In the technologically advanced 21st century, irrational numbers continue to play a crucial role, according to Carrie Manore. She's a scientist and a mathematician in the Information Systems and Modeling Group at Los Alamos National Laboratory.
"Pi is an obvious first irrational number to talk about," Manore says via email. "We need it to determine area and circumference of circles. It's critical to computing angles, and angles are critical to navigation, building, surveying, engineering and more. Radio frequency communication is dependent on sines and cosines which involve pi." Additionally, irrational numbers play a key role in the complex math that makes possible high-frequency stock trading, modeling, forecasting and most statistical analysis — all activities that keep our society humming.
The list could go on. "In fact, in our modern world, it almost makes sense to instead ask, where are irrational numbers NOT being used?" Manore says.