When you hear the words “rational” and “irrational,” they may bring to mind the relentlessly analytical Spock from “Star Trek.” But if you’re a mathematician, you probably think about ratios between whole numbers and square roots.
In the field of mathematics, where words sometimes have special meanings very different from everyday usage, the difference between rational and irrational numbers It has nothing to do with emotions. Since there are infinite irrational numbers, you better have a basic understanding of them.
Properties of Irrational Numbers
“When remembering the difference between rational and irrational numbers, think of one word: ratio,” explains Eric D. Kolaczyk. He is a professor in the department of mathematics and statistics at Boston University and director of the university’s Rafik B. Hariri Institute for Computer and Computational Science and 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 says in an email. “Otherwise we would say it is unreasonable.”
You can express a whole number or a fraction (parts of whole numbers) as a ratio by using one whole number, called the numerator, on top of another whole number, called the denominator. You divide the denominator by the numerator. This might give you a number like 1/4 or 500/10 (aka 50).
Irrational Numbers: Examples and Exceptions
Irrational numbers, unlike rational numbers, are quite complex. As Wolfram MathWorld explains, these cannot be expressed as fractions, and when you try to write them as a number with a decimal point, the numbers go on and on without ever stopping or repeating a pattern.
So what kind of numbers behave so wildly? Those that basically describe complex things.
pi
Perhaps the most famous irrational number is pi, sometimes written as the Greek letter π, meaning “p”, which expresses the ratio of the circumference of a circle to the diameter of that circle. As mathematician Steven Bogart explained in a 1999 Scientific American article, this ratio will always equal pi no matter the size of the circle.
Since Babylonian mathematicians tried to calculate pi about 4,000 years ago, successive generations of mathematicians have continued to struggle, trying to find increasingly longer strings of decimal expansions with non-repeating patterns.
In 2019, Google researcher Emma Hakura Iwao managed to increase the number of pi to 31,415,926,535,897 digits.
Some (But Not All) Square Roots
Sometimes a square root, the factor of a number that when multiplied by itself gives you the number you started with, is an irrational number; Unless it’s a perfect square, which is an integer like 4. root of 16.
One of the most notable examples is the square root of 2, which corresponds to 1,414 plus an infinite sequence of non-repeating digits. This 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’?
“We actually usually use the word ‘rational’ to mean something more based on logic or something like that,” Kolaczyk says. “Its use in mathematics appears to have appeared in English sources around the 1200s (according to the Oxford English Dictionary). If you trace the concepts of ‘rational’ and ‘ratio’ to their Latin roots, you’ll find that in both cases the root generally means ‘reason.’ It’s about execution.”
What is clearer is that both irrational and rational numbers play important roles in the advancement of civilization.
Mark Zegarelli, a math teacher and author who has written 10 books in the “Beginners” series, explains that while the history of language probably dates back to the origins of the human species, numbers appeared much later. Hunter-gatherers probably didn’t need much numerical precision, he says, other than the ability to roughly estimate and compare quantities.
“They needed concepts like, ‘We don’t have any more apples,’” Zegarelli says. “They didn’t need to know, ‘We have exactly 152 apples.'”
But as people began to cultivate land by establishing farms, building cities, and traveling far from home to produce and trade goods, they needed more complex mathematics.
“Let’s say you’re building a house with a roof and the distance from the base to the highest point is the same length,” Kolaczyk says. “What is the length of the roof surface from the top to the outer edge? It’s always a factor of the square root of 2 of the rise (length). And that’s an irrational number.”
The Role of Irrational Numbers in Modern Society
According to Carrie Manore, irrational numbers continue to play a vital role in the technologically advanced 21st century. He is a scientist and mathematician in the Information Systems and Modeling Group at Los Alamos National Laboratory.
“Pi is the first irrational number to talk about,” Manore says via email. “We need it to determine the area and perimeter of circles. It’s critical for calculating angles, and angles are critical for navigation, construction, research, engineering and more. Radio frequency communications depend on sines and cosines involving pi.”
Additionally, irrational numbers play an important role in the complex mathematics that enable much of high-frequency stock trading, modelling, forecasting and statistical analysis; All these are activities that make our society hum.
“In fact,” adds Manore, “in our modern world it almost makes sense to ask the question instead: ‘Where are the irrational numbers? Negative Used?'”
This article has been updated with AI technology, then fact-checked and edited by a HowStuffWorks editor.
Now This Is Interesting
From a computational standpoint, “we almost always use approximations of these irrational numbers to solve problems,” Manore explains. “These approaches are rational since computers can only perform calculations with a certain precision. While the concept of irrational numbers is ubiquitous in science and engineering, it could be argued that we have never actually used a true irrational number in practice.”
Original article: Differences Between Rational and Irrational Numbers
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