Differences between rational and irrational numbers

When you hear the words “rational” and “irrational,” they may remind you of the relentlessly analytical Spock from “Star Trek.” However, if you are a mathematician, you probably think about relationships between integers and square roots.

In the realm of mathematics, where words sometimes have specific meanings that are very different from everyday use, the difference between rational and Irrational numbers It has nothing to do with emotions. Since there are infinitely many irrational numbers, you would do well to gain a basic understanding of them.

Properties of irrational numbers

“When remembering the difference between rational and irrational numbers, think of one word: proportion,” 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 Computing and Computational Science and Engineering.

“If you can write a number as a ratio of two whole numbers (e.g., 1 over 10, -5 over 23, 1.543 over 10, etc.), then we put it in the rational number category,” Kolaczyk says in an email. electronic. “Otherwise, we say it is irrational.”

You can express a whole number or a fraction (parts of whole numbers) as a ratio, using a whole number called the numerator on top of another whole number called the denominator. You divide the denominator by the numerator. That can give you a number like 1/4 or 500/10 (aka 50).

Irrational numbers: examples and exceptions

Irrational numbers, unlike rational numbers, are quite complicated. As Wolfram MathWorld explains, they cannot be expressed using fractions, and when you try to write them as a number with a decimal point, the digits go on and on, without stopping or repeating a pattern.

So what kind of numbers behave in such a crazy way? Basically, those that describe complicated things.

Pi

Perhaps the most famous irrational number is pi, sometimes written as π, the Greek letter meaning “p,” which expresses the relationship between the circumference of a circle and 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.

The story continues

Since Babylonian mathematicians attempted to calculate pi almost 4,000 years ago, successive generations of mathematicians have continued to work, creating increasingly longer chains of decimal expansion with non-repeating patterns.

In 2019, Google researcher Emma Hakura Iwao managed to extend pi to 31,415,926,535,897 digits.

Some (but not all) square roots

Sometimes a square root (that is, a factor of a number that, when multiplied by itself, produces the number you started with) is an irrational number, unless it is a perfect square that is a whole number, like 4, the square. root of 16.

One of the most striking examples is the square root of 2, which results in 1.414 plus an endless string of non-repeating digits. That value corresponds to the length of the diagonal inside a square, as first described by the ancient Greeks in the Pythagorean theorem.

Why do we use the words “rational” and “irrational”?

“In fact, we typically use ‘rational’ to refer to something more reason-based or similar,” Kolaczyk says. “Its use in mathematics seems to have arisen as early as the year 1200 in British sources (according to the Oxford English Dictionary). If you trace both ‘rational’ and ‘ratio’ back to their Latin roots, you will find that in both cases the root has to do with with ‘reasoning’, in general terms”.

What is clearer is that both rational and irrational numbers have played an important role in the advancement of civilization.

While language probably dates back to roughly the origin of the human species, numbers appeared 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 don’t have any more apples,'” Zegarelli says. “They didn’t need to know, ‘We have exactly 152 apples.'”

But as humans began to till plots of land to create farms, build cities, and manufacture and trade goods, moving further and further from their homes, they needed more complex mathematics.

“Suppose you build a house with a roof whose elevation is the same length as the span from the base to its highest point,” Kolaczyk says. “How long is the extent of the roof surface from the top to the outside edge? It is always a factor of the square root of 2 of the elevation (stroke). And that is also an irrational number.”

The role of irrational numbers in modern society

In the technologically advanced 21st century, irrational numbers continue to play a crucial role, according to Carrie Manore. She is a scientist and mathematician in the Modeling and Information Systems Group at Los Alamos National Laboratory.

“Pi is the first obvious irrational number to talk about,” Manore says via email. “We need it to determine the area and circumference of circles. It is fundamental for calculating angles, and angles are fundamental for navigation, construction, surveying, engineering and more. Radio frequency communication depends on sines and cosines involving pi”.

Additionally, irrational numbers play a key role in the complex mathematics that makes high-frequency stock trading, modeling, forecasting, and most statistical analysis possible – all activities that keep our society going.

“In fact,” adds Manore, “in our modern world, it almost makes sense to ask, ‘Where are the irrational numbers? No being used?'”

This article was updated along with artificial intelligence technology, then verified and edited by a HowStuffWorks editor.

Original article: Differences between rational and irrational numbers

Copyright © 2023 HowStuffWorks, a division of InfoSpace Holdings, LLC, a System1 company