by When you hear the words “rational” and “irrational,” they may bring to mind the relentlessly analytical Spock in “Star Trek.” If you’re a mathematician, though, you probably think about ratios between whole numbers and square roots.
In the field of mathematics, where words sometimes have specific meanings that are very different from everyday usage, the difference between rational and irrational numbers which has nothing to do with feelings. Since irrational numbers are infinite, it’s good to have a basic understanding of them.
Properties of Irrational Numbers
“When you remember 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 Rafik B. Hariri Institute for Computing and Computer Science & Engineering.
“If you can write a number as a ratio of two whole numbers (eg, 1 over 10, -5 over 23, 1,543 over 10, etc.) then we put it in the rational number category,” says Kolaczyk in email. “Otherwise, we say it’s irrational.”
You can express a whole number or fraction — parts of whole numbers — as a ratio, using a whole number called a numerator on top of another whole number called a denominator. You divide the denominator into the numerator. That can give you a number like 1/4 or 500/10 (otherwise known as 50).
Irrational Numbers: Examples and Exceptions
Irrational numbers, in contrast to rational numbers, are quite complex. As Wolfram MathWorld explained, they cannot be expressed as 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, ones that describe complex things.
Pi
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 Babylonian mathematicians attempted to calculate pi nearly 4,000 years ago, successive generations of mathematicians have kept at it, coming up with longer and longer strings of decimal expansions with non- repeated.
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 is an irrational number — that is, a factor of a number that, when multiplied by itself, produces the number you started with — unless the square is a perfect square that is an integer, such as 4. root 16.
One of the most prominent examples is the square root of 2, which works out to 1.414 plus an infinite string of non-repeating 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’?
“We tend to use ‘reasonable’ to mean something more like based on reason or the like,” says Kolaczyk. “Its use in mathematics seems to have increased as early as the 1200s in Welsh sources (according to the Oxford English Dictionary). The root has to do with ‘reasoning,’ more or less.”
What is more clear is that irrational and rational numbers played important roles in the advancement of civilization.
Although language probably dates back to the origins of the human species, numbers came much later, explains Mark Zegarelli, a math instructor and author who has written 10 books in the “For Dummies” series. Hunter-gatherers probably did not need much numerical precision, other than the ability to roughly estimate and compare quantities.
“They needed concepts like, ‘We have no more apples,'” says Zegarelli. “They didn’t know, ‘We have exactly 152 apples.’
But as people began to accumulate plots of land to create farms, build cities and manufacture and trade goods, traveling further from their homes, they needed more complex mathematics.
“Consider building a house with a roof whose rise is the same distance as the run from the base at the highest point,” says Kolaczyk. “How long is the stretch of the roof surface itself from top to outer edge? Always factor the square root of 2 of the rise (run). And that’s an irrational number as well.”
The Role of Irrational Numbers in Modern Society
In the technologically advanced 21st century, irrational numbers continue to play a vital role, according to Carrie Manore. She is a scientist and mathematician in the Information Systems and Modeling Group at Los Alamos National Laboratory.
“Pi is obviously the first irrational number to talk about,” Manore says via email. “We need to determine the area and circumference of circles. It is critical to calculating angles, and angles are critical to navigation, construction, surveying, engineering and more. Radio frequency communication relies on sines and cosines of with pi.”
Furthermore, irrational numbers play a central role in the complex mathematics that make possible high-frequency stock trading, modeling, forecasting and most statistical analysis—all the activities that keep our society humming.
“Indeed,” says Manore, “in our modern world, it almost makes sense to ask instead, ‘Where are irrational numbers? no using?””
This article was updated in conjunction with AI technology, and fact-checked and edited by a HowStuffWorks editor.
Now That’s Interesting
In computing, “we’re almost always using approximations of these irrational numbers to solve problems,” Manore explains. “These approximations are reasonable because computers can only calculate to a certain degree of accuracy. Although the concept of irrational numbers is ubiquitous in science and engineering, it could be argued that we are not actually using a real irrational number in reality .”
Original article: Differences between Rational and Irrational Numbers
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