What’s the Difference Between Rational and Irrational Numbers?

Finding Square root by division method
Finding Square root by division method

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.

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