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Making Sense of Irrational Numbers

Learn about irrational numbers and why they make sense.

Did you know not all numbers make sense? Well, at least not all numbers are rational. There are such things as irrational numbers. So what are these numbers? Who discovered them? Where do we find them? How are they used?

What. The dictionary definition of an irrational number is one that cannot be written as a ratio of two integers. That doesn’t make a whole lot of sense. Another way to say it is “a number that can’t be written in decimal form.” Take 2.5 as an example of a real number. Irrational numbers, like Pi, can’t really be written like that—the numbers in the decimal go on forever. Although Pi is often written as 3.14, true Pi just keeps going.kid holding notebooks

Who. Hippasus of Metapontum was the first person to write about irrational numbers. He did it in Greece during the fifth century BC. And for doing so, he was thrown into the sea to drown. He was part of a group called the Pythagoreans who believed the universe was all numbers and their ratios. Numbers that didn’t make sense had to be evil, and Hippasus had to go.

Where. The most famous irrational number is Pi, 3.14159265359… If you go to you can learn all the facts you want to find out about Pi. The fact that Hippasus discovered was that the side of a right triangle that has two equal sides meets at 90 degrees. It is easier to explain what Hippasus found as the square root of two.

Where. We already said irrational numbers show up in Pi, which is the ratio of the area of a sphere to its diameter. And Hippasus’ right triangle, square root of 2 things. There is also Phi, sometimes called the “golden ratio,” which is adding the two previous numbers to create the next, 0, 1, 1, 2, 3, 5, 8… Then there is “e”, which is named for Leonhard Euler, a Swiss mathematician. We use “e” as the base of natural logarithms.

Why. Pi is used to find out information about spheres. If you wanted to know how many times to fold a piece of paper to get 64 layers, you would use “e.” Square root of two helps us with the Pythagorean theorem, which helps us to build things with corners that meet up at right angles.

So that’s it. Irrational numbers might seem strange, but they can be very useful in real life. Now that you know a little bit about them, see where you might be able to find irrational numbers in real life. Try measuring a room in your house and see if the numbers add up and the corner is actually 90 degrees. Find out the surface area of a basketball. Or try folding a piece of paper a bunch of times!

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