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TagIrrational numbers and infinity


4. How Almost Any Number Can Encode the Library of Congress

That’s a weird, counterintuitive — but quite real — consequence of the concept of infinity in math

We are used to dealing with simple numbers, like ½ and 2. Most numbers are not that simple. Most numbers, like 0.847859028378490… go on forever and ever without repeating or showing any pattern. Note that such numbers, called irrational numbers, have an infinite number of digits. And there are a lot of them. The number 0.847859028378490… for example differs from the number 0.847859023378490… (See if you can spot the difference.) If two numbers differ only at the billionth decimal and are otherwise the same, they are different numbers. Because an irrational number is infinitely long — and we have seen in the first three posts that weird things happen with infinity — we’d expect something weird to happen with irrational…

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3. In Infinity, Lines and Squares Have an Equal Number of Points

We can demonstrate this fact with a simple diagram

In previous posts, we have established that two sets are of the same size if there is a one-to-one correspondence between the elements of both sets. Applying this principle to Cantor’s theory of infinity leads us to the weird but valid conclusion that the number of points on a line segment is the same as the number of points in a square. To show that this is true, here is a picture of a unit length line segment and a unit square. Let’s choose a point on the line segment. Let’s say 0.6917381276543… . It’s shown with a big blue dot on the line segment on the left. If this point corresponds to an irrational number, it goes on forever…