Common Core Standards: Math
2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
A long, long, time ago in a place called Ancient Greece, the life of the people was very different from ours. There were no electronics: no DVDs, no Xboxes, no cell phones—not even email. There wasn't all that much to do other than invent the Olympics and think about math. So, once the Olympics were over, lots of people sat around and talked about math.
One of those ancient Greeks was named Zeno. He came up with the following notion:
"Before I can get to the pie on the other side of the room, I must first walk halfway there." (Sounds reasonable, right?) "But before I can do that," he said, "I must go a quarter of the way." (Still good with us.) "But first," Zeno said, "I must go an eighth of the way, and before that, a sixteenth."
Okay, where's he going with this? Well, nowhere, really. According to Zeno's Paradox, it's impossible to ever move at all, since one must always complete half the trip before he or she can go the rest of the way. That renders Zeno immobile and pie-less. How sad.
Similarly, it might be difficult for your students to see exactly what those pesky irrational numbers are equal to. Well, here's where Zeno's Paradox actually gets us places.
Let's say we want to approximate the square root of 2. We know it's between 1 (which is the square root of 1) and 2 (which is the square root of 4.) So it's got to be one-point-something.
If we square 1.5, we get 2.25. So the square root of 2 is a little less than 1.5. But if we square 1.4, we get 1.96. Since 2 is between 1.96 and 2.25, the square root of 2 is between 1.4 and 1.5. We could keep doing this all day, getting closer each time to the true value of the square root of 2, which is an irrational number.
Students should be able to come up with similar approximations for just about any irrational number they might encounter. The goal here is for students to know more or less where irrational numbers lie on the number line, not to have them find the exact location of π. (It's probably still across the room from Zeno. It's not like he moved it or anything.)