To "bisect" something means to cut it in half. The bisection method searches for a solution by bisecting: narrowing down the search area by half at each step. The idea is as follows.
We've drawn the situation below. The search area is marked in green. There could be other solutions as well, but we only need one! You now continue searching. If you don't, you narrow down your search area by half again.
Proceeding in this way, you'll either find a solution, or get very close to one. How close? As close as you like. However closely you want to approximate a solution, you'll be able to do it with the bisection algorithm. You don't know that of course; you're trying to find the solution! See the figure below. Having described the idea of the bisection method, we'll next discuss the theory behind it more rigorously. The bisection method relies upon an important theorem: the intermediate value theorem.
This theorem is a very intuitive one. If you're on one side of a river, and later you're on the other side of the river, then you must have crossed the river!
Portions of the graph must appear as shown below. But beware! Now, although we described the left endpoint being below the river i. All the theorem says is that there is at least one. If you're on one side of the river, and later you're on the other side of the river, then you must have crossed the river — you might have crossed it many times, but you certainly crossed it at least once! We now describe the bisection algorithm in detail.
In other words, what is. Let's now turn to some examples. We'll obtain some approximations to some well-known irrational numbers using the bisection method. As you can see, the bisection method can be a rather repetitive and time-consuming process! The best answers are voted up and rise to the top.
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