Pi Is a Ratio. So How Can It Be Irrational?

So, pi is an irrational number, which means it can't be a ratio, expressed as like this over that. However, pi is the circumference over the diameter, which is a ratio. It certainly is true that pi is definitionally the ratio of a circle's circumference to its diameter. That is, for some circle, if we measure the longest distance across that circle, the distance that goes through the center from end to end, and we compare that to the distance around the circle, the thing that we call the circumference, the result of that comparison, when we divide the circumference by that diameter, is always pi. So, if pi is definitionally a ratio, how is it that we can say pi is an irrational number? The first thing to say is we can't really define an irrational number as a number that can't be put into a ratio, because there's no such thing as a number that can't be put into some ratio. Consider other irrational numbers like, say, the square root of two. two is definitely irrational. Pythagoras killed a guy over showing that it was irrational. But, it's certainly not the case that we can't put it in a ratio, because we can just ratio over one. Every number can be expressed as itself divided by one. So, if what we mean by irrational is it can't be put into a ratio, then we're just saying there are no irrational numbers. Our definition for irrational numbers is not that they can never be put into a ratio, but that they can't be expressed as the ratio between two integers. Going back to root two, for example, one of the problems with root two is that we can get really close to its value with a ratio between integers, but we never get exactly its value. It's close, but not quite equal to 14 over 10. And, it's even closer, but not quite equal to 141 over 100. 1,414 over 1,000. We can express a rational number that is as close to root two as you could possibly want, and yet is not exactly equal to root two because we can show root two is irrational. We can show it's impossible to express root two as the ratio between two integers. Similarly with pi, although there are many ratios we can use to express a number very close to pi, 22/7 for example is a very common approximation for pi, and in fact it's an even better approximation for pi than 3. 14. That is, the ratio 22/7 is even closer to pi than the ratio 314/100. But 22/7 still isn't exactly pi, and no matter how you try to approximate it with a rational number, a ratio between two integers, you'll never express exactly what pi is. And that's what makes pi an irrational number.