Gabriel: Hey Lisbeth! I understand why you're confused, but let me clarify a couple of things about π and the relationship between rational and irrational numbers.
So, let's consider Claim A, that π is actually a rational number. That’s not true. In fact, π has been proven to be an irrational number, which means that it can’t be expressed as a fraction of two integers. This is a well-established fact in mathematics, and if you try to write π as a fraction, you will realize that you can only get an approximation, like 3.14. But no matter how close you get, you can never express it exactly as a fraction.
Now, let’s look at Claim B, which is that the circumference is actually irrational. And this is indeed a possibility! The circumference of a circle is directly related to its diameter via the formula C = πD, where C is the circumference and D is the diameter. Since π is irrational, it follows that if you use a rational diameter, the resulting circumference must also be irrational. Essentially, multiplying a rational number by an irrational number yields an irrational product.
So, to wrap it all up, the reason Gabriel's slope of approximately 3.14 gives us the impression of representing π is because we’re using an approximation for an irrational number. But we can't say π is rational—it’s not! This means that when we measure things like the circumference of a circle, we often end up with irrational results when related to π. Does that help clarify things?
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