How to divide irrational numbers?

Many times students just accept what they are told in math class without much questions; it's perhaps boring, they don't want to take time to investigate and delve deeper, or whatever the reasons.

The teacher feeds them with knowledge and they take it in: "Oh, okay, there's such a thing as Pi. Oh, okay, some numbers are irrational."

But a question such as this shows that the person is wondering, wanting to understand the material more. So it's a good question!

I will divide the 'division' question to two parts.

1) When exact answer is desirable, often we do cannot technically divide. For example, let's take Pi ÷ 2. Well, we just write it as Pi/2 or π/2 and leave it at that.

If we can simplify the answer, we do that. For example, √15/√5 can be simplified to √3.

We could simplify √2/2 this way: since 2 = √22, then √2/2 = √2/(√22) = 1/√2.

But often the expression √2/2 is left as is, since there is a convention that we should consider a root in the numerator to be "prettier" than a root in the denominator. Or I don't know why it is; I just remember this little rule from school math.

2) When you need a numerical answer, then you use the decimal approximation of the irrational number, and divide normally as you would decimals.

For example, to find Pi/2, you take the decimal approximation to Pi as, say, 3.14159, and go 3.14159 ÷ 2 = 1.570795.


Anonymous said…
Just wandered onto your blog, and saw this post -- one of the reasons for rationalizing the denominator, as far as I know, involves the fact that when computing with a slide rule, it's much easier to handle a radical in the numerator. At least, that's what my mother said.
Anonymous said…
the convention for leaving a irrational in the numerator comes from the days before calculators. It was a more accurate answer to do the long division with the irrational as the numerator, then you can approximate to the number of desired decimal places.

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