Buying the HP-75 has turned into a new, and likely passing, obsession. While it's burning, I've spent a lot of time browsing the HP Museum Forum. There are a lot of math posts, as one might expect. So I've been pulling some books out of my library and looking at "recreational" math on programmable and scientific calendars.
And when I wasn't browsing the forums, or noodling around with my HP-41CX, I pulled one of my "popular literature" mathematics books, Mathematical Sorcery, by Calvin C. Clawson, off the shelf and thumbed through it looking for something to read while sitting in the recliner. In Chapter 8, Extending the Form, I spotted some continued fractions, which was one of the topics Dr. Drang explored, so I decided to read that.
Clawson goes some way to "show his work" in explaining how continued fractions can be used to find a square root. I like to think I'm an educated person, but he used a term I was unfamiliar with as it appeared in this context: "Our next step is to write the term within the parenthesis as a fraction and multiply its numerator and denominator by its conjugate."
I couldn't follow what he was doing, so I went back to my library and pulled the Penguin Dictionary of Mathematics, off the shelf and looked up conjugate. There are many entries beginning with "conjugate," but none in the context of a fraction. I was no further enlightened.
So before turning out the lights last night, I figured I'd ask ChatGPT, "What is a fraction's conjugate in mathematics?" I received this helpful reply:
In mathematics, the conjugate of a fraction refers to a way of changing the sign between the terms in the fraction. The conjugate of a binomial fraction is obtained by changing the sign of the second term. For example, if we have a fraction (a + b)/(c - d), its conjugate would be (a + b)/(c + d).
The conjugate of a fraction is commonly used when simplifying or rationalizing expressions involving square roots or complex numbers. By multiplying the numerator and denominator of a fraction by the conjugate of the denominator, we can eliminate square roots from the denominator or simplify complex numbers in the denominator. This technique is helpful in various mathematical operations, such as simplifying expressions or solving equations.
Knowing that ChatGPT gives clear, confident answers that are often wrong, this morning I pulled the tome Mathematics, From the Birth of Numbers, by Jan Gullberg, from the library and looked up "conjugate" in the index. There were no entries with respect to "fractions," but one of the two entries was binomial surds, which I knew referred to irrational numbers, like some square roots. So I went to that page, and it was largely a discussion of rationalizing and simplifying expressions containing surds. So I knew ChatGPT wasn't blowing digital smoke up my butt.
It all became clear. You multiply by the conjugate identity to eliminate the square root in the denominator. I could follow along after that.
Out of curiosity, I checked the entry for rationalize in the Penguin dictionary, and while it mentions multiplying by an identity with an opposite sign, it never specifically mentions "conjugate."
Anyway, ChatGPT can be helpful, at least pointing you in the right direction.Originally posted at Nice Marmot 08:57 Saturday, 19 August 2023