Math Help
20:57 Saturday, 19 August 2023
Current Wx: Temp: 81.57°F Pressure: 1015hPa Humidity: 89% Wind: 3.44mph
Words: 578
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.
Of course, I'm still keeping up with my RSS feeds, and a couple of recent entries from Dr. Drang seemed timely, almost to the point of synchronicity.
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.
✍️ Reply by emailComing Around
22:11 Saturday, 19 August 2023
Current Wx: Temp: 86.49°F Pressure: 1013hPa Humidity: 78% Wind: 3.44mph
Words: 176
The other day, I mentioned something that I've long believed. That technology isn't the solution to anything. At best, it's a tool. Tools don't solve problems, they help people solve problems. And most of our problems have to do with us. Our human nature.
We've grown enamored with our tools and our facility with making them. We have more faith in technology than we do in ourselves.
In the early days of blogging, the blogosphere was filled with internet triumphalists. "This changes everything!" Markets were conversations, everything was miscellaneous, gatekeepers were being overthrown, marginal costs were going to be zero for digital goods, yada, yada, yada...
It was all bullshit.
Bullshit. Bullshit. Bullshit.
I understand the euphoria, the enthusiasm. I understand the endocrine rewards of novelty and attention. I understand how the intoxication skews thought and reasoning.
And in a competitive, capitalist, attention-based economy, I understand how bullshit came to be dogma.
You don't find many of the triumphalists recanting these days. I don't expect many will.
So this was something of a refreshing read.
✍️ Reply by emailHistory Lesson
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This morning's history lesson from Heather Cox Richardson.
Left to right: Laura Cornelius Kellogg, Harriet Tubman, Martha Coffin Wright and Sojourner Truth, in Seneca Falls, NY.
✍️ Reply by emailRising
08:21 Wednesday, 19 August 2026
Current Wx: Temp: 64°F Pressure: 1013hPa Humidity: 82% Wind: 5.28mphWords: 521
Work continues on the new house. Hopefully, we'll be pouring concrete on Monday. The weather has been fantastic and they're making good progress. This level is more demanding than the basement because there are more window bucks and they all have to be precisely aligned.
I think we finally have the HVAC design nailed down. Now I need to land a contractor. We've decided to use an air handler that can be mounted on the ceiling of the mechanical room. We have 9'-4" of height to work with below the trusses. Even accounting for mounting hardware and ducting, I'm confident we can get the buffer tank below the air handler, which really compresses the footprint, which means the hot water heater will likely fit in the mechanical room. I was considering putting it in the basement laundry, but now I think I'm going to put the water softener in there. I'll have plenty of room for the two tanks and the brine tank, and the filters would easy to change without having to contort myself.
Of course, this is all just hopeful, informed speculation until equipment starts arriving.
But I slept well last night, for the first time in a while.
It's a beautiful day today. The air is crystal clear, and quite cool. I really enjoyed this morning's walk. I skipped yesterday because I hardly slept Monday night.
I figured out how to donate blood around here, so I'm scheduled for a donation at the end of this month at the fire station. I want to get in the groove of doing that again.
The IIc Plus is at my feet here. I'm kind of struggling with figuring out how to get disk images onto the virtual drives built into the MegaFlash. I've ordered some ADTPro cables for the IIc and IIc Plus, and hopefully that'll avoid any conflicts with the number of devices on the SmartPort. I'm also going to take a stab at the TFTP network transfer.
I don't think I'm going to be able to use the database portion of AppleWorks to create an RSS feed, since the maximum field length is 70 characters. But I'll figure something out. Coincidentally, or perhaps not, Christopher Drum just took a long look at AppleWorks in his Stone Tools series. I used AppleWorks very productively for many years, including at sea using an Apple IIc with an ImageWriter printer and the little ET green-screen monitor. I think I had a 4MHz Zip Chip installed in it, but I don't think I ever had any memory expansion.
I have an appointment to get my ears lowered this morning, which means I can stop by Jerlando's on the way home and get a turkey sub.
My GLP-1 prescription has been rejected twice, which is according to plan from what I understand. I've spoken to the pharmacy and messaged my doctor. Hopefully we can get this thing over the finish line and I can start shedding some pounds and stop obsessing over Finger Lakes ice cream. (And cheese, baked goods, cannolis, beer, the list goes on.)
The beat goes on...
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