So the question has been posed, just exactly how much is a tiny tish, as in a tiny tish less than 7.1 square miles.
The answer is related to the very important numerative concept of significant digits (for which there IS a Wikipedia article). One of the joys of learning to operate a slide rule is that the process forces you to learn about and to actually confront the concept of significant digits.
Yesterday I wanted to take a nice close up of the slide rules but discovered that the battery on my good camera was discharged. I had to use the cheap, put it in your pocket and go for a bicycle ride digital. The picture was fine, but today I have a much better one.
So, the rule of thumb for performing calculations is that the result you get at the end can only have value and accuracy to the level of the accuracy of the data input at the start of the calculation. The rule for calculations involving a measurement is that the result can only be "significant" to the same level of accuracy as the lowest level of accuracy of measurements input.
In this case the measurement input was a radius of 1.5 miles. That is two significant digits. Therefore, any result with more than two significant digits contains spurious digits introduced by calculation.
The only reliable result is the one produced by the slide rule, about 7.1 square miles. With a radius measured at 1.5 miles an area of 7.1 square miles is the most correct answer.
However, in examining the slide rule, we can see that pi on the B scale lines up with something just a "tiny tish" less than 7.1 on the A scale.
There is no way to know just exactly how much a tiny tish actually is. Everyone gets to make their own definition. I think that in this case .0314 or about pi divided by 100 is an excellent definition.
Further affiant saith not. That's lawyer talk.
7 comments:
I used the word "gazillion" in an accounting lecture once -- an upper level class. It was meant to convey "a very big, non specific, number." One of my better students raised her hand and asked me "Just how many zeroes are there in a gazillion?" (This was before Bush was even president.)
I haven't thought about significant digits in a while. I love the internal consistency of math, and that's one of the concepts that keeps that train on track. (Is that a metaphor?)
Yes, that is a metaphor. That's my favorite part of this post (except maybe for the phrase "There is no way to know just exactly how much a tiny tish actually is").
I was interested in your final piece of lawyer talk. I hadn't heard the phrase before, but I looked it up. There is a nice explanation in a Michigan Bar Association Article that I stumbled upon and which amused me greatly. Is it "further affiant sayeth not" or "further affiant saith naught?" There seems to be just a tish of a difference.
OMG. A regular numbskull like me would need about 4.67 energy drinks to keep up with the conversations at your holiday gatherings!! How fun! Happy Pagan-Ritual-Holiday-Fest to you and yours!
Next to the buggy whip, do you also have a butter churn?
Pi has a large number of significant digits, assuming I have this straight, so it doesn't limit the number of digits in your product. So if the radius of the circle is 1.50 mi. instead of 1.5, can you claim greater accuracy to more digits when you square it and apply the Pi thing to the result? (Just asking. I think so, though.)
Yes. 1.50 has 3 significant digits and is therefore a measurement of considerably more accuracy than 1.5. A circle with a radius of 1.50 miles contains 7.07 square miles, a tiny tish less 7.1.
I think it is obvious enough intuitively. If you describe a distance as 1.50 miles you have almost certainly measured that distance much more carefully than the measurement involved in the off handed remark that I walk to places within a mile and a half of where I live.
Perhaps even 10 times more accurate.
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