Extending the range of calipers

[mklotz]

I did very well with geometry in high school. I was an A+ student. I struggled with Trigonometry though.
It wasn’t until I began my first year in Trade school the geometry really began to click for me. Suddenly I saw how it could be applied and used in the real world! I became an ace with geometry, so much so that I became the go to person if anyone had a geometry problem they deemed unsolvable. My ego became inflated! Then came CAD and now we were all on equal footing again. Everything seems to go full circle in the end. New technology replaces old an si goes evolution. Nothing is forever. Enjoy today!

The problem with CAD, and programs like it, is the fact that, while you can solve a particular problem, you learn nothing about math from it.

Mathematics is learned in a progression; each field builds on what was learned in the previous. Back when the schools actually taught useful material rather than woke psycho babble, the progression looked like

arithmetic
algebra
geometry
trigonometry
analytic geometry
calculus
advanced specialties, e.g. probability, statistics, matrix algebra, etc.

Once you introduce a tool like CAD, you eliminate the algebra and problem solving practice that you would get by following this progression.

[mklotz]

There was a check for the digital calculators that involved multiplying the number 12345679 by a whole number multiple of 9 that was greater than 0 and less than 10. If the calculator gave a result that was the multiplier, supposedly the calculator was working correctly. 12345679 x 81 gave you 999999999, 12345679 x 18 gave you 222222222, etc. I doubt it checks all the functions even on a 4-function calculator, but I never ran into a calculator that didn't get the correct results. I was told that one of the spreadsheet programs, I think an early version of Excel didn't get it right, but my brain has fried a number of times since then. I barely remembered how to do this check myself. And the 2007 version does get it right.

Trust but verify!

Bill

In the days of calculators with seven segment digit displays, it was occasionally necessary to check that all the segments still functioned. This was done by dividing 80 by 9, which action would light every segment producing 8.888888...

This works because 1/9 is an infinite decimal whose expansion looks like 0.1111111... Multiplying that by 8 produces 0.8888888... To eliminate the leading zero, multiply by 80 to produce 8.888888...

The same trick can be used to produce a string of any desired digit. For example, 30/9 produces a string of 3s, 3.333333...

[Toolmaker51]

Flashback, recalling the 80 x 9 = 8.888888 test. It wasn't verifying accuracy of calculation, but testing LED & LCD function to illuminate 100%. 8.888888 needs 49 elements. A random couple fizzle out, result changes a little, say [8.888889], or a lot [9.8888888]. No telling implications of that. Best thing about calculators, proving a calculation by running it again or changing operation etc is much quicker.
Goes without saying, that doesn't mean correct...
After writing, went back to alter font color; honoring those red displays, ala Texas Instruments. Mine would still work, isn't used anymore, just a little 'remember when?' thing. Whatever you do, make sure leaky 9v Duracell's aren't.

[mklotz]

Flashback, recalling the 80 x 9 = 8.888888 test. ...

80 / 9, not 80 * 9

Accuracy checks might include...

10 raised to the log(3) power - does it equal 3 ?

sin^2 + cos^2 should equal 1 for any angle

[Toolmaker51]

Whoops. X still isn't /, nor \; despite modest similarity

[mklotz]

Whoops. X still isn't /, nor \; despite modest similarity

Mathematicians seldom use 'x' to indicate multiplication because it's easily confused with the letter 'x' often used to denote the unknown in an equation. It is, however, used to indicate the cross product of two vectors.

Other options for multiplication include:

Simple juxtaposition ... ab - most frequently used
A raised dot between the two quantities - can't find a font that has that (also used to indicate the dot product of two vectors)
In coding, an asterisk ... a*b

The obelus (÷) is never used by mathematicians to indicate division. The forward slash (/) is preferred. The backslash (\) is sometimes used to indicate integer division where only the integerial part of the quotient is retained.

[WmRMeyers]

Mathematicians seldom use 'x' to indicate multiplication because it's easily confused with the letter 'x' often used to denote the unknown in an equation. It is, however, used to indicate the cross product of two vectors.

Other options for multiplication include:

Simple juxtaposition ... ab - most frequently used
A raised dot between the two quantities - can't find a font that has that (also used to indicate the dot product of two vectors)
In coding, an asterisk ... a*b

The obelus (÷) is never used by mathematicians to indicate division. The forward slash (/) is preferred. The backslash (\) is sometimes used to indicate integer division where only the integerial part of the quotient is retained.

Though I have taught mathematics, I have never claimed to be a mathematician. I assure you all of of that! I used the x for multiplication, which on computers is often indicated with an asterisk or *.

What I find interesting about this discussion is that I totally misapprehended the purpose of the test I described. Nor do I have any faintest clue as who originally told me about it.

Bill

[Toolmaker51]

Though I have taught mathematics, I have never claimed to be a mathematician. I assure you all of of that! I used the x for multiplication, which on computers is often indicated with an asterisk or *.

What I find interesting about this discussion is that I totally misapprehended the purpose of the test I described. Nor do I have any faintest clue as who originally told me about it.

Bill

Not an issue. We have a zillion winkles up there; it's thought using them averts senility. Wrong isn't a tenth the problem gone is.

[mklotz]

I was surprised that no one asked how I arrived at the formula I used in the original post. It's not obvious from casual inspection how one can calculate the radius knowing only the chord and the sagitta.

So, I documented the derivation and it's shown in this diagram...



The starting point for the derivation uses something termed the "Intersecting Chords Theorem". Despite the fact that it's discussed in Euclid's Elements, many folks are unaware of this very useful geometry tool. Basically, it says that if two chords intersect inside a circle, the product of the two parts of each chord are equal. Again, this isn't terribly obvious so the proof is given here...



The proof uses the fact that two triangles are shown to be similar. Similar triangles are not the same as congruent triangles. In congruent triangles both the corresponding sides and angles match. In similar triangles, only the angles match. All 30-60-90 triangles are similar but they're not all congruent.

Similar triangles do have a useful property that's exploited in the proof of the intersecting chords. The ratios of corresponding sides in two similar triangles are equal. Returning to the 30-60-90 example, the ratio of the side opposite the 30 degree angle to the hypotenuse in both triangles is the sine of 30 degrees regardless of the scale of the triangles.

[WmRMeyers]

I was surprised that no one asked how I arrived at the formula I used in the original post. It's not obvious from casual inspection how one can calculate the radius knowing only the chord and the sagitta.

So, I documented the derivation and it's shown in this diagram...



The starting point for the derivation uses something termed the "Intersecting Chords Theorem". Despite the fact that it's discussed in Euclid's Elements, many folks are unaware of this very useful geometry tool. Basically, it says that if two chords intersect inside a circle, the product of the two parts of each chord are equal. Again, this isn't terribly obvious so the proof is given here...



The proof uses the fact that two triangles are shown to be similar. Similar triangles are not the same as congruent triangles. In congruent triangles both the corresponding sides and angles match. In similar triangles, only the angles match. All 30-60-90 triangles are similar but they're not all congruent.

I have to take it on faith that SOMEBODY knows what they're doing. But it sure ain't me! Almost have to have told this story here before, but once again: Asked my Algebra teacher why I needed to learn this algebra crap in 10th grade. He said "So you can graduate from high school." I said "wanna bet?" Despite the fact that I could not do the math, he passed me, barely, in both algebra I & II. Did me no favors, either. 12th grade/senior year, had to take one more math credit, and they had one for practical math, including how to balance a checkbook. I managed to pass that one with decent grades. Though it took until the introduction of Microsoft Money for me to be able to keep my checking account balanced. They stopped producing of MS Money in 2007, but I still use it. https://answers.microsoft.com/en-us/...1-14763116568e Enlisted in the USAF in the Fall of 1973, after just barely graduating based on how stupid I was in the 9th grade, and working in several fields in the USAF I learned a bunch of uses for the algebra that my teacher didn't mention. I suppose it's because he didn't believe that I would find it useful. Could also be that he was tired of dealing with ID10T problems. For years I was mad at him about that. I wanted to be an astronaut, and if he'd told me I needed algebra (and a bunch of other math) to do that I might have applied myself earlier. At this point in my life I don't suppose I could reasonably blame him. I am pretty sure he died the day he found out I was teaching math, though.

Partially because I wanted to be a better teacher than he was, I told that story to my students, and often to their parents, quite a bit. Some folks just can't seem to learn higher maths, and I'm apparently one of them, but I don't know if early intervention might have helped me do better long term. Trying to learn such things as an adult is often more difficult than for the young. When I was stairstepping myself through the rock math classes at Rose State College, one of my classmates was an older lady (compared to my age back then, when I was in my 40's) who had taken each of her rock math courses three times, and was finishing up her second try at the Intro to Algebra class we were both in. She was a grandmother who'd dropped out of school in her teens. It took her three tries to get through each class, and learn all the basic info she needed to pass to the next level class. She knew that when she finished this class, she only had to take it one more time...

My oldest daughter got ASVAB scores sufficient to try the US Navy's Nuclear Power school. Wanted to do something Dad hadn't already done. She was the only student in her class of 38 who didn't have calculus and calculus-based physics in high school. The Navy expects a 70% fail rate in that class, and only 8 students graduated. She was the 8th. She said that the Khan Academy https://www.khanacademy.org/ got her through that class. She was the 8th graduate, and 8th in her class. She got a letter of commendation out of it. She was the only student so ill-prepared for the class to actually graduate in Navy history. Pretty sure she got her math brain from her mom, not her dad. And her stubborn from both of us.

Bill