I take your point and I think my confusion arises out of regarding anything not ‘perfect’ as in sqrt of 2 = an irrational number, less accurate than say sqrt of 4 = 2. I guess because in the former case there is always more work to do while in the latter case it’s all tied up and finished. Maybe it’s just a psychological prejudice.
[quote=“Francis_Vaughan, post:76, topic:791924”]
0
Bolding mine.
Your use of the word “calculated” is the core problem. What do you think you mean by this word? It seems you mean “expressible as a decimal number.” Which as I noted above is identical to saying “expressible as a rational number.” So why? Why insist that calculation of a value requires that it be expressed as a rational number? Apart from being taught in school that numbers are written as decimals there is absolutely no reason to insist on this. Decimal numbers, or any fractions (ie rationals) are just a convenience that are easy to use in many applications. There absolutely no reason to insist that these are somehow the “right” form of number, or that they are anything other than an accident of history.
Now we get to the core problem. What do you mean buy true? You seem to again believe that the only true values are those that are expressible as decimals. Why?
We have discussed earlier, almost all numbers are not so expressible, and only an infinitesimally small set of numbers are so expressible. By your logic almost no number has a true value.
By calculation you mean express as a decimal value. Most mathematicians would not regard this as calculating its value. We might determine a quick approximation to its decimal representation for engineering purposes, but the value of the square root of two is just that - the square root of two. Anyone doing mathematics (as opposed to arithmetic) will carry that value around in expressions until they (might) be able to combine terms in some manner to make it vanish (or not, as the case may be.) There may be a point where some arithmetic is performed to determine a useful expression of the value for some purpose. But that isn’t mathematics.
You’ve really got me thinking now. I want to go over this some more and I appreciate your politeness and patience with me. I think you might be right about how schooling places too much emphasis on rational numbers, possibly because they are more likely the ones the average person will use in life. 
Put it this way: There are many different ways of expressing numbers, some of which are better-suited for some purposes than others. For instance, when you’re counting things, you often use tick marks. Like, if I’m tracking the number of eggs my mom’s chickens have produced each day, then every time I take an egg out, I make a mark. For convenience, I make every fifth mark go through a group of four. So if the chickens are having a good day and give us a total of seven eggs, the final tally will look something like [del]||||[/del] || .
This is a perfectly valid way of representing numbers. And when what I’m counting is the eggs produced by the chickens in my mom’s backyard, it’s a very convenient and appropriate way of representing them. But it’s not well-suited to all sorts of numbers. If instead of my mom’s backyard chickens, I were dealing with all of the eggs produced in a year by Cal-Maine Foods, well, I could count the eggs with tick-marks, but it’d be horribly inconvenient, because the numbers are so large. And if I want to measure anything that’s not a positive integer, tick marks don’t really work at all. If I can’t easily represent 1.3 billion using tick marks, and I can’t represent four and a half using tick marks at all, does that mean that those numbers aren’t exact? Of course not; it just means that I’m not using the most appropriate system of representing those numbers.
Well, place value numbers with radix can represent many more numbers. They still don’t work well for all numbers, though. In any base, there will be some rational numbers that can be represented in a finite amount of space, and some that can’t be, with no fundamental distinction between the two (there’s no fundamental reason, for instance, why “one fifth” is better or more important than “one third”). And there are some numbers, like pi and sqrt(2), that can’t be finitely represented in a place value system with any base. Does that mean that those numbers aren’t exact? No, it just means that we’re using an inconvenient notation for them.
But there are notations for those numbers that are much more convenient, and which can be used to represent them exactly in a finite amount of space. In fact, we’ve been using those notations throughout this thread. I don’t have to call that number “3.14159265358979323846…” . I can just call it “pi”, and I’m done. All it takes is a single character (or two, if I don’t have a Greek keyboard), or a single syllable when spoken, and I’ve told you the number. The notation works just fine.
Warning: Deep into the mathematical weeds, for this next paragraph. Read at your own risk:
It turns out, though, that in any sort of notational system, there are always some numbers that can’t be represented in a finite amount of space. Which numbers? I can’t tell you, of course: That’s the whole point. But not only do such numbers exist, but they’re infinitely more numerous than the numbers that we can express. Fortunately, no such number is ever of interest (because if it were of interest, we could use the property that makes it interesting to define it).
Very good. I’m getting a fantastic education and now I realize why mathematics can be so interesting. TYVM. 
I think what you’re missing is that the numbers don’t go on forever, the decimal representations of those numbers go on forever. Pi is an exact number, so is the square root of two, so is e. But if you insist on writing the number as a decimal, that decimal representation goes on forever.
In summary, I think in this discussion you should use different terms for the number and for its decimal representation. In your writing, be careful to use “pi” and the “decimal representation of pi” and it will help you clear this up.
If the decimal representation of a number ended, then that number is rational. Any decimal representation that either stops (ends with repeating zeros) or repeats is provably a rational number. Pi and the square root of two are proved NOT to be rational, therefore we know they can’t repeat or stop.
As an aside, is there a good reason that there is so much fascination about the number pi, and much less about the number e? Pi is the rock star, e is its little-known brother, but I can’t think of any reason that pi is favored to the public.
Are there statistics supporting your claim? It never occurred to me that one could rank numbers by popularity.
In any case, one important fact is that pi is an ancient concept, while e is not. Also, everybody studies geometry, while perhaps analysis is less generally taught.
How do we know that pi is an exact number? Because it appears in millions of equations with exact answers. If pi were an approximation, there would be millions of answers and those equations would be useful except to the degree that outside measurements were brought in.
You may argue that pi cannot be used in an equation that forms an integral. That’s not correct. What’s called Euler’s Identity, named after the 18th century mathematician Leonhard Euler, combines three different components with no exact decimal representation and yet yields an integral.
e[sup]i*pi[/sup] + 1 = 0
This is often called the most beautiful equation in mathematics. It uses no more than high school algebra. It utterly depends upon pi and e and i to have exact values on the complex number plain. Most people neither understand it or believe it at first. It’s worth study, though. It forever breaks your addiction to arithmetic and opens up the world of mathematics as something larger and more spectacular.
It may be worth pointing out that irrational numbers were discovered before decimal notation was. The ancient Greeks knew about irrational numbers, and that the square root of two was one, but they didn’t use decimal notation to write down or imagine their numbers. They thought of numbers as corresponding to lengths of line segments (according to which sqrt(2) is a number), but also as ratios of whole numbers (according to which it is not), which is why the irrationality of something like sqrt(2) bothered them.
Right, almost everybody encounters (and “uses”) pi fairly early on in their schooling, since it shows up in formulas for the circumference of a circle, the area of a circle, etc. People don’t encounter e until much later, if at all; and it really requires calculus to see how it’s defined or what’s so special about it.
Other famous irrational numbers include the “golden ratio” phi, and Euler’s gamma.
Because if we did arrive at a termination, we could write it as a ratio of integers (i.e. a fraction) and it would therefore be rational.
For example, if pi were exactly equal to 3.14159, we could write it as 314159/100000. If there were more decimal places (but still only finitely many), the denominator would just be a bigger power of 10.
Which of course gets us to another class of numbers. The Constructible Numbers.
Given a straight edge and a compass, and finite steps, any length you can construct is a constructible number.
Which terns out to mean any number you can make starting with the numbers 0 and 1, and the operations, add, subtract, multiply, divide, and square root.
Now here you can still buy an argument about what constitutes a number.
Probably so, although arguably, some mathematicians have had similar psychological prejudices. I’m thinking of Leopold Kronecker, to whom is attributed the quote “God made the integers, all else is the work of man,” and who is apocryphally supposed to have said “What good your beautiful proof on [the transcendence of] π? Why investigate such problems, given that irrational numbers do not even exist?”—see discussion here.
Pi.
[QUOTE=CurtC]
As an aside, is there a good reason that there is so much fascination about the number pi, and much less about the number e? Pi is the rock star, e is its little-known brother, but I can’t think of any reason that pi is favored to the public.
[/QUOTE]
What pi is is obvious; it’s how you get the circumference of a circle. People understand circles; circles are, well, all around us. They’re important. We deal with circular things all the time. The comparison between how far a circle is around, and how far it is from one side to the other, is an obvious one to think about. Kids get circles, they like circles. Pi is a natural thing to want to know based on the most simple observations of the world.
e is “the base of the natural logarithm.” I have a degree in economics and so had to take and pass calculus and statistics, and even I am still not super clear on what e is. You really have no reason to ever think about e until you actually get to that level of understanding mathematics; it’s not something that is a visually obvious concept, like pi.
Fair enough, about pi being more popular than e.
Next question - do the things we know about pi also apply to e? Is it normal for example?
In any field where you solve differential equations (like electrical or mechanical engineering) “e” is an extremely important constant. Any basic circuit theory student (should) be able to visualize “e” from the settling time of a first-order system.
Both are suspected to be, but neither one has been proved to be. From Wikipedia:
Yes, I know that, Marvin. What CurtC wanted to know is why Pi is more popular.
How many people have to solve differential equations? A lot in total terms, but it’s a small percentage of all people.
How many people see CIRCLES everywhere? Everyone who can see.
Oh yes they do exist!
Consider the number represented by “…999.000…”. Let’s add one, which we will represent as “…0001.000…”, to it. And the answer is “…000.000…”. What do we usually call the number which, if we add one produces zero? Yes, “…999.000…” is another representation of negative one.
But wait, there’s more!
Another representation for one is “…000.999…”. Let’s add that to our representation of negative one “…999.000…”. And, you see where this is going? One added to negative one is always zero, or equivalently “…999.999…”.
While this may seem bizarre, it’s actually related to how signed integers are often stored on a computer. See Two's complement - Wikipedia . And it’s also comes up in physics. See Negative temperature - Wikipedia .
Yes you can consider -1 = …999 as a sort of 10-adic expansion, but beware that p-adic numbers are not the real numbers you learned about in your introductory analysis class.