Or sets all the way down, or any of a variety of other objects all the way down.
Yes, the decimal system is some what flawed.
That’s why we were taught to work in fractions in school. Most of my teachers wanted the final answer in a fraction reduced to it lowest terms. Converting it to a decimal introduces rounding errors.
Some teachers preferred answers left as compound fractions because they are easier to plug into another calculation.
I shudder anytime I see a kid grab a calculator and convert fractions to decimal values before working the problem.
3/8 + 1/4+ 2/3 + 1/2
9/24 + 6/24 + 16/24 + 12/24 = 43/24 = 1 19/24
.375 + .25 + .667 + .5 = 1.792
The 2nd example is not as precise. Although the actual answer is close enough for most applications.
Pi is troublesome because it can’t be expressed as an exact fraction. We use 22/7 as an approximation.
Well, it depends on where the numbers came from in the first place. If the 1/2 came from some formula where it was known that it was exactly 1/2, then it’s good to keep it as a fraction. But if you put a ruler down next to something and measured it as 1/2 inch, then the decimal is just as good as the fraction. And while 22/7 is usually a reasonable approximation for pi, 3.14 is nearly as good, and 3.141 is better. Really, if you’re in a context where you’re writing 1 19/24 as your answer instead of 1.792, then you should also be writing π instead of 22/7.
Okay, I stand corrected although I’m baffled by ‘real analysis!’
Thanks Andy, it’s all very confusing but I’m a bit wiser now. 
It’s like people used to complain about the metric system.
“It’s so complicated!”
“Why?”
“Well, it’s a mile for me to walk to school. But in the metric system it’s 1.60934 kilometers! I weigh 200 pounds, but in the metric system that’s 90.7185 kilograms! I want a gallon of milk at the store, but in metric it’s 3.78541 liters! It’s so much more complicated!”
It’s not a flaw but it seems to show to my inexperienced eyes that integers are simply inadequate for certain ‘quantities.’ To me, if you can never stop approaching pi or the sqrt of 2, etc. you cannot regard this as equivalent to using integers. The fact, that for any practical purposes it really doesn’t matter, still does not alter ths.
You can also have complex analysis. Which involves complex numbers. As opposed to real analysis - real numbers. The properties of real numbers versus complex numbers are remarkably different, so much so that each has a separate category for itself. Complex analysis has some quite astounding results. Very useful ones it might be said.
Number theory is a fun area. Even such simple things as the properties of prime numbers are mostly number theory. Hence Goldbach’s conjecture, unique prime factorisation theorem and so on.
I don’t want to give the impression I don’t think irrationals are useful and have a right to exist but why are many people here trying to tell me they are the same as integers? Someone posted earlier that .99999999999’ is equivalent to 1! How does *that *work?
Well thank you aceplace, I was beginning to think I was the only one here who saw it this way. 
Of course they’re not the same as integers. Neither are rational numbers, right?
And if 0.99999… is not equal to 1, what’s the difference? I mean that literally.
1 - 0.999999… = what?
It equals zero. If there is zero difference between two numbers they are the same number. Therefore, 0.99999… = 1.
If you want to argue that way out there at the infinity-th decimal point there’s a little 0.000…001 left over, why isn’t it a 0.000…0001 left over? And of course, there is no infinity-th decimal point, because infinity doesn’t work that way.
And sure, lots of people have tried to work out theories of infinitesimals, but to include them as regular numbers that you can do arithmetic with either leads to all sorts of contradictions or requires you to give up on common sense arithmetic.
Like, is there a difference between 2 and 2 + an infinitesimal? How about 2 + 2 infinitesimals?
Anyway, tangent. You don’t want to go there.
OK, but let’s ask what you’re trying to do here.
Are you a carpenter trying to figure out how much wood you’ll need to face the side of a 2 foot diameter wooden cylinder?
Then you’re not making measurements to the millionth decimal place. In this case any approximation of pi that is more accurate than your most inaccurate measurement will be fine. You’ve got a 2 foot cylinder, not a 2.000000000000000000000000 foot wooden cylinder. In this case using 3.14 as your value for pi is fine, because you don’t have measurements more precise than 1 part in a thousand.
But if you’re doing trigonometry you don’t ever want to calculate using the decimal expression of pi. If your answer is precisely 4π/3, that’s the answer you should give. If you’re doing some complex equation, leave pi in there as an exact value instead of approximating it. And if you need a real world carpenter answer at the very end, only do the calculation using an approximate value of pi at the very end, when you know your significant figures and therefore know how precise a value for pi would be useful.
That’s just because you didn’t use the proper representation for 2/3.
.375 + .25 + .666… + .5 = 1.271666…
There’s no loss of precision if you use the proper representation.
It’s long puzzled me why. 22/7 is not a particularly good approximation - it differs from the correct value of pi by about 1 part in 800.
Whereas (unless you have a serious objection to 3-digit integers) you can use 355/113, which differs by less than one part in 3.5 million.
(Mnemonic: start with 113355; split this into 113 355 and re-arrange appropriately.)
Oh, the irony of typos…
22/7 is not considered particularly accurate now, but 4000 years ago it was.
I have never encountered anyone using 22/7 to approximate π for any serious purpose, but it would be interesting to hear anecdotes from anyone who has, even if it was for a back-of-an-envelope calculation.
OK, I’m going to take another stab at this. The square root of 2 has an exact value, and that value is the square root of two. Same with pi. Pi has an exact value, and that value is pi. Yes, these values are not integers. That doesn’t mean they’re not exact. If you multiply the square root of two by the square root of two, you get exactly 2. If you have a square with sides of exactly 1 unit, what is the length of the diagonal? Does it have an exact value? Does the value of the diagonal vary depending on how closely you look at it? Does it get bigger then smaller then bigger than smaller then bigger then smaller? Or is it always the same?
The answer is that it is always the same. Now, it might be fair to say that the value of that diagonal is not a number, but an idea. Lots of ancient mathematicians agreed with that statement. Numbers are for counting things, like one thing and two things. If you can’t express it in terms of counting numbers, then it’s not a number. Like zero is not a number, or negative numbers are not numbers, they’re ideas, but not numbers. Fractions are numbers, and ratios of numbers are numbers, but anything else is right out.
And if you complain that you can’t write the value of the square root of two without being imprecise, that’s not true. You can write it precisely: the square root of two. You can’t write a decimal representation of that number precisely, but so what? Did we make a rule that if you can’t write a precise decimal representation of a number then it isn’t’ a number? Where in the rulebook does it say that?
And then there’s Analytic Number Theory, which uses complex analysis to study the whole numbers. (Arguably the most famous and important unsolved problem in mathematics, the Riemann Hypothesis, is a problem in analytic number theory.)
And the eventual proof of Fermat’s Last Theorem went off into fields of mathematics that didn’t even exist in Fermat’s time.
I don’t think that I’d call the Riemann Hypothesis the most famous or most important unsolved problem, though. The most famous is probably the Goldbach conjecture, followed by the twin prime conjecture. And the most important is probably whether P = NP.
Not only is 355/113 an excellent approximation to pi, it cannot be improved with small numbers. To find a better fractional approximation you must go all the way to 52163 / 16604, and it is only very slightly better. (I know this is related to pi’s continued fraction form, but is there a reason for that?)
P ≠ NP (though this is Gödel-undecidable). This may be less of a computing obstacle as quantum computers begin to serve as Turing’s Oracles.