[olau]

I was taught something of the same.

But I think it was misguided. I'll note that 1/3 is not a number, it's a calculation. So more complicated.

And fractions are generally much more complicated than the decimal system. Beyond some simple fractions that you're bound to experience in your everyday life, I don't think it makes sense to drill fractions. In the end, when you actually need to know the answer to a computation as a number, you're more likely to make a mistake because you spend your time juggling fractions instead of handling numerical instability.

Decimal notation used to be impractical because calculating with multiple digits was slow and error-prone. But that's no longer the case.

[volemo]

> I'll note that 1/3 is not a number, it's a calculation. So more complicated.

1/3 is a calculation the same way 42 is a calculation (4*10^1 + 2*10^0). Nothing is real except sets containing sets! /j

[DemocracyFTW2]

Yes, true. *BUT* 1/3 is a fraction with denominator 3. 1/5 is a fraction with another denominator, and 1/7 has yet another. So how much is 1/3 + 1/5 + 1/7? You can't just add up, you first have to multiply to get to common ground. The decimal expansions of these use the same base and are readily comparable.

[tsimionescu]

How would you add 0.(3) + 0.2 + 0.(142857)?

Computations are exactly the place where fractions shine over (repeating) decimals.

The disadvantage of fractions is that there is an infinite number of ways to represent each rational number - 1/3 is the same number as 2/6 or as 818171/2454513. Also, comparisons are harder as well. It's easy to tell that 1/3 is bigger than 1/7, but is 2/3 greater or smaller than 5/7? Here you have to do a computation to really tell.

Irrational numbers have a similar problem, btw. The square root of two is the same number as the fourth root of 8, but you can't tell this without performing some computations.

[DemocracyFTW2]

> How would you add 0.(3) + 0.2 + 0.(142857)?

Well I don't suggest that adding 1/3, 1/5 and 1/7 isn't more precisely done by keeping them as fractions and multiply them out to get (35+21+15)/105=71/105. In this case it's relatively easy to get an idea of the resulting value but the next computation with other fractions could give me 83742/36476 as you say which is harder to judge without doing the division as well, so the price for doing fractions as non-decimal fractions is that in order to get the sum of three numbers I'll generally have to perform 2 multiplications (ab)c to find a common denominator, than do 2*3=6 different multiplications with three different numerators to get the normalized numerators, then do 2 additions to get the resulting numerator, and finally 1 division to get a normalized answer. That's a whopping 8 multiplications, two additions and 1 division for the sum of three numbers.

If I am given the fractions as above I could also do three divisions to get their decimal expansions, followed by 2 additions to get the decimal expansion of the normalized result; this result will be imprecise in the general case but it can be as precise as I want to.

If I am given the decimal expansions right away then I can do 3+2+1=6 immediately to get 0.6 as an approximate answer which is not bad given that the correct answer is more like 0.6761904... and all I did was looking at the figures. The slightly harder 33+20+14 is already much closer with it's result, 0.67. There's no denying the fact that many mathematical problems are better done with fractions than with decimals but when doing things like physical measurements, decimal expansions are IMO more practical.

[anthk]

No, they arent. Adding periodic decimals can yield terrible results. Just... don't.

[tsimionescu]

This is ultimately a matter of definitions, and neither defining the fractions nor the decimals as the "true" representation of rationals is ultimately more or less correct.

But, operations on fractions are definitely easier than operations on decimals. And fractions have the nice property that they have finite representations for all rational numbers, whereas decimal representations always have infinite representations even for very simple numbers, such as 1/3.

Also, if you are going to do arithmetic with infinite decimal representations, the you have to be aware that the rules are more complex then simply doing digit-by-digit operations. That is, 0.77... + 0.44... ≠ 1.11... even though 7+4 = 11. And it gets even more complex for more complicated repeating patterns, such as 0.123123123... + 0.454545... (that is, 123/999 + 45/99). I doubt there is any reason whatsoever to attempt to learn the rules for these things, given that the arithmetic of fractions is much simpler and follows from the rules for division. The fact that a handful of simple cases work in simple ways doesn't make it a good idea to try.

[anthk]

Rationals are numbers, not calculations. They can evaluate to themselves as members from a set.