# For you maths-people as I'm confused.

How is it possible that I can write (for example) 1/3 and be 100% accurate while I couldn't write the same number in decimals being just as accurate?

It has to do with our choice of system. But what is it exactly? I don't seem to grasp it or isn't there anything to grasp?

matt grime
Homework Helper
it's nothing to do with a deficiency in the representative powers of the system, but with your intent to use a word such as "accurate" and hope it means something mathematically for it to translate as 'looks pretty' and 'doesn't take up too much space on the paper'.

Thanks for clearing that up... though I still think it's weird.

Lorentz said:
It has to do with our choice of system. But what is it exactly? I don't seem to grasp it or isn't there anything to grasp?
There isn't anything to grasp, depending on the system you use, you will not be able to accurately display certain fractions, like for example 1/3 in both binary and decimal system you would need an infinite amount of numbers. That is an entire issue when doing math in the computer and one that has already frequently caused me problems. It is not weird, just plain annoying.

Can't you write it in decimals like this:

$$\frac{1}{3} = 0.\bar{3}$$

Aren't both ways 100% accurate?

Integral
Staff Emeritus
Gold Member
Or you could write it

$$.1_3$$

That's base 3. The existence of the real number line is independent of the any representation of points on the line. In a Real Analysis course, where the properties of the Real Line are explored in depth, such representations are not used, thus the results apply to all representations.

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TALewis said:
Can't you write it in decimals like this:

$$\frac{1}{3} = 0.\bar{3}$$

Aren't both ways 100% accurate?

Actually, I don't think that's true. If you multiply your equation there through by 3, you'll see why.

matt grime
Homework Helper
please say you're not about to start another pointless 1 is not equal to 0.999... thread. they are equal, end of story

Yeah:

$$1=0.\bar{9}$$

I think that's an often asked question. But it's true. Or that's what they tell me.

Zurtex
Homework Helper
Stevo said:
Actually, I don't think that's true. If you multiply your equation there through by 3, you'll see why.
Before you go there, it is true and there are many many proofs that as stated above $1=0.\bar{9}$. If you are interested please try and learn them and see why they are right then just trying to argue them wrong illogically.

Ok, I apologise for making that comment. I found a proof, I'm convinced!

TALewis said:
$$1=0.\bar{9}$$

Does this notation mean: 0.999999... (going on infinately) ?

You can't get exactly 1/3 exept through drawing eg.

Triangle with two 1cm sided sides and a hypotonuse the Hypotonuse will be exactly = to Sqrt of 2

Lorentz said:
Does this notation mean: 0.999999... (going on infinately) ?

Yes, that's the so-called "bar" notation. The line indicates a repeating sequence of numbers. Here is another:

$$\frac{1}{11} = 0.\overline{09}$$

Here two digits 09 repeat continuously (0.09090909...).

Integral
Staff Emeritus
Gold Member
As mentioned above, in every base, there are some points on the Real line which cannot be represented in a finite number of digits. A very interesting example of this is:
$$.1_{10} = .000 \overline {1100}_2$$

The implication of this is that any time .1 is used by a computer in a calculation it MUST be rounded off.

Accurate Representation

TALewis said:
Can't you write it in decimals like this:

$$\frac{1}{3} = 0.\bar{3}$$

Aren't both ways 100% accurate?

Of course, you could write 1/3 as .3 with a bar over the three to represent an unending decimal, but this is again just a mater of "what looks pretty," and has nothing to do with the calculations in a computer. You see that 0, and 5 are divisable in base 10, and thus, 1/2 = .5. Now if you want to do math to the base 3, then we have 1/3=.1 (base 3) same as we have in base 10, 1/10 = .1 (base 10). By changing bases we solve such problems!?

jcsd
$$0.\bar{3} = 0.\dot{3}$$
$$0.\overline{30} = 0.\dot{3}\dot{0}$$
$$0.\overline{307} = 0.\dot{3}0\dot{7}$$