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January 23rd, 2019, 05:14 AM   #61
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Originally Posted by v8archie View Post
Or you could try to understand what recurring decimals actually mean and also the limitations of calculators.

It's rather easier than trying to guess which answers are rational and which aren't.
@V8archie - If you can explain to me what recurring decimals do actually mean then I would be grateful. I’m here to learn, not antagonise.

If I can find all these answers in a text book, please point me in the right direction and save your effort.

So, if a recurring decimal is not a real number, rather a representation, what real number does say 0.333 represent. It represents a 1/3

So, I’m assuming here that all fractions are real numbers and all recurring decimals represent an expressible fraction.

I also assume you can flit between representative numbers and real numbers in the same calculation. For example, if I divide 1 by 3 I come up with the representative decimal 0.333 but if I go on to multiply that by 6 I arrive back at the real number ‘2’.

However, if I work fractionally the first calculation is 1/3 and my second is 2.
So for what use does the representative decimal serve if a fraction does the same job?

If decimal representation is by its very nature ‘approximate’ then how is this anomaly allowed to function when a real number fraction does the same job but is perfect (not approximate).
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January 23rd, 2019, 05:52 AM   #62
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OK this seems a step in the right direction.Any infinitely recurring decimal is simply stricken from mathematical language and always expressed fractionally.
Even on calculators. We may need a linear method of expressing fractions and then forgot the whole decimal system between integers?
You are mixing up math and applied math.

Decimal representations are frequently necessary approximations when solving practical problems in the physical world.

Math tells us that the ratio between the hypotenuse of a right isoceles triangle and either of the other sides cannot equal the ratio of ANY imaginable pair of integers. (The proof is elegant.) In other words, we can't use integers at all. This is far more dramatic than the fact that 1/3, which can be represented in terms of integers, can be added, subtracted, multiplied, and divided without use of decimal notation, and can be represented in a finite number of digits in, for example, the duodecimal system of notation.

But the irrationality of the square root of 2 has no practical ill effects, none at all. We have no practical way to determine whether something even is a perfect triangle, let alone a perfect right isosceles triangle. In practice, we cannot measure the sides exactly. What we can do is to say that something is approximately a right isoceles triangle with sides that have an approximate length. In the world of practicalities, we are almost always dealing with approximations. When our approximations are good enough, we find no measurable differences from the results predicted by the idealizations of pure mathematics.

You are a project manager. Do you measure time by the microsecond? If a project is due on the 14th, and it is not complete until 2:33 a.m. on the 15th does anyone get fired or even yelled at? Do the accountants raise a fuss if you are 33 cents over budget? Do you get a promotion if you are 33 cents under budget?

Math deals with perfect things that can only be imagined. It gives results that can be applied to the real world when our approximations are good enough.

The mathematician Kronecker said "The dear God made the whole numbers; everything else is human stuff." If your religion is of the mystical bent, such as Platonism, you can interpret that to mean that fractions et al. are necessarily imperfect because they are not divinely created. No one ever divided a cake into exact one thirds.

In the real world, if the will says to divide the estate equally among three heirs and the estate is worth one million dollars, the division cannot be done perfectly. But it can be done approximately. One third is an idea. It exists in the mind. It cannot always be realized in the physical world. Somehow, however, I doubt that the heir who got $333,333.32 is all that unhappy.
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January 23rd, 2019, 06:12 AM   #63
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Somehow, however, I doubt that the heir who got $333,333.32 is all that unhappy.
thank you - that is a great explanation. Whole numbers are king.

Although I think that the 2 heir's who received $ 333,333.33 wouldn't resent the other who got $ 333,333.34
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January 23rd, 2019, 06:14 AM   #64
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@V8archie - If you can explain to me what recurring decimals do actually mean then I would be grateful. I’m here to learn, not antagonise.

If I can find all these answers in a text book, please point me in the right direction and save your effort.

So, if a recurring decimal is not a real number, rather a representation, what real number does say 0.333 represent. It represents a 1/3

So, I’m assuming here that all fractions are real numbers and all recurring decimals represent an expressible fraction.

I also assume you can flit between representative numbers and real numbers in the same calculation. For example, if I divide 1 by 3 I come up with the representative decimal 0.333 but if I go on to multiply that by 6 I arrive back at the real number ‘2’.

However, if I work fractionally the first calculation is 1/3 and my second is 2.
So for what use does the representative decimal serve if a fraction does the same job?

If decimal representation is by its very nature ‘approximate’ then how is this anomaly allowed to function when a real number fraction does the same job but is perfect (not approximate).
I shall let Archie answer in detail.

All recurring decimals represent a fraction.

$\dfrac{1}{4} = 0.2500000000...$

$\dfrac{1}{7} = 0.142857142857142857....$

What mathematicians call a "real number" include all sorts of things that cannot be observed in the physical world, for example

$\dfrac{\pi}{\sqrt{2}}.$

"Real" to a mathematician means "imaginable, and actually imagined before the 16th century." It is what the lawyers call a term of art that has a historical rationale.

Finally, mathematicians prefer fractions to decimals. When, however, it comes to the brute work of computing with what are frequently approximations to begin with, decimals are easier to work with. No one likes finding common denominators.
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Last edited by JeffM1; January 23rd, 2019 at 06:18 AM.
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January 23rd, 2019, 06:25 AM   #65
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thank you - that is a great explanation. Whole numbers are king.

Although I think that the 2 heir's who received $ 333,333.33 wouldn't resent the other who got $ 333,333.34
LOL

I screwed up my own example. That's funny.

I won't fix it lest others miss the joke.

In practice, the one who got the extra penny would probably end up paying for the drinks and so come out behind. Such complications do not bedevil the austere beauties of pure mathematics.
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January 23rd, 2019, 06:54 AM   #66
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Originally Posted by sintan View Post
@V8archie - If you can explain to me what recurring decimals do actually mean then I would be grateful.
You will need to learn about the theory of limits principally. What follows is an imprecise overview of what the theory of limits, what the real numbers are and what decimal notation means.

Essentially, we can imagine infinite sequences of numbers.
Some of these infinite sequences converge to a single value. An easy example is that the sequence $$\frac11, \, \frac12, \, \frac13, \, \frac14, \, \cdots$$ converges to zero. That is, the terms of the sequence get closer and closer to zero - and we can get the sequence arbitrarily close to zero by selecting sufficient terms. One definition of the real numbers is that they are the set of limits of all convergent infinite sequences.

If we add together the terms of an infinite sequence, we get a series - an infinite sum. $$\frac11 + \frac12 + \frac13 + \frac14 + \cdots$$
Now, it's not actually possible to add together an infinite number of terms - if you are using a calculator, you never get to press the "equals" button - but by considering the partial sums of this series, we can create a new sequence$$\frac11, \, \frac11 + \frac12, \, \frac11 + \frac12 + \frac13, \, \cdots$$ and we define the value of the infinite series, or the more loosely the result of the infinite sum, to be the limit of the sequence of these partial sums as we take more and more terms. (As long as such a limit exists).

The decimal representation of numbers is precisely a representation of some of these sequences. So the number $a_0.a_1a_2a_3a_4a_5\ldots$ represents the infinite sum $$a_0 + \frac{a_1}{10} + \frac{a_2}{100} + \frac{a_3}{1000} + \frac{a_4}{10000} + \frac{a_5}{100000} + \cdots$$
More specifically, the number $0.33333\ldots$ represents the infinite sum $$0 + \frac3{10} + \frac3{100} + \frac3{1000} + \frac3{10000} + \frac3{100000} + \cdots$$

We can show that every single one of these infinite sums (depicted by a decimal representation) converges, and thus for each one the value to which it converges is the value that the decimal representation represents.

In particular, we can say that \begin{align}\text{if} \quad S_n &= 1 + x + x^2 + x^3 + \cdots + x^{n-1} + x^n \\ \text{then} \quad xS_n &= \phantom{1 + {}} x + x^2 + x^3 + \cdots + x^{n-1} + x^n + x^{n+1} \\ \text{and by subtracting the second line from the first} \qquad \\ S_n - xS_n &= 1 + \phantom{x + x^2 + x^3 + \cdots + x^{n-1} + x^n} - x^{n+1} \\ (1-x)S_n &= 1 - x^{n+1} \\ S_n &= \frac{1-x^{n+1}}{1-x}\end{align}
Now, when $-1 \lt x \lt 1$, the value of $x^{n+1}$ gets smaller and smaller as $n$ increases so we can say that the limit of the partial sums $S_n$, which is the infinite sum, is $$1 + x + x^2 + \ldots = \frac1{1-x}$$

Thus, in the case of our $0.333\ldots$ we get \begin{align}0 + \frac3{10} + \frac3{100} + \frac3{1000} + \frac3{10000} + \frac3{100000} + \cdots &= \frac{3}{10}\left(1 + \frac{1}{10} + \left(\frac{1}{10}\right)^2 + \left(\frac{1}{10}\right)^3 + \cdots \right) \\ \text{here we have $x=\frac1{10}$ so} \qquad &= \frac{3}{10}\left(\frac{1}{1-\frac1{10}}\right) \\ &= \frac{3}{10} \left(\frac{1}{\frac{9}{10}}\right) \\ &= \frac{3}{10} \cdot \frac{10}{9} \\ &= \frac39 \\ &= \frac13 \end{align}

You can find an introduction to limits in any beginners book on Calculus.
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January 23rd, 2019, 07:54 AM   #67
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Uh, yeah. So how does that work? You want to show the notation to somebody else no? How does somebody else see this if it isn't made out of atoms? How do you write it down if it isn't made out of atoms?
You don't write it down as nobody can write a perfect line segment of any length. I asked why you think we have to be able to and got ignored. It should be obvious you and Maschke are implementing physical limitations that I am not. These are theoretical line segments of length $x$ inches where $x \in (0,1)$. If we cared to, we could let each line segment be a symbol for $\frac{1}{3}$.

The only point is that we can assign whatever symbol(s) we like to a number so as to create a notation for it. You asked if there are uncountably many symbols we could assign. I said yes. So did Maschke (just not in this thread). That's it. Nothing more to see here aside from what I considered to be an amusing conversation.
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January 23rd, 2019, 08:35 AM   #68
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As an addendum to the above, you can find the fraction that represents a recurring decimal quite easily:

Given
$$x = 2.46358358358 \ldots = 2.46\overline{358}$$
We multiply by the a power of ten that gets one of the recurring sets of digits to the right of the decimal point: in this case, $100$.
\begin{align}x &= 2.46\overline{358} \\ 100x &= 246.\overline{358} \end{align}
And then by a power of 10 that gets the recurring part to the left of the decimal point, in this case 1000.
\begin{align}100x &= 246.\overline{358} &(1) \\ 100000x &= 246358.\overline{358} &(2)\end{align}
We still have the recurring part to the right of the point because that's what recurring means - the digits repeat for ever.

Now we subtract the first line $(1)$ from the second $(2)$, causing the decimal parts (to the right of the point) to cancel each other:
\begin{align} 99900x &= 246112 \\ x &= \frac{246112}{99900} \\ \text{factorize} \qquad &= \frac{2^5 \times 7691}{2^2 \times 3^3 \times 5^2 \times 37} \\ \text{and cancel} \qquad &= \frac{2^3 \times 7691}{3^3 \times 5^2 \times 37} \\ &= \frac{61528}{24975}\end{align}
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January 23rd, 2019, 09:19 AM   #69
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Originally Posted by AplanisTophet View Post
. Nothing more to see here aside from what I considered to be an amusing conversation.
More amusing than the OP, for sure.

But a notation is inherently a physical thing. A notation is not something mathematical. A notation is something to communicate a mathematical idea to an audience. So we are bound by physical rules.
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January 23rd, 2019, 10:22 AM   #70
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A notation is something to communicate a mathematical idea to an audience. So we are bound by physical rules.
If my half-hearted examples of ways to create uncountably many symbols need to abide by physical rules, then I don't know how to make uncountably many off the top of my head. Perhaps my idea of letting each element of $\mathcal{P}(\mathbb{N})$ be a symbol for $\frac{1}{3}$ suffices, but then again, we could never write out each element of $\mathcal{P}(\mathbb{N})$, so I trust that fails your test also.

On the other hand, I would like Maschke to clarify how we come up with uncountably many symbols then. Was that just a hand-waivy assertion taken from a published text?
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