DNAunion

DNAunion: I was thinking about something I heard concerning how we would communicate with an ETI if we ever picked up on their signal, and it was said that we would start by using mathematics since it is a universal language (can't remember where I heard or read that). But is it?

The first thing I thought about was pi: 3.1415926535..... pi is a non-repeating, non-terminating number: an irrational number. But would it be for an alien civilization too?

Here's what got me to thinking (I didn't spend too much time on this).

How is one-tenth represented in our normal system of mathematics? Simple, 0.1. The decimal terminates immediately, with just one digit, and it absolutely exact. But, in binary, you cannot represent one tenth exactly no matter how many decimal places you carry it out to. For example:

0.1 = 1/2: too large, have to use 0.0
0.0 = 0
0.01 = 1/4: too large, have to use 0.00
0.001 = 1/8: too large, have to use 0.00
0.0001 = 1/16
0.00011 = 1/16 + 1/32 = 3/32 = 0.09375
0.000111 = 3/32 + 1/64 = 7/64: too large
0.0001101 = 3/32 + 1/128 = 13/128: too large
0.00011001 = 3/32 + 1/256 = 25/256 = 0.09765625
0.000110011 = 25/256 + 1/512 = 51/512 = 0.099609375

and so on. You can keep getting progressively closer and closer to one tenth, but you cannot actually ever represent it exactly using binary. In binary, one tenth is a non-terminating (and non-repeating?) number.

So is it possible that in some other base system, which aliens might use instead of our base 10 system, some numbers that we consider to be irrational numbers (such as pi) actually terminate?

Wizardry

Whether or not a particular number can be represented as the sum of a finite number of fractions with denominators integral powers of a particular number (a base-n "decimal" system) is a completely different issue from whether or not a number is rational.

For example, 1/7 and 1/10 are both rational numbers (as evidenced by the fact that they can be written in the preceding form), but neither can be written as a terminating binary number and 1/7 can't be written as a terminating decimal in base 10. However it can be so written in base 7 as 0.1.

Pi, e, the square root of two etc. are all different; they can't be written in the form a/b, no matter how a or b are chosen. Now, if irrational numbers can't be written as the quotient of two integers, then it follows that they can't be written as the sum of a finite number of rational numbers, since if they could then we could just follow the usual procedure for adding fractions:

c/d + f/g = (cg + fd)/dg
c/d + f/g + h/k = (cgk + fdk + hdg)/dgk

and wind up with a rational number, which we obviously can't do because we already said we were dealing with irrational numbers.

Therefore, it is impossible to represent pi (or any other irrational number) as a terminating decimal using an integral base.

Derec

Irrational numbers are those which cannot be expressed as a quotient p/q where p and q are integers. So it would not matter what number base is used.

Pi is even more than just irrational, it is transcendental, meaning that it is not a root of any polynomial equation with integer coefficients.

Principia

This is why the definition of an irrational number is simply that it cannot be represented as the ratio of two <a href="http://mathworld.wolfram.com/IrrationalNumber.html" target="_blank">rational numbers</a> (or equivalently, two integers). Non-terminating is never a sufficient criteria. Non-repeating is impossible to apply, unless one wishes to calculate every single digit.

BTW, 0.1 does repeat in base 2. Here's the proof.
1/10 = 1/2 * 1/5. For the moment, let's ignore the 1/2 part, because in base 2 that just means moving the decimal over to the left one place.

Let's say after N binary digits, the binary floating-point representation repeats. Then:
2^N * (1/5) - (1/5) is an integer. Our job is to find the smallest positive N such that this is true.

N = 1 =&gt; Nope
N = 2 =&gt; Nope
N = 3 =&gt; Nope
N = 4 =&gt; Yes

So what are the four digits? Well, DNAunion has done all of the hard work for me. All I have to do is look 4 binary digits from the first 0 (remember the 1/2) after the floating point. It is 0011.

So 1/10 = 0.0_0011_ (where _x_ means x repeated).
Time for DNA to brush up on his number theory. An irrational number is irrational in any base, aliens or not.

[ December 01, 2002: Message edited by: Principia ]</p>

wade-w

An irrational number is a number which cannot be expressed as the ratio of two integers. The non-repeating, non-terminating behavior is a consequence of this, not the definition.

That said, the answer to your question is no. An irrational number will be non-repeating and non-terminating no matter what base you use. Unless you used pi as your base, in which case the symbol "10" in this number system would be an exact representation of pi, but all other irrationals such as e would still be non-repeating, non-terminating, so that's really a trivial case.

I think that what was meant by mathematics being a "universal language" is that numbers such as pi or sqrt 2 are universal, and any advanced race will be familiar with those values. After all, the ratio of the circumference to the diameter of a circle or the length of the diagonal of the unit square are going to be the same on Rigel or in the lesser magellanic cloud as they are here on earth.

DNAunion

DNAunion: I think you might have missed what I was saying. Let’s start with this. I “defined” an irrational number as a number whose decimal representation is a non-repeating, non-terminating decimal. (NOTE: Decimal representation here does not deal with base 10, it just means that the expression of the number contains a decimal point, .) Do you consider that to be incorrect?

DNAunion: I perceive your comment about 1/7 to imply that I would have considered it to be an irrational number because it is a non-terminating decimal. If so, let me point out that I didn’t say an irrational number was a number whose decimal representation was non-terminating. 1/3 has a non-terminating decimal representation but it is not irrational. I included two properties: both non-terminating AND non-repeating. The decimal representation of 1/7 may be non-terminating, but it is a repeating decimal, and therefore, is not irrational.


Anyway, I guess I should have spent just a few more minutes on this before posting. I am sure I would have remembered the other way of “defining” an irrational number: a number that cannot be represented as n/d with both n and d being integers (and of course, d cannot be 0). Since 1/10 can be represented as a fraction of integers even in binary (it’s just 1/1010), then it is rational.

I should have just spent a couple more brain cycles before posting. Like I said originally...I didn't put too much thought into the matter.

[ December 01, 2002: Message edited by: DNAunion ]</p>

Answerer

Guys, what do all of you call numbers that have a value of 1/0 or 0/0? Undefined? Or infinite?

Principia

0/0 is <a href="http://mathworld.wolfram.com/Indeterminate.html" target="_blank">indeterminate</a>. 1/0 is undefined, as best as I recall.

wade-w

That is incorrect. An irrational number is a number that is not rational. A rational number is a number that can be expressed as the ratio (hence the name) of two integers.

Undefined. Infinty is not a number.

DNAunion

DNAunion: Obviously.

DNAunion: Okay. But then I have one more question: a thought experiment.

Here is a number, that Bill calculated to 30 decimal places before tiring, that he hands to me.

0.273240653682303929829016482931...

I know nothing about how it was derived. He asks me to determine whether it is rational or irrational.

Is there a way to tell if it is rational or irrational using the "n/d where both n and d are integers" method, without knowing the full decimal representation (if one even exists)?

[ December 01, 2002: Message edited by: DNAunion ]</p>