## Oh dear!

Tuesday 30 November 2004 at 7:38 pm | In Articles | 3 Comments

At the risk of repeating myself, I just have to post some questions a student appears to have been asked to answer by his teacher. I say appears because the student was seeking help and may have misunderstood the questions, but I doubt it. They are horrifying.

In the following questions, say which of

a) 0    b) 1    c)    (d) indeterminate

 1 5 9 13 2 6 10 14 3 7 11 15 4 8 12 16

Yes the questions do purport to be about the normal real number system. Shocking, isn’t it, that a student could be asked such drivel?

PS Just come across Not Infinity! which gives a good explanation of this type of misuse.

1. This kind of question is not of much interest to mathematicians, but I suspect it is endlessly fascinating to students. I frequently get questions along these lines: “can you show me why 0.99… is equal to one.” Everyone likes to have “one little fact” up their sleeve to trot out at cocktail parties and the like, and if it involves a grand-sounding idea like infinity. All the better!

Comment by patrick — Wednesday 1 December 2004 9:20 pm #

2. I recently took the GRE exam. Guess what was on it?

Comment by Bill Tozier — Thursday 10 February 2005 3:28 am #

3. Interesting post on infinity. Your explanation “not infinity” says this “What the mathematician means when he (uh, it) says that 1/∞ = 0, is that if you take 1 and divide it by a very large number x, it will be “close to” 0, and that the larger the number, the closer 1/x will be to 0.” An interesting conjecture, but if you actually go and read something like Introudction the Analysis of Infinities by Leonard Euler, I repeat LEONARD EULER, you can find him write the following “If a=0, we take a huge jump in the values of a^z. As long as the value of x remains positive, or greater than zero, then we always have a^z=0. If z=0, then a^0=1. However if z is a negative number, then a^z takes on an infintely large value; for example, if z=-3, then a^z=0^(-3)=1/(0^3)=1/0, which is infinite.”
Your post also talks like mathematicians don’t qualify as people when he said “the mathematician means when he (uh, it) says that 1/∞ = 0″ Mathematicians do qualify as people and homo sapiens.
Additionally, as Ian Stewart points out, calculations like this “limx→∞ 1/x = 0″ treat infinity just like it was a number.
This idea doesn’t quite work “The set of natural numbers is not “built up” by repeating some operation forever. You can never reach infinity by repeating something finite. That’s why it’s called infinity, duh.” Because it draws too much of a distinction between infinity and finiteness. The concept of an infinitesimal (a conception of infinity) allows for one to find the infinite within the finite.
“(And it is called that, precisely because it defines what infinity is.)” No, the axiom of infinity only gives one conception of the plurality which we call infinity.
“You cannot attain to infinity; you need to be given an already-completed infinity by a Higher Power.” But when you integrate equations like (x^2)dx from 0 to 6, you attain to an infinity of terms for that interval.
“But there are such things as infinite quantities, and it is in fact possible to define consistent arithmetic on them.”
Yes, using crisp set theory. But that still didn’t exhaust the idea of infinity in mathematics, because other mathematical ideas like that of fuzzy set theory, neutrosophic set theory, non-standard analysis, etc. emerged after crisp set theory.
“So when dealing with infinite quantities, don’t blindly assume that the rules of finite arithmetic automatically apply to them.”
But, a key rule does hold in both regular arithmetic and crisp set theory. Both regular arithmetic and set theory say that you either have an answer that works within the set of real answers or it doesn’t (a characteristic function which maps into the set {0, 1}).
“An awful lot of sets seem to be the same size as N.”
No, not even close. Neutrosophic sets don’t have the same size as N. Fuzzy sets don’t. Chaotic sets, I conjecture, don’t. Randome sets, I conjecture, don’t. A sets of real numbers doesn’t. I conjecture that the set of complex numbers doesn’t have the same size. I conjecture that the set of fuzzy numbers doesn’t have the same size, nor that fuzzy complex numbers have the same size. Fuzzy real numbers don’t have the same size. The set of planets in the solar system doesn’t have the same size as the integers (it doesn’t work out as a whole number, because you have planets like Pluto which people don’t agree on being either a planet nor a non-planet, making such have a membership value for its membership function of neither 1 nor 0). The set of tall men doesn’t have a similar measure of cardinality, as you have a fractional number for finite parts of the set. The set of readable books in your room, also doesn’t work out as the same as that of the natural numbers, since you can read books to fractional degrees that you don’t get or don’t make sense, while that of the natural numbers has a integer cardinality for finite subsets of it. Etc.
“That’s the topic for the next article. We will see that there isn’t just one infinity; there are actually many more!”
Well, that does work out. But, then we would preferably talk of infinities or the pluarlity of infinity. Yet, not even the mathematical community, in general, currently has the fortitude and courage to do this.

Comment by Doug Lefelhocz — Wednesday 28 June 2006 6:01 am #