is 24 too late to becoma a math whiz?

Mathematics is like no other science in that the divergence of the field is far greater than any other science. The highest level of abstract mathematics is really for the young...The Fields Medal which is considered the highest math honor as there is no Nobel prize (Wolf Prize may beconsidered)is awarded only to those under 40. Some people willargue that anyone over the age of 40 is too bogged down by prejudices to truly coe up with something revolutionalry enough to justify a Fields medal.

However, this is only th ehighest level of Mathematics. There are very few prerequsites in Business school for mathematics. Prior top starting B school, I believe students are expected to attend "Math Boot Camp" for those with a non-science backfground.

Here is another question. Prove that there are more irrationalthan rational numbers.

Are you "going back to school" because you don't have an undergrad degree or to get your masters? To get an undergrad math degree, you definitely don't have to be a "math whiz", just moderately talented. However, I don't know that someone with a math B.S. is that employable. As someone else mentioned, if you're starting with nothing, you might want to go into statistics. At least take some applied math and minor in another science. If you already have some of your undergrad classes out of the way, take a more theoretical proof-based class as soon as possible. That will tell you quickly whether you have the potential to succeed in grad level math or not.

math dude wrote:

Here is another question. Prove that there are more irrationalthan rational numbers.

Premise: a question is a grammatically symbolic object that ends with a "?".

Hypothesis: you are trying to be interesting on a cynical running board by talking about math.

Proof: your question lack a "?".

Therefore, you aren't actually asking a question, i.e., typing before thinking.

QED: you don't know how to write using letters.

ASAP: go back to math.

Correction: I should have said "problem" instead of "question".

Addition: You are an idiot.

You wrote the wrong thing.

I'm an idiot.

Very syllogistic.

I agree with your second statement.

math dude wrote:

Here is another question. Prove that there are more irrational than rational numbers.

I'm sure I could have given you a more elegant proof during my course in real analysis, but here's a try:

(1) There are countably many rational numbers

(2) There are uncountably many real numbers in the interval (0,1)

(3) Therefore, since the union of two countable sets is countable (trivial proof), there are uncountably many irrational numbers in (0,1)

The proof for (1) isn't that hard, you just write the irrationals in matrix form, starting with 1/1 in the top left, and go like this:

1/1 1/2 1/3 1/4...

2/1 2/2 2/3 2/4...

3/1 3/2 3/3 3/4...

.

.

Then to count them you start at 1/1, go to 1/2, then 2/1, then 3/1, 2/2, 1/3, 1/4, 2/3, etc...snaking back and forth.

I'll leave (2) for someone else to prove.

I doubt any natural aptitude that you might have had for anything intellectual has disappeared. Math lets you do amazing things; if you're interested, then go after it.

Very good, this essentially leads to the discussion on levels of differing infinity.

Essentially create a decimal matrix such that A1 does not equal A2 which does not equal A3. A3 does not equal A1 or A2.

These rows of infinitely long decimals can theorhetically account for every rational number possible.

The end result is that the diagnol of this matrix creates a decimal that can not exist on the infinite list of rows of infinitely long decimals.

Thus we have shown that there is a 1 to 1 correlation with the list and all Natural numbers.

Therefore, the diagnol decimal formed proves there is at least on number not accounted for on the infinite list of Natural numbers.

The idea of uncountability may seem like a silly mental exercise, but is actually significant in theoretical computer science. A regular language has a countable number of statements, whereas a context-free language does not. The idea of uncoutability was first proposed by Cantor in the 1800's, way before Turing laid the foundation for theoretical computer science during WWII.

Nice call, What I just described is a proof originally constructed by Gregor Cantor himself.

I seem to remember reading that Cantor tried for quite a while to prove that the irrationals were countable, and that it didn't immediately occur to him that they were not. His ideas also were not immediately accepted by his contemporaries.

Correct again, he actually spent several years in a Russian mental institution because of all of his theories regarding infinity.

I believe he derived several of his theories in the mid 19th century, and not a single contemporary backed his findings. Thus he was deemed crazy.

Several decades later almost all of his theories relating to infinity and the differing sizes of infinity have been accepted.

He is probably one of the last true mathematical geniuses of modern time.

Some number of years ago a 49 year-old student of mine completed a degree in electical engineering. He was a student of mine in multivariable calculus, differential equations and vector analysis. He had obviously been out of school for quite a while, and had come back to school because of a work- related shoulder injury. His 18 year-old classmates did seem quicker at times, if only because they were fresh out of high school, but I don't think they were any brighter. They just had a quicker recall of certain benchmark facts that us educators often mistake for a measure of intelligence. This student kept in contact with me, and went on to do quite well in comprehending the Dirac Delta function, Fourier coefficients, linear algebra, and a whole spectrum of fairly intense math concepts that are generally beyond the comprehension of all but the best students. Had this student lost some of his "sharpness"? I don't think so. On the younger side of the scale, I have a lot of students in their early to mid 20's, and there is little difference between them and their 18 year-old classmates.

Do mathematical abilities really decline significantly with age? Is it possible for a middle-aged person to be successful at the highest levels of math?

Math is like running and exercise - if you practice them (math and running) you retain a good level...with insignificant decline....especially with mathematics.

This is why you have Math Professors in their 60's and 70's still contributing in the field (of mathematics) and losing very little (if anything) to their younger peers.

What the Math Professors (the older ones over 40-50-60)

lose in speed, is made up in overall experience and perspective.

Math is a skill which needs frequent practice and exposure, to maintain ability.

The link below suggests that Newton wasn't even exposed to advanced mathematics until he was at least 20 years old. But once he began, he progressed rapidly.

grad student wrote:

math dude wrote:

Here is another question. Prove that there are more irrational than rational numbers.

I\\\\\\\'m sure I could have given you a more elegant proof during my course in real analysis, but here\\\\\\\'s a try:

(1) There are countably many rational numbers

(2) There are uncountably many real numbers in the interval (0,1)

(3) Therefore, since the union of two countable sets is countable (trivial proof), there are uncountably many irrational numbers in (0,1)

The proof for (1) isn\\\\\\\'t that hard, you just write the irrationals in matrix form, starting with 1/1 in the top left, and go like this:

1/1 1/2 1/3 1/4...

2/1 2/2 2/3 2/4...

3/1 3/2 3/3 3/4...

.

.

Then to count them you start at 1/1, go to 1/2, then 2/1, then 3/1, 2/2, 1/3, 1/4, 2/3, etc...snaking back and forth.

I\\\\\\\'ll leave (2) for someone else to prove.

Doesn\\\\\\\'t question 2 go something like:

Assume there are countably many real numbers in the interval (0,1).

So take any interval inside (0,1) such as (.259, .26), but we can find even more (uncountable) intervals inside (.259, .26).....

Yeah I know this is very crude but it has been a while.

I will throw this one out there as it was a favorite first day question of some of my professors.

Please prove that if a space is closed and bound that it is compact.