How difficult is Math Analysis? Some people tell me its easy and some say its rediculously hard......opinions?

Answer 1

It is challenging, especially since it is most students first exposure to rigorous proof-oriented mathematics. You would be wise to take an advanced calculus course first, which introduces you to the concepts of epsilon-delta proofs, sequences, etc etc.

Answer 2

Math analysis is fun. Math analysis is hard (because you have to actually understand how math works). However, once you do, you will make it appear easy (because you will find it fun).

Math analysis is the application of logic in its (almost) purest form. It is based on checking things that are either completely true or completely false, about a problem, then build a chain of reason from such rules.

It is also very helpful to solve problems where you only need to determine if an answer exists, without having to find the actual answer.

An easy example: is this number prime?

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If you had to find the actual factors, it would be a long calculation. However, you note that it ends with a 5 which makes it divisible by 5 (therefore, it cannot be a prime number).

A tougher example:

is the square root of 2 a rational number?
A rational number is one that can be expressed as a fraction of two integers: such as 7/17 or 36/13 or 5 (which is 5/1).

Any rational number can be written as p/q where p and q are integers. Also, any rational number can be reduced to its simplest state, where p and q are prime to each other (they share no common factor).

For example, 33/87 is a rational number, but the two numbers do share the factor 3 (they are both divisible by 3). So we can rewrite it as 11/29, which has exactly the same value and 11 and 29 are bot prime numbers (they cannot share common factors).

So, let us assume that the square root of 2 can be written as a rational number expressed as p/q, where p and q are relatively prime (they share no common factor):

√2 = p/q
square both sides:
2 = p^2 / q^2
2*q^2 = p^2

p^2 must be even (it is made up of a number, multiplied by 2).

If p^2 is even, then p must be even
(check it out: if a number is even, its square must be even; if the number is odd, its square must be odd).

So, we can write p as 2*r for some integer r (r being exactly half of p).
Because p is even, then q must be odd (otherwise they would share the factor 2 and that is forbidden by us making them relatively prime).

Rewrite our original formula:

√2 = (2*r)/q
2 = (2*r)^2 / q^2
2 = 4 r^2 / q^2
1 = 2 r^2 / q^2
q^2 = 2 r^2

And we discover that q must also be even, which is impossible.

We conclude that √2 cannot be expressed as a rational number p/q. And we did that without having to calculate the actual value.

The Greeks were already using this analytical proof that √2 is irrational, 2000 years ago.
Which is good, because they did not have calculators...

Answer 3

I recommend you read a book by Victor Bryant in you spare time. Read it before you go into the course because he motivates the subject very nicely and makes the transition form high school mathematics to university natural. I do agree with Steve, its an excellent idea to have an idea about limits, sequences and series.
If you are still in high school, the AP calculus BC would introduce
you to this topic.

Answer 4

well, for me math is very difficult, cause i used to hate numbers, but it depends to that person, if you like or more to say love numbers then it might be not that difficult for you,... i am not saying that math is boring, actually it always makes my nerve to palpate...