What's the difference between bachelor's, master's and Ph.D. in math

Is a Ph.D. title holder(math) necessarily better informed than a bachelor or master (math)?

Thank You.

There's a huge difference. A bachelor's requires a certain number of math credits. Someone with a BA or BS knows a lot of math and is pretty good at it. A master's requires at least one more year of math. In grad school, it's likely to be more theoretical, and require more classes about proving theoretical results than undergrad courses. Someone with a MA or MS is likely to know more math and more ABOUT math (theoretical underpinnings) than someone with a bachelor's degree. Some testing may be required in a couple of areas (sometimes called "comprehensives"). The tests aren't super hard, but someone with talent who has done well in his courses normally passes. But not everyone passes on the first try. Someone who can't pass on the number of allowed tries will be asked to leave the school. [I have an MA--I'm a Ph.D. program dropout, although I was doing well enough to go on to a second year--just didn't want to continue. A roommate of mine went on to finish his Ph.D., and he and I still talk math once in a while, but I can't hold a candle to him. I just talk to him about interesting things I come across.]

A Ph.D. requires at least 3 years (which would be the shortest time one could fulfill all requirements(, often 4-5, maybe more (there is a time limit), more credits than a master's, and a dissertation, which requires some advanced research in the candidate's area of specialization. It's like a term paper that takes two years of research and requires some original results, not just a regurgitation of something out of books and journals. Someone with a Ph.D. knows even more than someone with a master's, and has demonstrated the ability to do original research. At least, that's the normal expectation. Some dissertations may be of higher quality than others, but it's still not an easy thing to do. A Ph.D. candidate works with an experienced adviser in his field of specialization, who will guide and mentor him in finding a good problem to work on and seeing it through. Then the paper will be presented to a committee for review, and there will be an oral "defense" where he presents the results publicly to the committee and whoever else wants to show up, and they get to ask questions. However, that's pretty much a formality at that point, and it would be rare indeed for that to be a serious stumbling block.

Someone who's gone through the process farther than I have may have more up-to-date comments. I got my MA in 1972.

Idealistically, perhaps the PhD is better informed at some point in his or her life, but not necessarily forever. I've had to relearn math three times because of an illness, and it's becoming more difficult for me the third time around.

I think what you're trying to ask is if the PhD had to take more difficult courses or put more time into the subject, and that would likely be correct. I know persons who bicker over which better college they went to, or their children went to, so I'd say my answer is just my opinion.

Necessarily? No. It is possible for a Ph.D. to, say, get into an accident, and forget all of the math!

But generally, the more you study something, the more you know about it.