How many elements are there in the periodic table?

From Actinium to Zirconium, there are 103. (Source: Reader's Digest Illustrated Dictionary of Essential Knowledge)

I think there were 109 last time I looked but the naming and the supposed discovery of the last few tends to be controversial. These elements (like "unnilennium", no. 109) are produced as a few atoms at a time and are so radioactive they are gone before you can blink. As the aforementioned song says, there may be many others but they haven't been "dis-carvard".

As for learning them, a good chemist would know them in order of atomic number as they appear in the Periodic Table, starting with hydrogen. Or at least he would know where they appear in which column as this indicates their chemical properties.

114 -it is called Unuquadium: was discovered in Dubna's cyclotron with 18 Russian and 5 American physicists. I was created by bombarding Plutoniun-244 with a calcium-48 isotope, creating hot fusion after 40 days of bombardment. Some physicists feel there may be elements well excedding the 120 range.

Note that elements 113, 115, and 117 are not known, but are included in the table to show their expected positions. Elements 114, 116, and 118 have only been reported recently. But .... the team of Berkeley Lab scientists that announced two years ago (1999) the observation of what appeared to be Element 118 (heaviest undiscovered transuranic element at the time) has retracted its original paper after several confirmation experiments failed to reproduce the results. This means that the pages for element 118 and parts of the data for element 116 are wrong.
Element 110 now has a name: Darmstadtium with symbol Ds
http://www.webelements.com/

Through 2016, there are 118 recognized elements. Element 117 was the most-recently synthesized, in 2010, while ones like #116 and 118 were done back in 2000 and 2002.

At the moment, elements 113, 115, 117, and 118 do not have formal names, but Nihonium (Nh), Moscovium (Mc), Tennessine (Ts), and Oganesson (Og) are being considered.

https://iupac.org/iupac-is-naming-the-four-new-elements-nihonium-moscovium-tennessine-and-oganesson/