Here are a few more: (Everything between quotation marks is a false belief.)
Basic logic: Among students: "If A implies B then B implies A" (or "if A implies B then not A implies not B").
Even among mature mathematicians a frequent false belief is to forget that the conclusion of a theorem need not hold once the conditions of the theorem fail. Another common frequent belief is to assume that once the conditions fail then the conclusion must fail too.
Calculus: "The derivative of a differentiable function is continuous."
"An infinite series whose general term tend to 0 is convergent."
Geometry: "The circle is the only figure which has the same width in all directions." (Feynmann regarded this mistake as one reason for the space shuttle discovery disaster).
Polytopes: Often people believe that "given a convex polytope P you can slightly move the vertices to rational positions keeping the structure of the polytope unchanged."
(From Udi de Shalit): Some people believe that "if you hold a cube along a main diagonal, the remaining vertices all lie on a plane." Some even say that their number is 4.
Algebra (Also from Udi) "I have encountered many misconceptions about solvability by radicals. Some people think that 'the solution of an irreducible equation of degree 5 and higher, say over Q, is never expressible by radicals'. Some amature amateur mathematicians even say that 'equations of degree 5 and higher have no solutions'."
Probability: "If you play the casino patiently and carefully you will win in the long run" (and "you do not believe that?, this is my owm own experience on the matter!" and "Indeed when I am calm and patient I win, but when I lose my temper I lose big times".time".)
"an event which may occur has positive probabilityprobability": (not true for infinite probability spaces)
Various places: "If you want to prove that a certain infinite structure exists it is enough to show that there is no upper bound to the sizes of such structures."
Combinatorics: "This is a finite problem, surely you can solve it with a computer."
"Hall marriage theorem is very nice and I am surprised no combinatorialist bothered to extend it to a matching built from triples instead of pairs." (It is unlikely that a general characteriztion when a hypergraph built from triples has a perfect matching (of triangles) will be found.)
Computer science: "It is known that quantum computers can solve NP complete problems in polynomial time."
