<P> A second animated proof of the Pythagorean theorem . </P> <P> Some illusory visual proofs, such as the missing square puzzle, can be constructed in a way which appear to prove a supposed mathematical fact but only do so under the presence of tiny errors (for example, supposedly straight lines which actually bend slightly) which are unnoticeable until the entire picture is closely examined, with lengths and angles precisely measured or calculated . </P> <P> An elementary proof is a proof which only uses basic techniques . More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis . For some time it was thought that certain theorems, like the prime number theorem, could only be proved using "higher" mathematics . However, over time, many of these results have been reproved using only elementary techniques . </P> <P> A particular way of organising a proof using two parallel columns is often used in elementary geometry classes in the United States . The proof is written as a series of lines in two columns . In each line, the left - hand column contains a proposition, while the right - hand column contains a brief explanation of how the corresponding proposition in the left - hand column is either an axiom, a hypothesis, or can be logically derived from previous propositions . The left - hand column is typically headed "Statements" and the right - hand column is typically headed "Reasons". </P>

What are the 3 important kinds of mathematical statement