Producing Examples and Counterexamples Kun: “When encountering a new existing definition, the first thing every mathematician does is write down examples and counterexamples to help them understand it better.” (x*y) is faster than (x+x+x+...+x) with y 'x's ex: 10*10 faster than 10+10+10+10+10+10+10+10+10+10 ce: 5*0 faster than sqrt(x*x + y*y) and x*sqrt(1 + (y/x)*(y/x)) do the same thing ex: sqrt(1.414...*1.414... + 1.414...*1.414...) = 2 1.414...*sqrt(1 + (1.414.../1.414...)*(1.414.../1.414...)) = 2 ce: sqrt(0*0 + y*y) = |y| and 0*sqrt(1+(y/0)*(y/0)) = error