What is going on in the third example?
"Let a,b,m,n be integers, and suppose that b^2 = 2a^2 ..."
So, (b^2)/(a^2)=2 b/a = sqrt(2)
or, in other words, sqrt(2) is rational. then it goes and uses this in the rest of the example.
"Let a,b,m,n be integers, and suppose that b^2 = 2a^2 ..."
So, (b^2)/(a^2)=2 b/a = sqrt(2)
or, in other words, sqrt(2) is rational. then it goes and uses this in the rest of the example.
That's hilarious. It's the brick joke, in math. https://plantsarethestrangestpeople.blogspot.com/2012/01/bri... It's a great example, made terrible (for teaching) because the punchline is never explained.
Your excerpt left out key information, which is the secondary hypothesis that am+bn=1
Hint: what are a and b? I won't wait for you to find values that satisfy the hypotheses.
It's the same flavor as: assume a=b+1, and b=a+1. Then a-b=b-a You can prove that p=>q, sometimes, even if p is false. (Especially if p is false!)
A later chapter provides the punchline.
https://hrmacbeth.github.io/math2001/07_Number_Theory.html#t...
Nothing special is going on there, but it may be distracting if the joke doesn't land.