No.14405033
ViewReplyOriginalReport
Quoted By: >>14405043 >>14405051 >>14405063 >>14405121 >>14405183 >>14405270 >>14405307 >>14405359 >>14405481 >>14405738 >>14405774 >>14405799 >>14405845 >>14406070 >>14406242 >>14406322 >>14406522 >>14406523 >>14406541 >>14406601 >>14407293 >>14407510 >>14407614 >>14407892 >>14408223 >>14408363 >>14408615 >>14408777 >>14409944 >>14410308 >>14411303 >>14412196 >>14414864 >>14414909 >>14415035 >>14415124 >>14415305 >>14415642 >>14416101 >>14416165 >>14417641 >>14420180 >>14420195 >>14420204 >>14420276 >>14420277 >>14420284 >>14420430 >>14420800 >>14420803 >>14421949 >>14423004
You are an eighth grade math teacher, and you've just finished showing your class how to find increasingly close decimal approximations to by squaring decimals and testing if the result is less than or greater than 2. You tell them that if you continued this process forever, you would obtain an infinite decimal which is the exact square root of 2.
One of your students, David, raises his hand.
>That's bull-,
he says, stopping himself short of cursing in class.
>You said yourself that it's impossible for a terminating decimal have a square of exactly 2. At each step in this process you add a small amount to the number, but at each step the square will fall short of 2. Continuing forever won't change this. You'll always have a number whose square is less than 2. This is just like last year when Miss Keeki tried to tell us that 0.999... equals 1.
(Miss Keeki had given her class the common 10x - x = 9 argument, but David had never been convinced by it.)
Kate raises her hand and you let her speak.
>He might have a point. I saw a video on Youtube where this Australian math professor said that infinite decimals don't really work. Maybe there is no square root of 2.
Your mission, should you choose to accept it, is to prove, based on your preferred axioms or construction of real numbers, that there is in fact a positive real number whose square is 2. Bonus points if the proof would be understandable and convincing to an eighth grader like David.
One of your students, David, raises his hand.
>That's bull-,
he says, stopping himself short of cursing in class.
>You said yourself that it's impossible for a terminating decimal have a square of exactly 2. At each step in this process you add a small amount to the number, but at each step the square will fall short of 2. Continuing forever won't change this. You'll always have a number whose square is less than 2. This is just like last year when Miss Keeki tried to tell us that 0.999... equals 1.
(Miss Keeki had given her class the common 10x - x = 9 argument, but David had never been convinced by it.)
Kate raises her hand and you let her speak.
>He might have a point. I saw a video on Youtube where this Australian math professor said that infinite decimals don't really work. Maybe there is no square root of 2.
Your mission, should you choose to accept it, is to prove, based on your preferred axioms or construction of real numbers, that there is in fact a positive real number whose square is 2. Bonus points if the proof would be understandable and convincing to an eighth grader like David.
