We were sitting in our new hut eating eel stew. The stew was quite nice, although as I'm a vegetarian ceoliac I was a wee bit unsure about it. I could see that the other mes were having similar difficulties, still needs must. Least of our worries so to speak.
"Why is it that zero point nine nine recurring and one are the same number?"
"They're both rational, right?"
"And there can be no irrational number between them, yes?"
"So they must be the same as two different rationals always have an irrational in between."
"I've never seen a proof of that."
All our heads went down, we were thinking. Our spoons [where did they come from!?] flopped into our bowls.
"Lets do a triangle argument, say we have two rationals, one bigger than the other, they're repeating decimals that differ at some point in the mantissa." [We knew what we meant.]
"So at some point we have something like 1234 as opposed to 4321."
"Change the 1 to a to a 2 and the it's bigger than the first but still smaller than the second."
"Then you can change everything that follows up or down as you like..."
"So you have an irrational between the two numbers."
eels
We were sitting in our new hut eating eel stew. The stew was quite nice, although as I'm a vegetarian ceoliac I was a wee bit unsure about it. I could see that the other mes were having similar difficulties, still needs must. Least of our worries so to speak.
"Why is it that zero point nine nine recurring and one are the same number?"
"They're both rational, right?"
"And there can be no irrational number between them, yes?"
"So they must be the same as two different rationals always have an irrational in between."
"I've never seen a proof of that."
All our heads went down, we were thinking. Our spoons [where did they come from!?] flopped into our bowls.
"Lets do a triangle argument, say we have two rationals, one bigger than the other, they're repeating decimals that differ at some point in the mantissa." [We knew what we meant.]
"So at some point we have something like 1234 as opposed to 4321."
"Change the 1 to a to a 2 and the it's bigger than the first but still smaller than the second."
"Then you can change everything that follows up or down as you like..."
"So you have an irrational between the two numbers."
"Very well done boys, but how does that help?"