I’m not a math person at all, so I’m not really debating your proof, but it seems to me that if 0.9• = 1, then what does 0.1• equal? It “fits” perfectly into the “space” between 0.9• and 1, but if 0.9•=1 then 0.1• should equal 0, right? Except it doesn’t, because 0.1<0.1• and 0.1 definitely isn’t 0.
I definitely understand why some religious people think numbers are a tool of Satan.
Think of it with fewer digits.
If we have 0.999 and 0.111, then yeah, each 1 neatly complements each 9. But you have to consider that each of them result in a 1 the next digit upwards (as 09 + 01 = 10). So 0.999 + 0.111 equals 1.110. What you actually want here is 0.001, which just complements the last 9, which returns a 1 for the next 9, which returns a 1 for the next 9 and so on. 0.999 + 0.001 = 1.000. If you increase the digits, the zeros between the decimal point and the one increase too, causing the number to get smaller and smaller. If continue this infinitely long, the number becomes 0.
I’m not a math person at all, so I’m not really debating your proof, but it seems to me that if 0.9• = 1, then what does 0.1• equal? It “fits” perfectly into the “space” between 0.9• and 1, but if 0.9•=1 then 0.1• should equal 0, right? Except it doesn’t, because 0.1<0.1• and 0.1 definitely isn’t 0.
I definitely understand why some religious people think numbers are a tool of Satan.
0.1111… is equal to 1/9. 0.0000… is trivially equal to 0.
My calculator agrees with you but it still doesn’t make intuitive sense to my brain. 0.1• still seems like it “fits” neatly into 0.9• to create 1.
Ah, well. Good thing I’m not designing bridges, right?
Think of it with fewer digits.
If we have 0.999 and 0.111, then yeah, each 1 neatly complements each 9. But you have to consider that each of them result in a 1 the next digit upwards (as 09 + 01 = 10). So 0.999 + 0.111 equals 1.110. What you actually want here is 0.001, which just complements the last 9, which returns a 1 for the next 9, which returns a 1 for the next 9 and so on. 0.999 + 0.001 = 1.000. If you increase the digits, the zeros between the decimal point and the one increase too, causing the number to get smaller and smaller. If continue this infinitely long, the number becomes 0.
0.999• + 0.000• = 1.000•
It’s infinite ones - as you expand it, the illusion offered by the single digit disappears. 0.999 + 0.111 is 1.11, so 0.999… + 0.111… is 1.111…