Does 1.9999... equal 2.0
A few decades ago, when I was in high school, a math teacher showed the class this "proof" that 1.9999..... is equal to 2.0. That's 1. followed by an infinite number of 9's is equal to 2.0.
10 x 1.9999... = 19.9999...
Ten times 1.9999... minus one times 1.9999... will equal 9 times 1.9999...
So 19.9999... - 1.9999... = 18
Nine times 1.9999... divided by 9 will equal 1.9999...
So 18 / 9 = 2.0
Therefore 1.9999... equals 2.0.
About a year later I came up with this thought.
Add a few nines after the "..." to make the original number 1.9999...999
It still is 1. followed by an infinite number of nines, so it should still be the same number as 1.9999...
Now do the math.
10 x 1.9999...999 = 19.9999...990
19.9999...990 - 1.9999...999 = 17.9999...991
17.9999...991 / 9 = 1.9999...999
Meaning that the original 1.9999...999 is equal to 1.9999...999, which is true.
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In algebra we have two variables, a and b, which are different from one another. A = 1.9999... and b = 2.0 and the math teacher "proved" that a = b. My putting the infinite number of nines somewhere in the middle of all the nines that follow the decimal point simply proved that a = a, which is true.
Anyway, it's up to you to decide whether 1.9999... is equal to 2.0 or not.
By Rod Heilfron
Send email to: rodneyj123@cox.net
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