Here's two proofs of why 0.9999999... is equal to 1
Proof 1:
p = 0.9999...
10p = 9.9999...
9p = 10p - p
9p = 9
p = 1
p started as 0.9999... but is now 1.
Proof 2:
1/3 = 0.333333...
2/3 = 0.666666...
1/3 + 2/3 = 1
0.333333... + 0.666666... = 0.999999...
Therefore 0.9999... = 1
Information links:
Proof 1 -
http://www.purplemath.com/modules/howcan1.htm
Proof 2 -
http://david.callanan.ie/
Proof 1:
p = 0.9999...
10p = 9.9999...
9p = 10p - p
9p = 9
p = 1
p started as 0.9999... but is now 1.
Proof 2:
1/3 = 0.333333...
2/3 = 0.666666...
1/3 + 2/3 = 1
0.333333... + 0.666666... = 0.999999...
Therefore 0.9999... = 1
Information links:
Proof 1 -
http://www.purplemath.com/modules/howcan1.htm
Proof 2 -
http://david.callanan.ie/
Recurring decimals are unique to the base you are using. For example, try to write 1/3 in binary. Now try to write 1/5 (0.2) in binary. See what happens !!
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