Here's two proofs of why 0.9999999... is equal to 1

p = 0.9999...

10p = 9.9999...

9p = 10p - p

9p = 9

p = 1

p started as 0.9999... but is now 1.

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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