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Repeating and infinite string of decimal digit 0.999... is the real number as its proved to equals the integer 1

Geometric proof for this formula

0.9999 can be written as

0.9999... = 0.9 + 0.09 + 0.009 + 0.0009 + ...

Further this can be written a

0.9999 = 9/10 + 9/100 + 9/1000 + 9/10000....

= 9/10 + 9/10(1/10) + 9/10(1/100)+9/10(1/1000)+...

= 9/10 + 9/10(1/10)1 + 9/10(1/10)2+9/10(1/100)3+...

Here **a = 9/10, r =1/10** which is common ratio.

As the size of the common ratio is comparatively less than 1, we can apply the infinite sum formula to derive the value

0.999... = (9/10)[1/(1 – 1/10)]

= (9/10)[1/(9/10)]

= (9/10)(10/9) = 1

Hence the formula proves that **0.99999 = 1**

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Daily Maths Topic Today - Do you know 0.9999 = 1?.