Geometric Sequences - Repeating Decimals to Fraction
Geometric Sequences - Repeating Decimals to Fraction - How it Works - Video
Theorems
Example 1
Example 1:
0.3333333 ... is 1/3.
To be sure let's use geometric sequences to find why this is the case.
We can rewrite 0.3333333 ... as a sequence 0.3 + 0.03 + 0.003 + ... where 0.3 is the 1st term, 0.03 is the 2nd term, 0.003 is the 3rd term, and ... .
If we change these decimals to fractions, we get 3/10 + 3/100 + 3/1000 + ...
Now, we can rewrite these as 3/10 + 3/10 * (3/10)1 + 3/10 * (3/10)2 + ...
We have found our common ratio which is 1/10. Do we see a pattern? We added 1 zero each time we added another term, which is why we have 10.
We can use the theorem on the sum of infinite geometric series. S = a1 / (1 - r). Our first term is 3/10 or 0.3 or a1 . Our ratio is 1/10 or 0.1 or r. If we substitute the decimals back into the formula and simplify it, our fraction is 1/3.
Example 2:
Example 2:
1.27272727 ... is what as a fraction. This one is a bit harder unless you have this memorized.
Let's use geometric sequences to find what this is as a fraction.
We can rewrite 1.27272727 ... as 1 + [0.27 + 0.0027 + 0.000027 + ...] where 0.27 is the 1st term, 0.0027 is the 2nd term, 0.000027 is the 3rd term, and ... . Notice after each new addition we have added 2 zeroes and this is because two numbers are repeated. If we had 3 numbers, then we would add 3 zeros in each new term.
If we change these decimals to fractions, we get 1 + [27/100 + 27/10,000 + 27/1,000,000 + ...]
Now, we can rewrite these as 1 + [27/100 + 27/100 * (1/100)1 + 27/100 * (1/100)2 + ...
We have found our common ratio which is 1/100. Do we see a pattern? We added 2 zeros each time we added another term, which is why we have 100.
We can use the theorem on the sum of infinite geometric series. S = a1 / (1 - r). Our first term is 27/100 or 0.27 or a1 . Our ratio is 1/100 or 0.01 or r. If we substitute the decimals back into the formula and simplify it, our fraction is 3/11. Now we can add our 1 to our fraction and we get 1 and 3/11 or 14/11.