Answer:
<u>The correct answer is that the smallest sequence or decimal period of 3/11 when is written as a decimal has two digits (0.27⁻⁻⁻ Notation bar above 27)</u>
Step-by-step explanation:
1. Let's recall what are repeating decimals, how do they should be written and how many kind of repeating decimals we have.
A. What are repeating decimals?
We call repeating decimals to all those numbers whose decimal representation eventually becomes recurring or periodic.
B. How do they should be written?
Let's use our example here, this way:
3/11 = 0.2727272727....... 0.27⁻⁻⁻ We use a notation bar or vinculum over the decimal period, that is the minimal sequence of numbers that is repeated over and over again. For our case, the decimal period is 27.
C. How many kind of repeating decimals we have?
1. Purely periodic decimal
We say we have a purely periodic decimal when its period starts immediately after the decimal point. Examples:
2/3 = 0.6666666.... 0.6⁻⁻ (Notation bar over the decimal period)
3/11 = 0.2727272727.... 0.27⁻⁻⁻
2. Mixed recurring decimal
We say we have a mixed recurring decimal when the recurring decimal does not start immediately after the decimal point. Examples:
7/15 = 0.4666666.... 0.46⁻⁻ (The notation bar or vinculum is written only above 6, because the 4 only appears once)
1/18 = 0.055555....0.05⁻⁻ (The notation bar or vinculum is written only above 5, because the 0 only appears once)
3. Irrational or non-periodic numbers
These are number than can't be written as fractions and they don't have a recurring decimal. The most popular example is π.
π = 3.14159265358979........
