What is the difference between terminating and repeating decimals? How does the end behavior of a decimal relate to rational number or irrational number?

Answer 1

Terminating decimals are decimals that have a finite number of digits after the decimal point, while repeating decimals have a pattern of digits that repeat indefinitely.

For example, the decimal 0.25 is a terminating decimal because it has only two digits after the decimal point. On the other hand, the decimal 0.333... (with a bar over the 3) is a repeating decimal because the digit 3 repeats infinitely.

The end behavior of a decimal is related to whether it can be expressed as a rational number (meaning it can be written as a fraction) or an irrational number (meaning it cannot be written as a fraction).

Terminating decimals can always be expressed as fractions. For example, 0.25 is equivalent to 25/100, which can be simplified to 1/4. This means terminating decimals are always rational numbers.

Repeating decimals can also be expressed as fractions. For example, 0.333... is equivalent to 1/3. In general, any repeating decimal can be converted into a fraction by writing the repeating part as a fraction and simplifying. Therefore, repeating decimals are also rational numbers.

On the other hand, irrational numbers, such as √2 or π (pi), cannot be expressed as fractions or terminating/repeating decimals. Their decimal representations continue indefinitely without forming a repeating pattern. The end behavior of irrational numbers is that they have infinite non-repeating decimal expansions, which sets them apart from rational numbers.