What is the difference between terminating and repeating decimals? How does the end behavior of a decimal relate to rational number or irrational number?
Terminating decimals and repeating decimals are two different types of decimal representations of numbers.
A terminating decimal is a decimal number that ends, or terminates, after a certain number of decimal places. For example, 0.75 and 3.25 are terminating decimals because they have a finite number of decimal places. In other words, the decimal representation of a terminating decimal will eventually reach a point where there are no more non-zero digits after a certain decimal place.
A repeating decimal, on the other hand, is a decimal number that has a repeating pattern of digits after a certain point. For example, 0.333... and 1.666... are repeating decimals because the digit "3" and "6" respectively repeat indefinitely. The repeating pattern is denoted by placing a bar above the digit or digits that repeat.
The end behavior of a decimal can provide insights into whether it is a rational number or an irrational number. A rational number is one that can be expressed as a fraction of two integers, while an irrational number cannot be expressed as a fraction and has an infinite non-repeating decimal representation.
If a decimal terminates, it can be expressed as a fraction with a denominator that is a power of 10. For example, 0.75 can be expressed as 3/4, which is a fraction of two integers. Therefore, terminating decimals represent rational numbers.
If a decimal repeats, it can also be expressed as a fraction. For example, 0.333... can be expressed as 1/3, which is a fraction of two integers. Similarly, 1.666... can be expressed as 5/3. Therefore, repeating decimals also represent rational numbers.
In contrast, irrational numbers have decimal representations that neither terminate nor repeat. For example, the square root of 2 (approximately 1.41421356...) is an irrational number because its decimal representation continues indefinitely without any repeating pattern.
In summary, terminating decimals represent rational numbers, repeating decimals represent rational numbers, and decimals that neither terminate nor repeat represent irrational numbers.