Which set of numbers is always irrational?(1 point)
Responses
positive decimals
positive decimals
terminating decimals
terminating decimals
repeating decimals
repeating decimals
non-terminating, non-repeating decimals
non-terminating, non-repeating decimals
The set of numbers that is always irrational is non-terminating, non-repeating decimals.
To understand why, we can start by defining these terms.
- Terminating decimals: These are decimal numbers that end, or terminate, after a finite number of digits. For example, 0.25, 3.5, and 7.0 are all terminating decimals.
- Repeating decimals: These are decimal numbers that have a repeating pattern of digits after the decimal point. For example, 0.333..., 0.666..., and 0.121212... are all repeating decimals.
- Non-terminating, non-repeating decimals: These are decimal numbers that neither terminate nor have a repeating pattern. For example, pi (π) is a non-terminating, non-repeating decimal, which approximates to 3.1415926535... but the digits after the decimal point continue indefinitely without any predictable pattern. Another example is the square root of 2 (√2), which cannot be represented as a fraction and has an infinitely long non-repeating decimal expansion: 1.4142135623...
While positive decimals can be either terminating or repeating, and non-terminating decimals can be either repeating or non-repeating, it is only the set of non-terminating, non-repeating decimals that is always irrational. Therefore, the correct answer is non-terminating, non-repeating decimals.