which of the following must describe an irrational number?
1- a number with a nonterminating decimal expansion
2- a number with a nonrepeating decimal expansion
3- a number with a repeating or terminating decimal expansion
4- a number with a non repeating and nonterminating decimal expansion
not #1: 1.333333... = 4/3
not #2, since 1.345 is nonrepeating (but terminating)
not #3: see #1, or 1.3 = 13/10
so, #4.
To determine which option describes an irrational number, let's break down each choice:
1. A number with a nonterminating decimal expansion: This option includes both rational and irrational numbers. Rational numbers, such as 1/3 (0.3333...), have nonterminating decimal expansions, but they are not considered irrational.
2. A number with a nonrepeating decimal expansion: This option is incorrect. Numbers with nonrepeating decimal expansions are rational, such as 1/7 (0.142857142857...).
3. A number with a repeating or terminating decimal expansion: This option also includes both rational and irrational numbers. Rational numbers can have repeating or terminating decimals, such as 1/2 (0.5) or 1/9 (0.1111...).
4. A number with a nonrepeating and nonterminating decimal expansion: This option precisely describes an irrational number. Examples of irrational numbers include √2 (1.41421356...) or π (3.14159265...).
Therefore, the correct choice is option 4 - a number with a nonrepeating and nonterminating decimal expansion.