California Assessment of Student Performance and Progress (CAASPP) Math Practice Exam

How can an irrational number be defined?

1. A number that can be expressed as a fraction

2. A number that has a repeating decimal expansion

3. A number that cannot be expressed as a ratio of two integers

4. A number that is always an integer

The correct answer is: A number that cannot be expressed as a ratio of two integers

An irrational number can be defined as a number that cannot be expressed as a ratio of two integers. This means that it cannot be written in the form of a fraction where both the numerator and the denominator are whole numbers. Irrational numbers are characterized by their non-repeating and non-terminating decimal expansions. For example, numbers like π (pi) or the square root of 2 continue infinitely without repeating a pattern. This aspect distinguishes them from rational numbers, which can be neatly expressed as fractions, whether they are whole numbers, finite decimals, or repeating decimals. In contrast to this definition, a number that can be expressed as a fraction does not qualify as irrational, as this describes a rational number. Similarly, a number with a repeating decimal expansion is also a rational number, as repeating decimals can be expressed as fractions. Lastly, an integer is a whole number, and all integers are rational since they can be expressed as a fraction with a denominator of 1, thus cannot be irrational.