Which number is irrational?
A. 0.3
B. [5
C. 0.777
D. 00.454445
The Correct Answer and Explanation is:
The correct answer is D. 0.454445….
Explanation:
An irrational number is a number that cannot be expressed as a simple fraction (i.e., a ratio of two integers) and has a non-repeating, non-terminating decimal expansion. Let’s analyze each option:
- A. 0.3: This number is rational. It can be written as the fraction ( \frac{3}{10} ), where both the numerator (3) and the denominator (10) are integers. The decimal expansion is terminating, and thus it is a rational number.
- B. [5: It seems like there is a formatting error in the option (it should be a number or a decimal), but from the context, if this refers to a valid rational number such as ( \frac{5}{1} ), it would be a rational number. It can be expressed as a ratio of two integers.
- C. 0.777…: This number is rational. The decimal repeats (the digit “7” repeats infinitely), and any number with a repeating decimal can be written as a fraction. Specifically, ( 0.777… ) is equal to ( \frac{7}{9} ), which is a ratio of two integers, so it is rational.
- D. 0.454445…: This number is irrational. The decimal expansion goes on infinitely without repeating a specific pattern, making it impossible to express it as a fraction. The non-repeating, non-terminating nature of the decimal expansion indicates that the number cannot be written as a ratio of two integers. This is a key characteristic of irrational numbers.
Summary:
- Rational numbers can be written as fractions and either have a terminating or repeating decimal expansion.
- Irrational numbers cannot be written as fractions and have a non-terminating, non-repeating decimal expansion.
Thus, option D, 0.454445…, is the irrational number.