Which of the following numbers are rational, and which are irrational? a. 2.375375 b. 3.0120123 c. √169 d. 21 e. 3.12 f. √735 g. 8 h. 0.72 i. 5.626626662
The Correct Answer and Explanation is:
Here is the classification of the numbers from the image.
Rational Numbers: a, c, e, g
Irrational Numbers: b, d, f, h
Explanation
To classify these numbers, we must first define rational and irrational numbers. A rational number can be expressed as a ratio or fraction p/q of two integers, where p is the numerator and q is the non-zero denominator. In decimal form, rational numbers either terminate or have a repeating pattern of digits. An irrational number cannot be expressed as a simple fraction. Its decimal representation is both non-terminating and non-repeating, meaning the digits continue forever without a predictable, repeating block.
Analysis of the Rational Numbers:
- a. 2.375375… This number has a clear repeating pattern, where the block of digits “375” repeats indefinitely. Any decimal that has a repeating pattern is a rational number.
- c. √169 The square root of 169 is 13. The number 13 is an integer, which can be expressed as the fraction 13/1. This fits the definition of a rational number.
- e. 3.1̅2̅ The bar over “12” indicates that these two digits repeat forever, making the number 3.121212… As a repeating decimal, it is a rational number.
- g. 35/x This number is shown as a fraction. Assuming the denominator (which is not fully legible) is a non-zero integer, the number is rational by definition, as it is a ratio of two integers.
Analysis of the Irrational Numbers:
- b. 3.0120123… This decimal continues forever, but the pattern does not repeat. The sequence seems to be constructed by appending the next integer (012, then 0123). Since the decimal is non-terminating and non-repeating, the number is irrational.
- d. 2π The constant pi (π) is a famous irrational number; its decimal expansion never terminates or repeats. When an irrational number is multiplied by a non-zero rational number like 2, the result remains irrational.
- f. √7 The number 7 is not a perfect square. The square root of any integer that is not a perfect square is an irrational number. Its decimal representation is endless and does not repeat.
- h. 5.626626662… This number displays a pattern where the quantity of the digit “6” increases between each “2”. This pattern is predictable but not repeating, so the decimal expansion is non-repeating and non-terminating, which makes it an irrational number.
