Question

1. circle the irrational number in the list below.

2.27 (\frac{5}{9}) (sqrt{15}) (sqrt{196})

1. lisa writes the follow

5.737737773…, 26, (sqrt{45})
a. which numbers are
b. which numbers are

1. construct arguments deena says that 9.565565556… is a rational number because it has a repeating pattern. do you agree? explain.

Explanation:

Step1: Recall Rational Number Definition

A rational number is a number that can be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q
eq0$. For a decimal to be rational, its decimal expansion must be either terminating or repeating (with a fixed, repeating pattern).

Step2: Analyze the Pattern in 9.565565556...

The number 9.565565556... has a pattern, but the pattern of digits (the number of 5s between 6s) is increasing: first "56", then "556", then "5556", etc. A repeating decimal requires a fixed - length group of digits to repeat indefinitely. For example, a number like $0.\overline{3}$ (where $\overline{3}$ means 3 repeats) or $2.14\overline{56}$ (where 56 repeats) has a fixed repeating block. In 9.565565556..., the block of digits that repeats is not fixed. The number of 5s between the 6s is changing each time: 1 five, then 2 fives, then 3 fives, etc. So, it does not have a repeating (fixed - pattern) decimal expansion.

Step3: Conclusion on Rationality

Since a rational number must have a terminating or a repeating (with fixed pattern) decimal expansion, and 9.565565556... does not have a fixed repeating pattern, it is not a rational number. So, we do not agree with Deena.

Answer:

No, I do not agree. A rational number's decimal expansion must be terminating or have a fixed - length repeating pattern. In 9.565565556..., the pattern of digits (the number of 5s between 6s) is not fixed (it changes from 1 five, to 2 fives, to 3 fives, etc.), so it does not have a repeating decimal expansion and thus is not rational.