Question

explain it!
sofia wrote a decimal as a fraction. her classmate nora says that her method and answer are not correct. sofia disagrees and says that this is the method she learned.
0.12112111211112...
$x = 0.\overline{12}$
$100 \cdot x = 100 \cdot 0.\overline{12}$
$100x = 12.\overline{12}$
$99x = 12$
$x = \frac{12}{99}$
lesson 1-2
understand
irrational
numbers
i can...
identify a number that is irrational.
a. construct arguments is nora or sofia correct? explain your reasoning.
b. use structure what is another nonterminating decimal number that can not be written as a fraction.

Explanation:

Step1: Analyze Sofia's decimal

The given decimal is $0.12112111211112...$, which is a non-repeating, non-terminating decimal. Sofia incorrectly treated it as a repeating decimal $0.\overline{12}$.

Step2: Recall rational/irrational rules

Rational numbers can be written as fractions, and include terminating or repeating decimals. Irrational numbers are non-repeating, non-terminating decimals that cannot be written as fractions.

Step3: Evaluate part A

Sofia's method only works for repeating decimals. Her decimal does not have a repeating digit pattern, so it is irrational and cannot be written as a fraction. Thus, Nora is correct.

Step4: Identify example for part B

Any non-repeating, non-terminating decimal is irrational and cannot be written as a fraction. A common example is the decimal expansion of $\pi$.

Answer:

A. Nora is correct. The decimal $0.12112111211112...$ is a non-repeating, non-terminating decimal (irrational number), so it cannot be written as a fraction. Sofia incorrectly applied the method for repeating decimals to an irrational number.
B. $\pi = 3.1415926535...$ (or another non-repeating, non-terminating decimal like $0.1010010001...$)