Question

Which of the following is equivalent to \(1.2 \overline{34} ?\) \\[ 1.23412341234 \ldots \quad 1.23434343434 \ldots \quad 1.23444444 \ldots \\]

Short answer

1.23434343434...

Step-by-step solution

Understand the Notation

Realize that the notation \(1.2 \overline{34}\) means the decimal 34 is repeating. This can be written out as 1.234343434...

Compare Options

Compare 1.2 \overline{34} with each of the choices:\(1.23412341234 \ldots\) \(\rightarrow 1234 is not repeating \,\) \(1.23434343434 \ldots\) \(\rightarrow 34 is repeating \,\) \(1.23444444 \ldots\) \(\rightarrow 4 is repeating\)

Identify the Correct Option

From the comparison, \(1.2 \overline{34}\) matches with \(1.234343434 \ldots\), which shows that 34 repeats.

Key concepts

notation of repeating decimals

In mathematics, repeating decimals are decimals that have one or more digits that repeat infinitely. This is often denoted with a bar over the repeating digits. For example, in the notation \(1.2 \overline{34}\), the bar over 34 indicates that this group of digits repeats indefinitely. Therefore, this decimal can be expanded to \(1.234343434...\). Understanding how to recognize and interpret this notation is essential for solving problems involving repeating decimals.

comparing decimals

Comparing decimals, especially repeating ones, involves ensuring you understand their patterns clearly. Let's take the options from our exercise and compare them: 1.23412341234... does not exhibit a repeating pattern for 34, while 1.234343434... does. For the option 1.23444444..., it is clear that the digit 4 repeats indefinitely. Understanding which digits repeat helps in comparing and distinguishing these numbers.

decimal patterns

Identifying decimal patterns is crucial. Repeating decimals have clear, discernible patterns. Take \(1.2 \overline{34}\), the repeating pattern is the sequence 34. Once you notice a repeating section, it becomes easier to recognize and expand the pattern. Using bars or other notations to highlight repeating sections helps in quickly identifying the pattern and in performing comparisons or conversions.