Decimal expansions are a way of expressing numbers in the base 10 system. Every real number can be represented in decimal form, which can be categorized into three main types: terminating decimals, non-terminating repeating decimals, and non-terminating non-repeating decimals.

1. Terminating Decimals:

A terminating decimal is a decimal that has a finite number of digits after the decimal point. For example, the number $0.75$ can be expressed as the fraction $\frac{3}{4}$, and its decimal expansion stops after two decimal places. Other examples include $2.5$, $0.125$, and $3.00$.

2. Non-terminating Repeating Decimals:

A non-terminating repeating decimal is a decimal that continues infinitely but has a repeating pattern. For instance, the decimal expansion of $\frac{1}{3}$ is $0.333...$, where the digit 3 repeats indefinitely. Another example is $\frac{2}{11}$, which is $0.181818...$. These decimals can be expressed in the form $0.\overline{a}$, where $a$ is the repeating part.

To convert a non-terminating repeating decimal into a fraction, we can use the following method:

1. Let $x = 0.\overline{a}$.
2. Multiply both sides by a power of 10 that shifts the decimal point to the right by the length of the repeating part.
3. Subtract the original equation from this new equation to eliminate the repeating part, and solve for $x$.

For example, to convert $0.\overline{3}$ to a fraction:

1. Let $x = 0.\overline{3}$.
2. Then, $10x = 3.333...$.
3. Subtracting gives $10x - x = 3$, or $9x = 3$.
4. Thus, $x = \frac{3}{9} = \frac{1}{3}$.

3. Non-terminating Non-repeating Decimals:

These decimals do not terminate and do not have any repeating pattern. An example is $\pi$, which is approximately $3.14159...$, and $\sqrt{2}$, approximately $1.41421356...$. These numbers are classified as irrational numbers because they cannot be expressed as fractions.

Relationship Between Rational and Decimal Expansions:

Rational numbers, which can be expressed in the form $\frac{p}{q}$ (where $p$ and $q$ are integers, and $q \neq 0$), will have either a terminating decimal or a non-terminating repeating decimal. In contrast, irrational numbers will always result in a non-terminating non-repeating decimal expansion. This distinction is crucial for identifying the type of number based on its decimal representation.

Converting Fractions to Decimal Form:

To convert a fraction to its decimal form, we can perform long division. For example, converting $\frac{5}{8}$ involves dividing 5 by 8:

• 5 divided by 8 is 0.625, which is a terminating decimal.

In contrast, converting $\frac{1}{6}$ through long division yields $0.1666...$, a non-terminating repeating decimal.

Understanding decimal expansions is essential in mathematics as they provide a clear representation of numbers, facilitating easier calculations and comparisons between different values.