The decimal representation of the fraction $\frac{1}{3}$ is $0.3333\ldots$, where the digit $3$ repeats infinitely.

When you divide $1$ by $3$, you obtain a repeating decimal. This indicates that the digit $3$ continues indefinitely without terminating. In mathematical notation, this is commonly represented as $0.3$ with a dot above the $3$ ($0.3\dot{}$) to signify the repetition. Alternatively, it can also be expressed as $0.33$ with a line over the two $3$s ($0.33\overline{}$).

To understand the reason behind this phenomenon, let’s examine the process of division. When dividing $1$ by $3$, you first determine how many times $3$ can fit into $1$. Since $3$ is greater than $1$, you introduce a decimal point and a zero, transforming it into $10$. The number $3$ fits into $10$ three times (since $3 \times 3 = 9$), leaving a remainder of $1$. You then bring down another zero, which again makes it $10$, and the cycle continues. Each iteration yields a $3$ and a remainder of $1$, which is why the digit $3$ repeats indefinitely.

This type of decimal is referred to as a recurring or repeating decimal. Recognizing these decimals is crucial in mathematics, as they frequently arise from fractions and can be beneficial for various calculations. Mastering the conversion between fractions and decimals is an essential skill in GCSE Maths, enabling you to solve problems more effectively.