Is 1/3 Equal to 0.333...?

Yes. The repeating decimal $0.333\ldots$ is exactly equal to $\frac{1}{3}$. The dots mean that the digit 3 continues forever; they do not mean that the decimal stops after some very large number of digits.

By contrast, a finite decimal such as $0.333$ is not equal to $\frac{1}{3}$. It is a close approximation, slightly smaller. The distinction is between an infinite repeating decimal and any particular finite truncation of it.

What does $0.333\ldots$ mean?

A decimal is a way of writing a number as a sum of place values. For example,

$$0.333\ldots = \frac{3}{10}+\frac{3}{100}+\frac{3}{1000}+\cdots.$$

This is an infinite geometric series. Its first term is $\frac{3}{10}$, and each term is one tenth of the term before it. Since the common ratio is $\frac{1}{10}$, the sum is

$$\frac{\frac{3}{10}}{1-\frac{1}{10}}=\frac{\frac{3}{10}}{\frac{9}{10}}=\frac{1}{3}.$$

So the repeating decimal is not merely close to $\frac{1}{3}$: its value is exactly $\frac{1}{3}$.

A limit-based explanation

You can also define the infinite decimal as the limit of its finite truncations. The first few truncations are

$$0.3,\quad 0.33,\quad 0.333,\quad 0.3333,\quad \ldots$$

After $n$ digits, the truncation is $0.\underbrace{33\ldots3}_{n\text{ digits}}$. As a fraction, it is

$$\frac{3}{10}+\frac{3}{10^2}+\cdots+\frac{3}{10^n} =\frac{1}{3}\left(1-\frac{1}{10^n}\right).$$

As $n$ grows, $\frac{1}{10^n}$ approaches zero. Therefore the truncations approach

$$\lim_{n\to\infty}\frac{1}{3}\left(1-\frac{1}{10^n}\right)=\frac{1}{3}.$$

This is the precise meaning of the ellipsis: the decimal denotes the limit of the finite decimals, not a last decimal digit that somehow needs to be reached.

Why multiplying by 3 also works

A familiar shortcut is to set $x=0.333\ldots$. Then

$$10x=3.333\ldots$$

Subtract the first equation from the second. The repeating tails match and cancel:

$$10x-x=3.333\ldots-0.333\ldots=3,$$

so $9x=3$ and $x=\frac{1}{3}$. This algebra is valid for the infinite decimal because it represents a real number, defined by the limit of its finite truncations.

Another equivalent check is that $3(0.333\ldots)=0.999\ldots=1$. The equality $0.999\ldots=1$ can be shown in the same way: the finite decimals $0.9, 0.99, 0.999,\ldots$ approach 1. Thus multiplying $0.333\ldots$ by 3 gives exactly 1.

A concrete comparison with a finite decimal

The finite decimal $0.333$ is

$$0.333=\frac{333}{1000}=\frac{1}{3}-\frac{1}{3000}.$$

It is therefore less than $\frac{1}{3}$ by exactly $\frac{1}{3000}$. Adding another 3 makes the approximation closer, and adding more 3s keeps reducing the gap. But no finite number of 3s makes the decimal exactly equal to $\frac{1}{3}$; the equality holds for the infinite repeating decimal.

Common mistakes

• Treating the dots as an unspecified stopping point. In $0.333\ldots$, the convention is that the 3s continue indefinitely. If a decimal stops, it should be written without the repeating ellipsis.
• Assuming a very close approximation is exact. $0.333333$ is close to $\frac{1}{3}$, but it is still finite and slightly smaller.
• Thinking an infinite decimal is an unfinished number. The notation names a well-defined real number through the limit of its truncations. Infinite decimals are not written by reaching a final digit; there is no final digit.

The same idea applies to other repeating decimals. For instance, $0.272727\ldots$ is a repeating decimal whose value can be found by treating it as a geometric series or using the multiply-and-subtract method.