Understanding the Approximation of 1/3 and 0.33 in Decimal Form

When dealing with fractions and their decimal representations, some values, like 1/3, can be a bit misleading. Let's delve into why 1/3 is close to 0.33 but not exactly 0.33.

Decimal Representation of 1/3

To truly understand why 1/3 is close to 0.33 but not exactly 0.33, we need to look at the decimal representation of 1/3.

Decimal Representation

When you divide 1 by 3, the result is:

[ frac{1}{3} 0.3333ldots ]

This is a repeating decimal often written as 0.overline{3}.

Comparison

The decimal 0.33 is a rounded approximation of 1/3. It can also be expressed as:

[ 0.33 frac{33}{100} ]

Understanding the Difference

The decimal 0.33 is actually slightly less than 1/3. To compare the two values, we can convert 0.overline{3} into a fraction:

[ 0.overline{3} frac{1}{3} ]

Let's break down why this is true:

0.33 is equivalent to ( frac{33}{100} ), which is approximately ( 0.3333 ) when converted back to a decimal. The true value of ( frac{1}{3} ) is approximately ( 0.3333ldots ) with 3s repeating infinitely.

Conclusion

Therefore, while 0.33 is a close approximation to 1/3, it does not equal it because 1/3 is an infinite decimal that cannot be precisely represented as a finite decimal. In summary, ( frac{1}{3} approx 0.33 ) but not exactly, as 0.33 is a rounded version of the true value.

Illustrative Example with a Calculator

If you enter 1/3/033 into your calculator, the answer is: 1.01010101010101010…, which is a repeating decimal. This shows that 1/3 is larger than 0.33.

Fractional Difference

Let's look at the difference between 1/3 and 0.33:

( frac{1}{3} 0.3333ldots ) ( 0.333ldots - 0.33 0 )

Since they are close, the difference would be zero if they were exactly the same. However, as they are, the approximation 0.33 is slightly less than the true value of 1/3.

Decimal Systems and Number Bases

The decimal system with base 10 is just one way to represent numbers. Different bases can yield different and sometimes more elegant representations.

Other Bases

In different number systems:

In base-3, ( frac{1}{3} 0.1 ) In base-6, ( frac{1}{3} 0.2 ) In base-9, ( frac{1}{3} 0.3 ) In base-12, ( frac{1}{3} 0.4 )

These values terminate nicely in different bases. However, for other ratios, the values change:

In base-3, ( frac{1}{4} 0.01overline{4} ) In base-6, ( frac{1}{4} 0.1overline{4} ) In base-9, ( frac{1}{4} 0.overline{2} ) In base-12, ( frac{1}{4} 0.3 )

These are all valid number systems, and the base number simply indicates the number of numerals available. In the decimal system, we have 10 numerals (0 through 9).

Terminating Decimals

With other denominators that divide into 10 or multiples of factors of 10, the decimal value terminates quickly. For example, ( frac{1}{4} ) in the decimal system is 0.25 because 4 divides into 10 twice and 20 five times.

However, in the base-10 system, 3 can never divide into 10 evenly, which is why the decimal value of 1/3 continues forever:

[ frac{1}{3} 0.333ldots ]

This does not mean that something cannot be divided into 3 parts evenly; it simply means that in a base-10 system, the representation is an infinite series.

Origami Example

For example, consider a square sheet of origami paper folded into thirds. A 24cm x 24cm sheet folded into thirds results in each section being 8cm x 8cm, which is one third of the whole. This division is accurate and reflects the concept of 0.3333…, but in practice, it works perfectly well for many applications.

Conclusion

The approximation 0.33 is a practical and sufficient representation for many purposes, but the true value of 1/3 is an infinite decimal. Understanding the nuances between these representations enhances our grasp of mathematics and its practical applications.