Exploring the Math Behind 1/3: Why Certain Numbers Have Exact Values While Others Do Not

Mathematics often reveals the intricacies that govern our understanding of numbers and their representations. The question of whether it is possible to have exactly 1/3 of some numbers but not others raises an interesting discourse on number systems and their representations. This article delves into the nuances of why certain representations of fractions cannot be written as exact decimals in base-10, while they can in other bases.

The Misconception: Why It Is Perceived as Impossible in Base-10

Few would dispute the notion that fractions like 1/3 pose a challenge when converted to decimal form in base-10. The result, 0.333..., with the digit '3' repeating infinitely, may seem to suggest that 1/3 cannot be accurately represented as a decimal. This is where the base-10 representation can be perceived as inadequate, leading some to question the nature of 1/3 itself.

Understanding the True Nature of 1/3

It is crucial to differentiate between a number and its representation. The number 1/3 itself is an exact value, independent of how we choose to represent it. The issue lies in the base-10 decimal system, which is not capable of producing an exact finite representation of 1/3. However, this does not diminish the exactness of the fraction itself. It is merely a deficiency in the base-10 representation:

For instance, consider the fraction 4/3. When represented in base-10, it appears as 1.333..., with the '3's repeating indefinitely. While we cannot write down the exact decimal representation of 4/3 in base-10, it does not mean that 4/3 is not exact. Instead, the problem arises from the limitations of decimal representation in this base.

Alternative Representations: Base-3 and Exact Values

One can represent 4/3 exactly by switching to a different base. In base-3 (ternary), 4/3 is represented as 1.1, with no repeating digits. Similarly, base-13, base-27, or any base that is a multiple of 3 can provide exact finite representations for 1/3. This demonstrates that the exactness of 1/3 is an inherent property of the number itself, not a limitation of the base-10 system.

Correcting the Question

It is important to note that the question as initially posed may have been misworded. The actual inquiry should focus on the conditions under which 1/3 can and cannot be expressed exactly in different bases. The revised question:

"Is it possible mathematically to have exactly 1/3 of some numbers and not others, if so, why?"

Adheres to a more rigorous framework and invites a deeper exploration of the number system and its limitations.

Understanding these principles can help dispel common misconceptions about fractions and their representations, leading to a more robust appreciation of the true nature of numbers and their mathematical properties.