# Explaining Why 0.999... is Equal to 1

Harper Quill

Updated Monday, September 18, 2023 at 11:16 PM CDT ## Understanding the Concept of Equivalence

When it comes to explaining complex mathematical concepts to young children, it can be challenging to find a practical approach that resonates with their understanding of numbers. One such concept is the equality between 0.999... and 1. In this article, we will explore different explanations provided by Reddit users and uncover the hidden truths behind this intriguing mathematical phenomenon.

## The Difference Between Two Numbers

To begin, let's address the question posed by Reddit user teh_maxh: "If two numbers are different, there must be a difference. What is 1-0.999…?" This question highlights a common misconception about the nature of decimal representation. While it may seem that there is a difference between 1 and 0.999..., the reality is that the difference is infinitesimally small, approaching zero.

## The Power of Multiplication

Reddit user etzel1200 offers a practical approach by suggesting the division of 1 by 3, resulting in 0.33,333.... When this number is multiplied by 3 again, it becomes 0.999,999,999..., which is equal to 1. This multiplication process demonstrates that even though the decimal representation of 0.999... seems to go on forever, it ultimately converges to the value of 1.

## Synonyms in Mathematics

In a creative analogy, Reddit user rasa2013 compares the concept of 0.999... to synonyms in language. They explain that just as happy and jolly are different words that mean the same thing, 0.999... is simply a synonym for 1. This analogy can help children understand that although the decimal representation may appear different, it represents the same value as 1.

## Finding the In-Between Number

Another approach, suggested by Reddit user BurnOutBrighter6, involves asking children to think of a number between 0.999... and 1. By emphasizing that if two numbers are different, there must be another number between them, this approach encourages children to realize that there is no such number. This realization leads to the understanding that 0.999... and 1 are, in fact, equal.

For older children who have a grasp of algebra, Reddit user laz1b01 provides a step-by-step mathematical proof. By assigning x as the value of 0.999..., they demonstrate that multiplying x by 10 results in 9.999..., and subtracting x from 9.999... leaves 9. This equation simplifies to x = 1, solidifying the equality between 0.999... and 1.

## Exploring Number Patterns

Reddit user cobalt-radiant introduces an interesting pattern involving fractions. By dividing numbers from 1 to 8 by 9, they demonstrate that the decimal representations follow a repeating pattern of 0.1111..., 0.2222..., and so on. When it comes to 9 divided by 9, the pattern suggests that the decimal representation should be 0.9999.... However, mathematical principles dictate that any number divided by itself is equal to 1, leading to the conclusion that 0.999... is indeed equal to 1.

## Base Systems and Equivalence

Reddit user KevTheToast raises an intriguing question about exploring the concept of equivalence by counting in different base systems. While this approach may provide alternative representations for certain numbers, it does not directly address the equality between 0.999... and 1. However, it opens up opportunities for further exploration and understanding of number systems.

## A Visual Proof

Finally, Reddit user Americano_Joe presents a visual proof that demonstrates the equivalence between 0.999... and 1. By adding three fractions, each equal to 0.3333..., the sum equals 1, which is also equal to 0.999... This visual representation provides a tangible way for children to understand the equality between these two seemingly different numbers.

Explaining the equality between 0.999... and 1 to young children requires a practical approach that aligns with their understanding of numbers. By utilizing different explanations from Reddit users, we have uncovered various ways to convey this concept. Whether through multiplication, analogies, algebraic proofs, number patterns, or visual representations, these explanations shed light on the hidden truths behind this fascinating mathematical phenomenon.