### On why 0.999... = 1

One reason why Dr. Edgar Escultura sought to trash the trichotomy axiom is because of an apparent paradox that **0.999... = 1**. But 0.999... = 1 is already proven **true** in mathematics.

However, I agree that the equation 0.999... = 1 is not intuitive, because in the real world we know that 0.999 is less than 1.

But take note that although 0.999 is less than 1, **0.999...** (with the three trailing dots) is **not**. Pay attention to the three dots.

The three dots in 0.999... means that the decimal point is followed by an **infinite number of 9's**. In the same manner, 0.333... means a number with an infinite number of 3's trailing its decimal point.

These nonterminating repeating decimals sometimes arise when dividing two counting numbers. Say, the fraction 1/3. If we want to obtain the decimal representation of 1/3, we simply divide 1 by 3. Using the manual division we learned in school, 1 divided by 3 is represented as

+------ 3 | 1

Since 1 is smaller than 3, we need to append a decimal point and a zero to 1, making the division look like this:

+------ 3 | 1.0

Now we can divide by treating "1.0" as "10":

.3 +------ 3 | 1.0 - 9 ------ 1

The division leaves a remainder of 1. We can still continue to divide by appending another zero to "1.0" and "bringing down" the zero to the remainder. The result is this:

.3 +------ 3 | 1.00 - 9 ------ 10

Treating the remainder as "10", we can divide it by 3 :

.33 +------ 3 | 1.00 - 9 ------ 10 - 9 ------ 1

The remainder is again 1. We can still append zeroes and obtain the following:

.333 +-------- 3 | 1.000 - 9 ------ 10 - 9 ------- 10 - 9 ------- 1

This again leaves a remainder of 1, so the process would continue indefinitely, in fact **infinitely**. So we can say that the decimal representation of 1/3 is 0.333... (with the three dots). Through this process, we can also deduce that 1/9 is 0.111... and 8/9 is 0.888... .

Now let's proceed to the proof that 0.999... = 1. There are many ways to prove it, but I'll show you two ways.

One way is by stating 1 as a quotient of two identical counting numbers, say 9/9. The number 9 divided by 9 equals 1, so we can say that

1 = 9/9

But we know that 9 = 1 + 8. So we can expand the numerator of 9/9 and obtain

1 = (1 + 8)/9

We can split (1 + 8)/9 into two, so it now becomes

1 = 1/9 + 8/9

We already know that 1/9 = 0.111... and 8/9 = 0.888... . So the equation now becomes

1 = 0.111... + 0.888...

We can add 0.111... and 0.888... and obtain this equation:

1 = 0.999....

Next, I will show you another way of proving it. I learned this technique in high school.

We already know in the previous discussion that 1/3 = 0.333... . But we also have a way of **obtaining the reverse**, that 0.333... = 1/3.

We simply let the variable **x** be 0.333... .

x = 0.333... [Equation 1]

Multiplying both sides of Equation 1 by 10, we have

10x = 3.333... [Equation 2]

Since Equation 1 and Equation 2 are equivalent, we can "subtract" Equation 1 from Equation 2:

10x = 3.333... [Equation 2] - x = 0.333... [Equation 1] ------------------ 9x = 3.000... Or 9x = 3 [Equation 3]

Dividing both sides of Equation 3 by 9, we have

9x/9 = 3/9

Or

x = 3/9 .

Reducing 3/9 into lowest terms, we obtain

x = 1/3.

Now let us apply this technique for proving that 0.999... = 1.

Let the variable **x** be 0.999... .

x = 0.999... [Equation 1]

Next, mutiply both sides of Equation 1 by 10:

10x = 9.999... [Equation 2]

Subtract Equation 1 from Equation 2:

10x = 9.999... [Equation 2] - x = 0.999... [Equation 1] ---------------- 9x = 9 [Equation 3]

Dividing both sides of Equation 3 by 9, we have

9x/9 = 9/9

Or

x = 1.

Therefore, since x = 0.999... and x = 1, then **0.999... = 1** .