Close only counts in horseshoes, hand grenades and… math???
What? No way?!! I can’t be serious, can I?
If you get frustrated with mathematical precision, today is your lucky day because for the first time in the history of this publication close is actually mathematically precise!
I’m going to show you how with basic math you can show your friends that the repeating decimal 0.999… is in fact equal to 1.
Yep, you heard me right. We can show this is true without anything fancy. No calculus, no limits, no advanced concepts whatsoever. So let’s jump in!
How To Express Repeating Decimals as Fractions
What many people don’t realize is that you can easily write any repeating decimal as a fraction.
So here’s what you need to know.
If you have a single-digit repeating value, write the repeating digit over a denominator of 9, like this:
If you have a 2-digit repeating value, write the repeating digits over a denominator of 99.
If you have a 3-digit repeating value, write the repeating digits over a denominator of 999.
See the pattern?
Given N repeating digits, the formula is:
Showing 0.999… Equals 1
Okay, here we go! Begin by setting 0.999… equal to its equivalent fraction.
But of course, 9 ÷ 9 = 1. So we have:
And we’re done. Short and sweet.
Still Not Convinced?
If you’re still wondering how this can be true, I feel you.
This is completely counter-intuitive.
It’s mind boggling.
I’m proposing that by appending 9’s to the end of 0.999… not only is it getting closer and closer to 1, but it is 1!?
Here’s another way of looking at it.
1/3 is equal to 0.33333… and 2/3 is equal to 0.66666…, so 1/3 + 2/3 must equal 0.3333… + 0.6666, right?
Sum both sides and there it is again!
My math buffs out there are probably a little disappointed right now because this is just too easy.
SO easy it’s, as they say in the math world, trivial.
I hear ya. I’m yearning for a little more math in my day too. So if you wanna stick around a little longer I’ll show you why this is true with a good ‘ole infinite series. You know some of that fancy math I said we weren’t going to do today, but hey I think it’s going to be worth it.
The Infinite Series Approach
Let’s start by breaking down what 0.999… represents in place value.
If you think way back to your elementary school days, you might recall a teacher explaining place value to you and saying something like,
We can write 0.999… in the same fashion, beginning like this:
Or as fractions:
If we add up the first 5 values, we get 0.99999 and if we kept writing out the decimal expansion we could write it out to infinity and get the exact value 0.999….
Manually writing out the decimal expansion to infinity isn’t feasible, that’s why there’s shorthand for it.
Start by factoring out 9.
Then rewrite the denominators as powers of 10.
Now use sigma notation to represent the infinite summation.
Side Note on Sigma Notation: If you’re new to sigma notation, let me break it down for you. The greek letter sigma ∑ is used in mathematics to represent repeated addition.
The first value in your summation is generated by plugging in the first indicated value for n, found by looking underneath ∑. In this case, that value is n=1, so we get (1/10)¹.
To get the next value you plug in the next integer, n=2, to get (1/10)². Then plug in n=3 to get (1/10)³, and so on. Keep on doing this until you get to the value above the ∑ symbol. In this case, that’s infinity, so there is no end.
And of course, all of these generated values are added together and, in our case, the entire summation is multiplied by 9.
Another way to think about the terms in our summation is that each successive term is obtained by multiplying the previous term by a common ratio.
That means we have a geometric series that converges to a/(1 – r) where a is the first value in the series and r is the ratio we multiply by to get the next term.
Convergence in a series simply means that the series will continue to get closer and closer to a specific value as you add more and more terms to the series. The series is getting infinitesimally close to the convergence value. On an infinite scale, convergence becomes equality.
In our case, we start with 1/10 and multiply by 1/10 each time, so both a and r = 1/10.
Complete the arithmetic on the right hand side.
Now we have formally shown that 0.999… converges to or equals 1.
Final Thoughts & Inspiration
If this feels strange to you, that’s good! You know that two different numbers, should in fact be different numbers. The counter-intuitive nature of this problem is inherent to the strangeness that accompanies working with infinity.
Even in a simple problem such as this one, you are moving from the confines of what is tangible to the human brain onto a scale that is in a way beyond our comprehension. As finite beings we can grasp at understanding the concept of infinity or foreverness, but can never truly experience it. Which means that often times what we know to be true in our finite world, turns out to behave differently on an infinite level.
Accepting the transition from easily discoverable and reproducible mathematics to what can only be tested and imagined in our minds is part of the beauty and wonder of mathematics.
So don’t take today’s lesson at face value and shrug it off.
Let there be a layer of fog between you and perfect understanding. Let your mind be blown by the fact that infinity behaves strangely. And let it make you crave for more understanding and most of all more mathematics.
Thanks for reading and pondering infinity with me today!