# Math trick for Palo Alto math lovers: Why .999… equals 1

We’re all smart in Palo Alto. We’re probably smarter than most other people. But some things, we just haven’t thought of before. Or maybe, you thought about them back when you took Calculus in college, but it’s been a while. Well don’t you worry, those of us who still use math in our daily lives love sharing fun mathy facts to the rest of our intelligent Palo Alto friends!

You’ve probably heard before that the repeating decimal .999… equals 1. And with a little long division, we can see that this is so. Or if you like, we can split it up into 1/3 and 2/3, two things that add to 3/3, which we know is 1.

2/3 is equal to .666… and 1/3 equals .333…

If we add these, we can see that

2/3 = .666…

+ 1/3 = .333…

3/3 = .999…

There we go! 1 = .999… – done! How easy was that? Most of us are satisfied with that as proof! So we must be correct.

What’s that? A select few sceptics are still in disbelief? How could .999… equal 1, you say? Isn’t it always .0…01 away from equalling 1?

Hmm… Well maybe we can try something else. Let’s break this up into what we really mean. How about a (geometric) series of numbers that add up to .999…, like this one:

.999… = .9 + .09 + .009 + .0009 + .00009 …

Okay, cool. Let’s rewrite these as fractions:

.999… = 9/10 + 9/100 + 9/1,000 + 9/10,000 …

Now, we’ll think about what defines a geometric series like this. These things take a sequence of numbers and add them all up. This sum that we have heads in some direction. Sometimes, it can lead toward infinity, like when adding a sequence like 1, 2, 4, 8, 16 … – we call this a diverging series. Other times, though, the sum of the numbers in a sequence head towards a particular number as you add each next term. For example, if we try to add the terms in the sequence 1/2, 1/4, 1/8, 1/16, 1/32 … Our first two terms add up to 3/4 (.75), the first three add to 7/8 (.875), the first four add to 15/16 (.9375), and the first five add to 32/33 (.9696…). Every term we add will bring us closer and closer to a fraction which equals one.

Because of this, we say that the series converges at 1, or that the sum of the infinite terms in the sequence is 1.

When we start adding the terms in our series here, we’ll get that our first two equal 99/100, our first three equal 999/1,000, our first four equal 9,999/10,000, and our first five equal 99,999/100,000. Here we can see that the more terms we add, the closer we get to equaling 1!

Though we may never reach one by adding the terms of this sequence, as we head toward infinite terms, we can show that the limit of the sum is 1.

So, call it crazy, or wizardry, or a flaw in the decimal system, but .999… and 1 are two different ways to represent the idea of the same real number!

## For more math tricks in the Palo Alto area:

Check out Mathnasium of Palo Alto-Menlo Park http://www.mathnasium.com/paloalto-menlopark/schedules