Well done Math Cadet! You not only learned that two seemingly different numbers are in fact equal!
You’ve finished day 1 of the Math Camp. Tonight, find a parent, sibling, teacher, or friend and see if they know that these two numbers are in fact equal. If they don’t, now you know how to prove it to them.
Looking forward to seeing you back here tomorrow where we discover an ancient connection between math and art!
What’s that? You still aren’t convinced? OK, let’s talk about it a little more.
First, and most importantly, it is 100% true that \(.\bar9\)=1. This isn’t some trick or some theoretical math problem. They are exactly equal.
Now, you might be thinking, “but if I subtract them, there will always be that tiny .000000…00001 at the end”. It certainly seems like that is true, but here are some ideas to think about that I think will help you fully understand. In the equation 1-\(.\bar9\), how many “0’s” are there before that final “1”? Well, since there are an infinite number of “9’s”, there will be an infinite number of “0’s” before that final “1”.
Now think about that, an INFINITE number of “0’s”. INFINITE!!! That means that you will never finish writing all the “0’s”, and since you can only write that “1” after all the “0’s”, you never actually get to write it. The number .0000…000001 is actually the same as 0.
Here’s another simpler way to think about it. I hinted about this at the beginning of today’s activity. As you can verify in your calculator, 1/3=.33333….
Now, simply multiply both sides by 3!
It was right there in front of you the whole time.
Well done today Math Cadet. See you back here tomorrow!