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2+2=4? You Don’t Know as Much Math as You Think!
Do you really know math? Well, you likely know 2+2=4 (if not you have bigger things to work on than reading this). But is this all there is too it? In high school and below, math classes focus on what is known as computational mathematics. They give you a something to find, and you find it. This can range from ‘find x if 2+x=5’ in 7th grade, to ‘sin(x)=1, what is x’ in 11th grade, and maybe even to ‘find the derivative of 3x+543’ if you get to calculus*! Our goal here is not to learn how to calculate each of these things, but instead to focus on the other type of mathematics. Unexpectedly, mathematics on a much higher level focuses on abstract thinking more than computation! Repetitive computation can get math a bad rep, but bear with me, it can get interesting really quickly! If that isn’t enough to motivate you to hear me out, there are related meme’s at the end!
Before we can get into the abstract part of this, we must quickly review the concept of repeating decimals. 0.33?, for example is 0.3333333333… with infinite 3’s following it. While this may seem random, it comes from a fairly simple division, 1 divided by 3 (don’t believe me? Try it yourself!). Knowing all the calculations is irrelevant for the purpose of this article, just know that the bar symbols the numbers below it repeating forever! See, this math stuff isn’t so bad.
Time for the beginning of the interesting part (no, no memes yet. Patience)! Is there a second way to write number one? This is confusing at first, so let’s take this slowly. What is 1 divided by 9 (if you want to find it yourself, put it into a calculator!)? It’s 0.1?. Next, think about 2 divided by 9. This one is 0.222??. Similarly, 3 divided by 9 is 0.3333? and 4 divided by 9 is 0.444?. Now, let’s skip to 8 divided by 9 (0.888?). What comes next? Well, 9 divided by 9 is one! But by the pattern we just saw, it should be 0.9999?. Something is wrong with math! No. Something is wrong with the pattern! No. Something is wrong with me! Maybe, but that’s unrelated.
Before that can become clear, consider what makes two numbers different from each other (No, not just that they are different! Think harder)? The answer is that their is always something between them. Between 1 and 2 is 1.5. Between 1.5 and 1.6 is 1.555. Between 1.5555555 (7 fives after the decimal) and 1.6666666 (7 sixes after the decimal) is 1.5555555555555 (14 fives after the decimal) . If you think of it this way, there is an infinite amount of numbers just between 1 and 2 (and almost any other numbers if you think about it)!
But what is between 0.9? and 1? For small amounts of repeats this is easy to find. For example, between 0.99 and 1 is 0.999. Between 0.999 and 1 is 0.9999. So for finite (this is a fancy way of saying it ends somewhere, no matter how far out) you can add another 9 at the end and have a number in between. Our problem, however, comes from the 0.9?? never terminating (there is no end to the 9’s after the decimal) . There is no number between 0.9 and 1, so we can say that they are the same number intuitively (for a formal mathematical proof, see the sources at the end of the article). This idea doesn’t just happen to the number 1. For example, 13.9? is the same as 14 by the same logic we used above!
Now, what is 2+2? Yes, it still is 4, but you can also say it is 3.9?, because that is the same thing as 4! Well, it is obviously much more convenient to write 4 than 3.9??, so that’s what everyone does, but it is important to know these subtle things, because it’s the small things that matter! As the saying goes, success is the sum of a million small actions.
I may be sitting here writing an article, but I can feel that some of you still doubt that this even matters at all, and might even hate yourself for wasting 5 (well, 4.9?) minutes of your life reading this. So I leave you with this problem: (Please, leave solutions in the comments! I will respond with the correct answer once 10 people comment.)
Write the solution to 2x3+5x5+5 in three forms, one of which uses infinitely repeating decimals, and two other forms. (extra credit: Now write this using fractions. Why does this work?)
If that still doesn’t convince you that this matters, next time someone asks you an annoying math problem (what's 9+10?), you can now answer with an annoying answer back(18.999999999999999999999999)! That alone is worth learning all of this. Ok, i’m sorry that was a bad excuse of a dead meme. But, if you read the whole article and didn’t skip to the end, you now know a little more mathematics than when you started.
*The answers are 3, 90 degrees, and 3 respectively for anyone curious :)