8÷2(2+2) = ? Mathematician Explains The Correct Answer

Hey, this is Presh Talwalkar reminding you to mind your decisions. You may remember me from videos about other viral math problems. Right now, the internet is going crazy over another problem, so it's time for the order of operations to save the day. What is 8 divided by 2 parentheses 2 plus 2 equal to? We'll calculate this using the order of operations commonly remembered by PEMDAS or BODMAS. We'll start with the parentheses. 2 plus 2 is equal to 4. We then have a division and a multiplication. When you have operators of the same precedence, you work from left to right. So we take 8 divided by 2 first. This evaluates to be 4, and then we complete the multiplication of 4 by 4, and we get 16. This is without a doubt the correct answer according to the modern interpretation of the order of operations. But let's do a little thought experiment. Let's travel back in time a hundred years. I actually found an academic paper discussing this very ambiguity in 1917. How would they solve this problem? They also would start with the parentheses, and 2 plus 2 would be equal to 4. But then they would interpret this expression a little bit differently. At that time, mathematical typesetting was limited, so if you had something like x divided by 2y, you would understand it's x over the quantity 2y. So for this expression, we would take 8 divided by 2 multiplied by 4. That's because it would be hard to write this expression in the limited typesetting of the time. So they would complete this as 2 multiplied by 4. This will be equal to 8, and then we have 8 over 8, which is equal to 1. This is not the correct answer today. So let me address some of the issues about this problem. Some people will say they learned it a different way. Please let us know a textbook or printed reference, because we always want to learn how math was taught over the years. What about the distributor property? It really doesn't have anything to do with the correct answer. 8 divided by 2 parentheses 2 plus 2 is definitely equal to 8 divided by 2 multiplied by 4. The question is whether to do 8 divided by 2 first, or 2 multiplied by 4. What about implied multiplication? Many calculators will treat it like regular multiplication, as I'll show later in the video. Is the question even well-defined? It's as well-defined as asking, what is the sum of angles in a triangle? Almost everyone is going to answer 180 degrees, but that's only according to the assumptions of plane geometry or Euclidean geometry. Similarly, 8 divided by 2 parentheses 2 plus 2 is equal to 16, according to the modern interpretation of the order of operations. Just as there are non-Euclidean geometries, there can be non-modern ways or other ways of interpreting this. But the modern interpretation is 16. To elaborate on this point, some people would say the expression is ambiguously written, so there is no correct answer. It is analogous to a poorly written sentence. Consider, I saw the man with binoculars. Did you use binoculars to see the man? Or, did you see a man who had binoculars in his possession? There's a point to be made here. Just as we should write precise sentences, we should write precise mathematical expressions. But in practice, we have to resolve ambiguity. Just as courts interpret and make rulings on ambiguous laws, calculators parse and evaluate seemingly ambiguous expressions. So how would they parse this expression? What happens when you actually put this into a calculator? The calculator will use its own program order of operations, and one way to visualize this is a binary expression tree. So let's take this expression, and we need to parse it into just binary expressions, where we take two things at a time. Most calculators will do this. One subtree will be 8 divided by 2, another subtree will be 2 plus 2, and then finally we multiply these two results, and we get the answer of 16. But there is another way you could interpret this expression, if you put a parentheses right here. This binary expression tree will look very different. At the very bottom we have 2 plus 2, we then take 2 multiplied by this result, and then we take 8 divided by this result. This would give the answer of 1, but that's not how most calculators will do it. Just to give you an example, let's input this expression into Google. How does it parse this expression? Well, it gives the answer of 16, and if you look very closely, it parses it exactly like the binary expression tree on the left. This is how most calculators and programming languages will interpret this expression. I love that the internet is so passionate about these problems. It will spur interest in mathematics, and definitely lead to mathematical discoveries. Thanks for watching, and for making Mind Your Decisions one of the best channels on YouTube. Thanks for watching, and thanks for your support.