6÷2(1+2) = _ _ Correct Answer Inside Finally Solved!! _ PEMDAS_BIDMAS is Wrong_!

OK, so in this video, we are looking at this problem six divided by two, one plus two, and it always seems to cause huge debates. I think it's great that people are having a proper good debate and discussion on Instagram, Facebook, YouTube. What's interesting is that even different calculators are giving different answers, mainly getting the answer nine or one. Even if I try my own two calculators, I get different answers. So which answer is actually correct? That's what we'll be looking to find out in this video. Let's try and actually work through this question, like how many people have been doing. Depending on where you live, you may use BIDMAS or PEMDAS. They pretty much mean the same thing, and they basically tell us in which order the operations should be performed. And this is something you would have learned at school. And, you know, I definitely have to teach this to my pupils. Starting off, we do the the brackets first. So one plus two is three. So we end up with six divided by two and in brackets three. And this is where it gets interesting. You see, the operation between the two and the three is multiplication. So it comes down to six divided by two times by three. Still working through BIDMAS or PEMDAS, they both say the same thing. When you have operations of the same precedence, you go from left to right. Same precedence here means one is not more important than the other. And in maths, division and multiplication have the same precedence. Since we are now left with multiplication and division here, BIDMAS or PEMDAS say we go from left to right. We then get six divided by two, which is three. And then three times three is nine. Is this correct? Well, let's think about it. Take one divided by two X. How do you see this? I'm pretty sure you see this as one over two X. But if you break down the operations, it's one divided by two times X. Using BIDMAS or PEMDAS, you go from left to right to give one divided by two, which is a half, then half times X is half X. Now, is this how you see one divided by two X? So you can see this way of working is already causing some big issues. If we then extend the problem to A B divided by X Y, I'm pretty certain you would see this as A B over X Y. But the PEMDAS BIDMAS convention says when you have operations of the same precedence, you go from left to right, which gives A B Y over X. So you can see the issue here. BIDMAS or PEMDAS is going against how most of us naturally see these sums and expressions. And this is why the original question is causing such issues. It goes against how many of us see maths being written. This is why we avoid using the division sign in higher level maths. It causes huge problems and creates massive amounts of ambiguity. Calculus students know that you will never see the quotient rule for differentiation written like this or this indefinite integration problem written like this. It's absolute madness. But this is how they are always written, free from ambiguity. So what's the deal with this problem? Well, I used the word convention before, and this is what it's really about. Convention makes our lives easier, but they are not precise or exact and can change. For example, a long time ago, this division sign meant everything on the left divided by everything on the right. But this is definitely no longer the case. Similarly, BIDMAS and PEMDAS are conventions which came about to make our lives easier, and we can clearly see they break down. We have to get more technical to see what's actually going on now. There is some consensus and it's to do with the type of multiplication we have with this problem. You will notice that between the two and the three, it's not an explicit multiplication sign. It's what is known as implied multiplication or multiplication by juxtaposition. Very often we naturally see this type of multiplication with higher precedence, like my AB divided by XY example. We know or feel BIDMAS or PEMDAS is wrong. And it is wrong because we see this as AB over XY. Again, this still doesn't solve the issue, which is why calculators will keep giving different answers. Some will be programmed to give higher precedence to multiplication by juxtaposition and some won't. When I press equals on this calculator, you can see it takes implied multiplication as higher precedence and even adds brackets to show that it's doing this. My other calculator clearly doesn't do this. I know some fields of applied mathematics explicitly say implied multiplication has higher precedence. This Wikipedia page mentions this, for example. And to be fair, I think most people would see implied multiplication as higher precedence in almost all cases. Now, if you are inclined to follow this and many people naturally will do, then you will feel the answer is one because you will always do the two times three first. I've actually used A is equal to one plus two in brackets here to make the point more explicit. You may feel this resolves the issue, but it doesn't because we will still find huge problems with this type of notation. How do you see these expressions? For example, it seems to just lead to new and an endless amount of more problems. For me, as a pure mathematician, I need to look at the problem with a greater degree of precision. These operations that are involved in this question, like addition, multiplication, division, et cetera, are called binary operations, which simply means two things combine in some way and become one, for example, take three plus four. The operation is plus three and four are two things. And under addition, they combine and produce one thing, which is seven. Using strict mathematical definition, every binary operation must be explicitly stated in order. And this is very important because not all binary operations are commutative. Essentially, what this means is all numerical and algebraic notation must be clearly written as a strict series of binary operations, something simple like two X squared has to be written like this. And relatively simple expressions will have to be written like this, which just looks like a horrible mess. This is why conventions are incredibly important. They are not exact or precise, but they make our lives easier. There's no actual mathematical basis or justification for bit mass or PEMDAS or taking implied multiplication as higher precedence other than making our lives easier, like who would want to write out a boatload of brackets each time? That's the thing. Conventions have evolved over time. If you look at the history of algebra, you see that in the past maths was written very differently to mean the same thing. In this example, this simple equation would have been written very differently until modern algebra came along. What we have today is a continuing evolution of how maths is written and presented. The fundamental maths does not change, but convention does. But because convention is not necessarily mathematically precise, we have to be very careful when we write mathematics. There's an infinite amount of problems which we run into if we are not careful. These problems, for example, how do you see 2 root 2 divided by 2 root 2? My calculators disagree again. And this is the main issue with this question and many others. The conventions which have come about, be it bit mass or PEMDAS or how we see implied multiplication, have failed in this example by creating ambiguity. We are not allowed to write ambiguous statements in maths, and this is the problem with the original question. The real issue here is not which answer is correct, but it is the way the question has been written using strict mathematical notation, clearly defining each binary operation. We can really only write this problem in one of these two forms. So what is the final answer then? The answer is this, no solution. Simply because of all the reasons we have discussed. This question is not written or presented correctly. And as I've said, we are not allowed to make ambiguous statements in mathematics where there is ambiguity. The writer must ensure this is removed in their presentation. You will find as you go on to study higher level mathematics, the questions you ask and how you ask them become incredibly important. These questions also have no solution, for example. I hope that's been useful. Anyway, thank you for watching. Please remember to like and subscribe and I'll see you in the next video.