Hi, it's calculator time and you know I love calculators. So let's take a look at this Twitter question that uh, popped up on my timeline from Nick's Craft. Who shows that like what the heck is going on here? Because the result from their casio from for a simple equation does not match their shoe phone. So what's going on? Let's try and explain it.

So let's get one of the latest Casios here and let's repeat this. Six divided by two and parentheses, open parentheses and 2 plus 1. right? So you might think that 6 divided by 2? Well, that's 3. And then, well, there should be a multiply in there.

and that's 3. of course, because you've got to do inside the bracket. So 3 times three should equal nine. But let's do equals on here.

And oh, it gives us the answer of one, What's going on? So let's try that on my Android shoe phone here. Six divided by two parentheses, two plus one. and it's already given us. Well, the answer there? Uh, without closing the brackets, it's already given us the answer of nine.

So what's why does the shoe phone not match the casio calculator? Well, it's an interesting question and it's not a bug. Now you might think that this is just an issue with I don't know how like these modern shoe phones calculate things. Well, let's get another calculator, shall we? And this is a Ti Ti-30xs multi-view So let's give this a go. Six divided by two parenthesis, two plus one.

Enter Uh, nine and well, we can do that again just to make sure. And again And again it's going to give us the answer of nine. It matches the shoe phone. What's happening? So what's going on here is obviously what's called operator priority or priority of calculations.

There's many different uh terms for it, but basically what we've got here is we're implying, uh, a multiplication sign between the two and the parentheses here. Instead of putting it in explicitly, we're actually employing it. And that's called implied multiplication or sometimes juxtaposition. uh, multiplication because they're next to it just by the nature that they're next to each other.

That's what it means. Anyway, we call it implied multiplication. So if we actually repeat this and we put the multiplication in there, okay, it does give us, um, still, the result of nine. So this Android calculator makes no distinction between an implied multiplication sign when it's not there, or if you specifically put it in, they're the same priority of operation.

So obviously, you can completely come a gutsy here if you don't know how your calculator works and know how to use it in order to give the result that you actually want. You should never assume that a calculator is going to work a certain way, because if you're used to this and you borrow your friend's Ti and it gives you a different result or vice versa, then, well, I don't know your rocket crash lands on Mars, but now watch closely. This new casio is actually really good. It helps out in this regard.

It's actually telling you specifically what it's going to do. So watch what happens here on the display as I press enter and evaluate that expression to see it. It added an extra parenthesis in here because this is what it's doing internally and this is Casio's new way of actually telling you, hey, look, I specifically don't want to confuse you here, so I'm going to show you what I'm actually doing internally. So it's added the parentheses here.

So what it's decided is that it's more important to evaluate this parentheses first before doing 6 divided by 2. And that's why you get the answer: 1. because 2 multiplied by 3 here is 6. Again, we actually have an implied multiplication in here, but because we've got the parentheses around here, it's it's not going to make a difference.

We've essentially got it. Does 2 plus 1, and I'll show you, uh, the order of operations in a minute. But it does 2 plus 1 first because anything inside a parenthesis takes priority. So it does that first.

So it does two plus one, which is three multiplied, implied, multiplied by two which is six and six divided by six is, of course, one. So this is not a bug, it's just the way that the calculator evaluates the expression and has an order of priority. In fact, this particular example seems to be so well known in the industry that Casio have actually included it in the manual. and here it is for the Fx991ex and you can see it specifically gives the example here that we've got and it specifically tells you how it's going to evaluate it and it specifically tells you it will deliberately include the parentheses in there.

But why does it do that? And this is what you have to look for in your calculator manual. For whether or not your calculator is actually going to do this calculation, Priority sequence could be called you know, order of operations or something like that. The priority sequence of input calculations is evaluated in accordance with the rules below. Uh, where when the priority of two expressions is the same, the calculation is performed from left to right.

So you might think that surely an implied multiplication in here is uh, basically the same priority as a divide here. Uh, but it's actually not. Look at this. Number One is, as I said, parenthetical expressions basically anything in the parentheses.

So two plus one uh will get evaluated first and then functions that have parentheses. Uh, so you know functions, sine, cos, tan, and all that. So you have to get all the way down to priority number seven here. until you find multiplication where the multiplication sign is omitted.

So they're saying that means implied multiplication or juxtaposition. Some manuals like Ti, for example, might actually use the terminology like implied multiplication, but that's what they mean. So that's priority 7. and you have to get all the way down to priority 10 here before you get to multiplication and division.

So you can see that implied multiplication actually has a higher priority than just regular multiplication, which is why it inserts an extra parenthesis there. Because it's telling you specifically that it has a higher priority gives a higher priority to the implied multiplication than it does to the division. But if we put in the specific multiplier sign, we tell it exactly what we want. Instead of using parentheses, we can go 6 divided by 2 and then multiplied by 2 plus 1.

We will get the answer 9 because we're specifically put in the multiplication sign instead of it implying the multiplication. So in this particular case, it's going well. Okay, you've used this multiplication sign. I know exactly what you're doing.

There is no implied multiplication, so it's going to do the multiplication first instead of using this higher priority. So why on earth would a calculator treat an implied multiplication with a higher priority than a regular multiplication? Well, it probably comes about from basic algebra, and you almost certainly learned this. This is the what's called the distributive uh property. and if you have A with a parenthesis and B plus C, that's actually equal to A times B plus A times C, the dot is the multiplication there.

Now, although the calculator doesn't actually like rearrange it and calculate it this way, I don't know maybe it might internally or something, but it certainly doesn't show you that. But the point is, this is how you would write an algebraic function on paper like this. So the calculator actually sees that and goes right. I'm going to put a bracket around like that, and I'm going to treat that as a function and that's going to take higher priority over any sort of, like implied multiplication.

And you'll see the same thing happen with something like this expression with an implied multiplication before the square root sign here. Now, the Ti gives you, uh, the result of, um, square root of two or one point, four, one, four. So let's uh, express this on the casio. You can see that once again, it added the parentheses in there to show you what it's doing and this is 0.707 So it's doing exactly the same thing.

but which one's right? Well, you saw that my Android shoe phone gave the result of nine and uh, this Ti gives the result of nine here. And if you use the Google calculator, it gives you a result of nine. And if you use uh, Wolf from Alpha, it also gives you a result of nine. So is there something weird going on with the Casios? Well, yes and no.

Now here's the original. Uh. Casio Fx991ex over here in the original Twitter post and it gives the result of one. And of course we've seen that the 991ex instead of Mx.

So it's the same series as the 991, but the Mx and Ex It also gives you one. But what if we do the Fx 991 Es not Ms not Ex. but E S what happens? Press enter Nine. What's going on here? This is nuts.

We've got three Fx 991 calculators. Two of them give a and give the result of one. Uh, which is obviously using that distributive property the higher priority for the implied multiplication. But the 991 Es works the same as the Ti.

And like those online calculators, huh? Well, this actually seems to come down to the markets that the calculators are sold in Now, I believe. um, the result of Nine Like this is a very specific American thing. It's a Yankee thing where, uh, like the educators in the Us, they actually through their various textbooks I don't know the history of how they were doing it or whatever. They actually give the same priority to a an implied multiplication as they do to a regular multiplication.

Hence, you get the result of nine. and I I assume that this, uh, Casio Fx991es is a model specifically for the American market. I believe. please correct me in the comments down below, but this is what I've been able to ascertain is that, and I've found evidence of this from Cassio themselves in another video which I'll uh, I just found which I'll link in and they got a response from Cassio themselves that saying yeah, this is basically a North American thing and if you want to actually sell calculators into that market, the American market, it's got to work like this and give you the result of nine.

And this is why calculators are certified for exams. you'll have various you know, like educational bodies actually verify calculators and certify them for a particular market. So let's take this one. We've got a Casio Fx82 Iu plus two.

Second edition. Uh goodness. Cassio. Um, anyway, this one is it's this is approved by the Board of Studies.

Uh, for the New Zealand Qa. for external examinations. I assume this is like Australian as well because this is the Au model. This is the Australian model.

so let's see what this puppy does. Six divided by two parentheses. Two plus one and it gives us the result. Sure enough, one and it added the extra parentheses in there.

so that works just like the other Casios I have. But this particular Es model is different and this is why this is not a bug. This is very deliberate. They deliberately choose the difference in the priority order of operations for different markets, and America seems to be different to the rest of the world.

But please leave it in the comments down below if what your calculator does in your particular country. But this sort of stuff is in the manual if you just hunt for it calculation priority sequence so you can know exactly what your calculator is going to do, it's going to follow this implied multiplication with a higher priority than it does for regular multiplication. But others like this Ti over here, they won't. Here's the manual for that.

It says that they're specifically the same. So there you have it. Oh, by the way, I think some of the more modern Ti ones actually, um, have a higher priority to the implied multiplication. So uh, yeah, I just don't have one here to show you.

But there you go. I hope you found that video interesting and you've got to be careful and know the order of priority of your calculations. But of course, the best way to avoid any issues with this For the calculator and every calculator will then work. The same is to not imply anything as you know, the expression assumptions are the mother of all.

you know what's right. So 6 divided by 2 multiplied, put the multiple in there and then parentheses 2 plus 1 and then you will get the result you will usually desire for nine. Um, so probably all mathematicians are probably going to say this one is correct. Uh, that gives you the answer of uh one.

But engineers and personally I'm like old school, I think it should give a result of nine. Just as an aside, uh, Cassio came up with V Pam back in the 80s. Visually perfect algebraic method it's called and this was a technique to try. and you know, express things exactly.

So if you put 10 times uh sign for example, you would have to press sign first before you actually, uh, put in the number whereas older non-vpam calculators which I greatly prefer and you'll hear me in videos all the time saying, ah, none of that V Pam rubbish. Um, because I'm I'm old school in that regard, then you would go 10 times 10 and then press the sign button. Whereas you know, Cassio decided, no, we want things to match what they are on paper, so it doesn't confuse the kids. so the calculators don't operate in a different way to what you're seeing down on the paper.

And this isn't necessarily related to the implied multiplication because as you saw here, okay, both of these are Vpam calculators, but they give a different result. So yeah, you can't rely on just Vpam and non-vpam actually are giving you this. It's calculators for a specific market. Speaking of non Vpam uh, calculators, right? We've got a really old school one here and we've got a more modern uh Fx 260 which I've done a review and Ted and of uh, by the way anyway, um, six divided by two parentheses.

Two plus one that equals two. Ah, oh god, we've got something different again. Try this on here: 6 divided by 2 parentheses, 2 plus 1. And what do we get? 2 as well.

So what's happening here is when you do 6 divided by 2 like this and then don't press enter, but you press the parentheses. What it's doing is actually eliminating the digit you just typed in and then it's going 2 plus 1, which evaluates to 3. and then it's going 6. Divided by 3 is equal to 2..

And we can see that in operation here. If we go 6 divided by 10 for example, parentheses 2 plus 1. it gives us the same result. It's ignored that 10.

It's wiped it out. So that's just how non-v Pam calculators work. So you could argue that's even worse than the argument we're having. Uh, between whether it should be nine or uh, one that we use the order of priority uh thing here.

That's just how these older school, non-vpam calculators work. Although, if we go back to my shoe phone here, this is using the real Calc app, which is what I actually use. It's supposed to have like a look and feel like a vocacio, uh, calculator, not like an exact emulator. But anyway, if we go 6 divided by 2 parentheses 2 plus 1, it does actually give us 9, just a throw spanner in the works.

I was going to throw this on the second channel, but I thought no, it's important enough to be on the main channel because you know you could really come a gutsy with this thing. And you know, calculations are important in engineering and you could get them wrong when you're doing calculations on the fly like this and you're implying something, but your calculator should give you the right result if you tell it specifically what you want to do. So anyway, if you enjoyed that video, I found it interesting, Please give it a big thumbs up. As always, discuss it down below and let us know if you've got uh, oddball calculators because um, once again, it's not a brand thing as you saw like casio.

Uh, do it in different ways depending on the market. Ti I believe do the same thing And yeah, I know all the Hp enter key fan boys are out there going. What's this? parentheses rubbish? Catch you next time.