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Order of Operations
Order of operations. Some folks might remember this topic as PEMDAS. In this video, we’re going to expand the view a little. The first idea we need to discuss is the idea of a mathematical grouping symbol. Any grouping symbol shows that certain operations are grouped together. What are the mathematical grouping symbols?
Well, by far, the most famous is parentheses. And, of course, anything inside parentheses has to happen before the stuff outside parentheses. Parentheses. Parentheses group those operations together. So that’s one grouping symbol, but it’s not the only one.
Another very common one are square brackets or straight brackets. These have the same meaning as parentheses. There’s really no difference. It’s much more stylistic difference. Some people prefer the square brackets instead of parentheses. What I prefer if I’m layering several layers of parentheses, I like to use brackets as one of the outer layers.
That’s just kind of a stylistic thing for example, something along these lines. In other words if I had parentheses instead of brackets, it would be exactly the same meaning. Another one is the square root sign. So this one is tricky cuz it has two meanings, it has the meaning of take the square root, but it also has the meaning of group things together.
If I have a bunch of things underneath the square root sign, those things are grouped together, they have to happen first before I take the square root, and then before I do any of the operations outside of the square root. No operations can pass through the square root, that’s very important. The square root symbol is a grouping symbol.
Another one is the long fraction bar. So for example if I have x plus 6 over x minus 4, that denominator x minus 4, that’s grouped together, that comes as a single unit. I can’t, for example cancel the two x’s there. Because anything that happens to that denominator, has to happen to the whole thing.
They are grouped together. So, this is another example of a grouping symbol. Another one, and this one doesn’t really even have an official name. I’m calling it the exponent slot, for lack of a better term. And what I mean by that, is all the stuff that would be up in the exponent. So, the x minus 7 here, that’s up in the exponent.
We have 3 to the power of x minus seven 7. That x minus 7 has to happen together. We have to do that first. Then we can compute the power, and then we can do any operations that are outside of the exponent situation. So the exponent slot is also a grouping symbol.
Notice the two meanings of each of these. The long fraction bar means both group and divide. It has those two meanings. The exponent slot means both group and to the power of. It has those two meanings. And of course, all this is perfectly when we have things written in proper mathematical formatting.
Everything is clear in proper mathematical formatting, but translating to plain text is tricky, what would I mean by this? Well when you write an email or when you post something to a forum, of course you’re writing in plain text. When you’re writing in plain text, you don’t have full mathematical formatting available, and so, this can lead to some problems, and here’s what I mean.
Suppose we have some expressions like this. And suppose we have to type these in email, or in a forum. We have to type them in plain text. Let’s say this first one, for example. Many people will make the mistake of writing it like this. And this is 100% wrong.
Now why is this wrong? So people look at this, they see 2 to the n plus 3, and they say, well that’s 2 to the power of n plus 3. We’ll I’m just gonna stick it in a caret but and of course the caret does mean 2 to the power of but the problem is what exactly does this red expression mean?
Does it mean that we’re doing 2 to the power of the whole thing and plus 3 or is this merely 2 to the n? We’re gonna calculate that power and then separately at the end, we’re gonna add 3. What’s actually being grouped together? And the problem here is, as we just said on the last page, the exponent slot has two meanings.
It has the meaning of, to the power of, but it also has meaning as a grouping symbol. The caret carries the meaning of, to the power of, but the caret is not a grouping symbol. And therefore we need to substitute in our own grouping symbol to, to encapsulate the grouping symbol meaning that was written on the left side of the equation.
So the correct way to translate this into plain text is like this, we have to supply our own grouping symbol to show how things are grouped. Similarly, we have an expression with a long fraction bar. Many people will make the mistake of writing this, like this. And of course, this is an unholy disaster. This is completely wrong because again, we don’t know what’s in the denominator.
We have 1 divided by and then is it just y, is it y minus 7, is it y minus 7 plus 3? We have no idea from the way it’s, it’s written right there. Once again, the long fraction bar has two meanings. It has the meaning, divided by, and it also has the meaning of a grouping symbol. The little slash, the inline slash, that carries the meaning of divided by, but that’s no longer a grouping symbol.
So we need to supply our own grouping symbol once again. So here it is written correctly. So, in some ways, this might strike you as an inconvenience. It’s sort of an accident of our current technological state that all these things, you know, we’re sending off to messages and plain text and plain text we can’t do that mathematical formatting.
I actually look at it as highly advantageous because when you have to translate mathematically formatted information into plain text, it forces you to think explicitly about the nature of the grouping symbols. And if you’re thinking explicitly about the nature of the grouping symbols, you’re thinking much more deeply mathematically.
So I would say even if you don’t have to send any emails, even if you don’t have to send in any support tickets to negotiate, if you don’t post in the forums, at least practice translating mathematical format information into plain text. Practice this and make sure you understand the grouping symbols because that’s something very important mathematically.
Okay. Now we can talk about order of operations. Many people call this PEMDAS, but I will call it GEMDAS. Now why am I calling it this? Because that first rank, that first priority, that’s not just parentheses. That’s all grouping symbols.
All grouping symbols have the same level of priority in their grouping, in the grouping function. So the grouping symbols come first, then exponents, then multiplication, multiplication and division, then addition and subtraction. multiplication and division are at the same level, addition and subtraction are at the same level.
So these are the four priority levels of GEMDAS. For example, let’s solve this. So of course we have to do what’s in the parentheses first, 7 minus 5 we get 2. Then, we have to do the exponents and now we just have a single number to a single exponent. So we’ll do the exponent 2 to 2 is 4 then the multiplication then the addition.
Another example. Here we have something that was presumably was grouped under a long fraction bar and it was written correctly. So first, we have to do what’s in the parentheses, 10 minus 7 is 3 then I’m gonna do some multiplication and division at the same time. I’m gonna multiply the 2 by 3, I’m also gonna divide the 12 by 3.
The final multiplication is the 9 times 4 and then add 6, and that’s the answer. I want to re-emphasize a point in the previous problem. Multiplication and division are at the same level of GEMDAS priority. So we have some choice about the order in which to do them. For example, suppose we have A times B divided by C times D. So, written like this it’s suggesting that we multiply the product A times B, we multiple the product C times D, and then we divide those two products.
So we could do it in that order. But we could also do it like this, we could take B, we could divide by C times D, and then once we’re done, we could multiply that whole ratio by A. Or we could do this, we could do A divided by D. Separately, we could do B divided by C. And then just multiply those two quotients.
Or we can do this, we can do A divided by C, multiplied by B, and then divide by D. So I hope by showing you these different orders, I’m giving you a sense of the different ways you can rearrange things. As long as your always multiplying by the same things and dividing by the same things for as those two don’t get swapped. You can swap around the order in any way that you want.
And given that you can change the order in any way you want, is there any particular goal you should have? Well, yes. Always choose to cancel before you multiply. And this is something we’re gonna to talk about in considerable depth when we talk about fractions in the upcoming videos.
Finally, if there are multiple layers of parentheses or grouping symbols, we have to work our way from the inside out. So, in other words, we take the innermost grouping symbol and then at that inner most level we do exponents, then multiplication and division, then addition and subtraction. Then we move outside to the next layer.
And we keep on moving out and repeating, all the GEMDAS priorities, all of them, at each level of grouping. So for example, here I have a long fraction bar. So we’re gonna do, we have to handle everything under the long fraction bar before we do anything else.
Within the long fraction bar, we have parenthesis, so that’s another grouping symbol. So first thing I have to do is those innermost parentheses, 13 minus 11, so that becomes 2. Now still under the long fraction bar, I have to do all that. I have to do 2 to the exponent.
Then I have to do the multiplication. Then I have to do the addition. Now we’re down to a single numbers. And now we can actually divide. We can divide 60 by 20. Of course that’s 3.
And then add 2, and we get 5. So that’s how we have to handle multiple layers of parentheses and grouping symbols. In summary, the very first thing we talked about is the very important idea of mathematical grouping symbols. Then we talked about group, translating grouping symbols into plain text.
And again, this is not a trivial operation. If you can do this correctly, if you can do that translation correctly, you understand some deep mathematical ideas. Then we talked about order of operations, which I called GEMDAS, grouping symbols, exponents, multiplication and division, addition and subtraction.
We emphasized that multiplying and dividing are at the same level of priority that gives us considerable choice about how we will rearrange, what order we’ll do the product and the quotients. And finally, with nested grouping symbols, we have to work from the inside out.
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