Okay, if you've done any form of maths in school, you've probably heard of BEMA, BEDMAS, PODMAS, PIMDAS or some variation of that acronym. Basically, it means the order of operations and is the cornerstone of basic arithmatic. Some very successful trolls out there have decided to post basic maths equations and trick people who have a poor understanding of it.
And so, I'm writing a brief note, explaining the order, why it's in that order, and hopefully clearing up some misconceptions.
Firstly, the acronym itself. The acronym isn't important, but I'm most in favour of BEMA because it cuts out division and subtraction, no longer implying that they come before or after addition or multiplication.
So!
1. Brackets (Or, Parentheses) ()
Brackets always come first. Why? Because mathematicians need a symbol that means "Do this first". Plenty of useful situations, such as 2×(x+3) and so forth. The (x + 3) always comes first. I should note that things outside the brackets are not part of it. This seems obvious, but apparently people seem to think that 2(x+3) means the 2 is part of the brackets. I'm not sure why. It's not. There's an invisible multiplication sign (This is identical to what I wrote earlier) and that means you do it in the multiplication step.
You can nest brackets and stuff too. Anything inside the brackets follows the normal order of operations so you do the innermost brackets first and move out.
There are also implied brackets, where the brackets aren't written because mathematicians are lazy. These mostly come in two situations.
Firstly, square roots. Anything inside the square root symbol is considered bracketed even though the bracket is nearly never written. I'd give an example but facebook notes don't let me.
Secondly, things inside the power. Say, 2-1, the -1 is considered all part of the power.
Secondly, division written as fractions. The numerator (ie the top line) and the denominator (ie the bottom line) are both in seperate brackets. So, for example:
2x + 3
3x - 1
This is the same as
(2x + 3)
(3x - 1)
I should point out now that 2x + 3 ÷ 3x - 1 does not have any implied brackets, and is, in fact, the same as 2 × x + 3 ÷ 3 × x - 1. This is important but I'll get into why later.
2. Exponents (Or, orders or indices) ^
These are normally written as a superscriptor the alternative symbol is ^. It's just powers and square roots. 23 or x2 or √4 come second. Why is this?
Basically, it's because of what a power is. It's repeated multiplication. 2^3 means 2 × 2 × 2. If you have, say, 3×23, what you're doing is 3 × 2 × 2 × 2, which is very different from (3×2)3 or 6 × 6 × 6.
Now here, we get problems from -12 because is this -1 × -1 or -(1 × 1). The answer is the latter. Exponents come before the inverse sign. If we meant -1 × -1, we'd write (-1)2.
Square root happens at this time too because the square root is just something to the power of a fraction.
3. Multiplication (And divison) ×÷
This seems to cause a lot of issue because of the whole implied brackets thing above. But, first things first, multiplication and division happen at the same time. One doesn't happen before the other, you do them both at exactly the same time. Why? Because multiplication and division are identical operations. The only difference is when you divide, you multiply by the inverse of a number. What's the inverse of a number? If your number is x, the inverse is either x-1 or 1÷x. Both are the same thing.
Go on, try it. Get a decent calculator and do, say 6 ÷ 2. You'll get 3. Then do 6 × 0.5 (0.5 being the inverse of 2). You'll get 3 again. They're both identical operations and that's why I don't like having BEMA explicitly state division. Division is just a funny way of doing multiplication.
Now then, why do these come in this bit? Because, like exponents, multiplication is just repeated addition. 6 × 3 is just 6 + 6 + 6. 23 is just 2 × 2 × 2 is just (2 × 2) × 2 = (2 × 2) × (2 × 2) = 2 + 2 + 2 + 2. Again, 2 + 6 × 3 is 2 + 6 + 6 + 6, which is different to 8 + 8 + 8.
Now, I mentioned 2 × x + 3 ÷ 3 × x - 1 earlier. Now, order of operations says that the middle bit is all done at the same time. So this isn't 3 ÷ (3 × x), it's 3 ÷ 3 × x. The first is equal to 1/x, the second is equal to x. That's a huge difference and the original troll maths was based around this distinction.
4. Addition (and Subtraction) +-
Okay, this is fairly obvious. Last, you do all addition and subtraction. Again, these happen at the same time because subtraction is just adding the negative of a number. 5 - 3 is the same as 5 + (-3). I don't think I need to go in to more detail here.
Doing things at the same time.
I've mentioned this a few times. Doing multiplication and division at the same time, and addition and subtraction at the same time. Obviously, that's a rather tricky thing to do. So, what I mean when I say this is you work from left to right. If you have 5 + 3 - 2, going from left to right, you do the addition first, 5 + 3 = 8, then subtract 8 - 2 = 6. For 5 - 3 + 2, you do the subtraction first, 5 - 3 = 2, then the addition, 2 + 2 = 4. Same with division and multiplication. 12 ÷ 4 × 2, do the division first to get 3, then multiply by 2 to get 6. Or, 12 × 4 ÷ 2, you do the multiplication first to get 48, then divide by 2 to get 24.
Now, for something more advanced (Feel free to ignore this if you aren't a nerd), the truth is, the order doesn't actually matter. To use a fancy maths term, addition and multiplication are commutative. Which means the order doesn't matter. If you remember that subtraction is adding by the negative of a number and division is multiplying by the inverse of a number, the order becomes unimportant. 5 + 3 + (-2) is the same as (-2) + 5 + 3 is the same as 3 + 5 + (-2). Likewise, 12 × (1/4) × 2 is the same as (1/4) × 2 × 12.
Aaand that's about all I have to say. Remember people, don't feed the troll and remember the order to do things!
And so, I'm writing a brief note, explaining the order, why it's in that order, and hopefully clearing up some misconceptions.
Firstly, the acronym itself. The acronym isn't important, but I'm most in favour of BEMA because it cuts out division and subtraction, no longer implying that they come before or after addition or multiplication.
So!
1. Brackets (Or, Parentheses) ()
Brackets always come first. Why? Because mathematicians need a symbol that means "Do this first". Plenty of useful situations, such as 2×(x+3) and so forth. The (x + 3) always comes first. I should note that things outside the brackets are not part of it. This seems obvious, but apparently people seem to think that 2(x+3) means the 2 is part of the brackets. I'm not sure why. It's not. There's an invisible multiplication sign (This is identical to what I wrote earlier) and that means you do it in the multiplication step.
You can nest brackets and stuff too. Anything inside the brackets follows the normal order of operations so you do the innermost brackets first and move out.
There are also implied brackets, where the brackets aren't written because mathematicians are lazy. These mostly come in two situations.
Firstly, square roots. Anything inside the square root symbol is considered bracketed even though the bracket is nearly never written. I'd give an example but facebook notes don't let me.
Secondly, things inside the power. Say, 2-1, the -1 is considered all part of the power.
Secondly, division written as fractions. The numerator (ie the top line) and the denominator (ie the bottom line) are both in seperate brackets. So, for example:
2x + 3
3x - 1
This is the same as
(2x + 3)
(3x - 1)
I should point out now that 2x + 3 ÷ 3x - 1 does not have any implied brackets, and is, in fact, the same as 2 × x + 3 ÷ 3 × x - 1. This is important but I'll get into why later.
2. Exponents (Or, orders or indices) ^
These are normally written as a superscriptor the alternative symbol is ^. It's just powers and square roots. 23 or x2 or √4 come second. Why is this?
Basically, it's because of what a power is. It's repeated multiplication. 2^3 means 2 × 2 × 2. If you have, say, 3×23, what you're doing is 3 × 2 × 2 × 2, which is very different from (3×2)3 or 6 × 6 × 6.
Now here, we get problems from -12 because is this -1 × -1 or -(1 × 1). The answer is the latter. Exponents come before the inverse sign. If we meant -1 × -1, we'd write (-1)2.
Square root happens at this time too because the square root is just something to the power of a fraction.
3. Multiplication (And divison) ×÷
This seems to cause a lot of issue because of the whole implied brackets thing above. But, first things first, multiplication and division happen at the same time. One doesn't happen before the other, you do them both at exactly the same time. Why? Because multiplication and division are identical operations. The only difference is when you divide, you multiply by the inverse of a number. What's the inverse of a number? If your number is x, the inverse is either x-1 or 1÷x. Both are the same thing.
Go on, try it. Get a decent calculator and do, say 6 ÷ 2. You'll get 3. Then do 6 × 0.5 (0.5 being the inverse of 2). You'll get 3 again. They're both identical operations and that's why I don't like having BEMA explicitly state division. Division is just a funny way of doing multiplication.
Now then, why do these come in this bit? Because, like exponents, multiplication is just repeated addition. 6 × 3 is just 6 + 6 + 6. 23 is just 2 × 2 × 2 is just (2 × 2) × 2 = (2 × 2) × (2 × 2) = 2 + 2 + 2 + 2. Again, 2 + 6 × 3 is 2 + 6 + 6 + 6, which is different to 8 + 8 + 8.
Now, I mentioned 2 × x + 3 ÷ 3 × x - 1 earlier. Now, order of operations says that the middle bit is all done at the same time. So this isn't 3 ÷ (3 × x), it's 3 ÷ 3 × x. The first is equal to 1/x, the second is equal to x. That's a huge difference and the original troll maths was based around this distinction.
4. Addition (and Subtraction) +-
Okay, this is fairly obvious. Last, you do all addition and subtraction. Again, these happen at the same time because subtraction is just adding the negative of a number. 5 - 3 is the same as 5 + (-3). I don't think I need to go in to more detail here.
Doing things at the same time.
I've mentioned this a few times. Doing multiplication and division at the same time, and addition and subtraction at the same time. Obviously, that's a rather tricky thing to do. So, what I mean when I say this is you work from left to right. If you have 5 + 3 - 2, going from left to right, you do the addition first, 5 + 3 = 8, then subtract 8 - 2 = 6. For 5 - 3 + 2, you do the subtraction first, 5 - 3 = 2, then the addition, 2 + 2 = 4. Same with division and multiplication. 12 ÷ 4 × 2, do the division first to get 3, then multiply by 2 to get 6. Or, 12 × 4 ÷ 2, you do the multiplication first to get 48, then divide by 2 to get 24.
Now, for something more advanced (Feel free to ignore this if you aren't a nerd), the truth is, the order doesn't actually matter. To use a fancy maths term, addition and multiplication are commutative. Which means the order doesn't matter. If you remember that subtraction is adding by the negative of a number and division is multiplying by the inverse of a number, the order becomes unimportant. 5 + 3 + (-2) is the same as (-2) + 5 + 3 is the same as 3 + 5 + (-2). Likewise, 12 × (1/4) × 2 is the same as (1/4) × 2 × 12.
Aaand that's about all I have to say. Remember people, don't feed the troll and remember the order to do things!
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