Order of operations?

Question

Ok, it's been taught to us in basic Algebra that you do multiplication and division first, then addition and subtraction (always left to right). My question is how did mathematicians come to this conclusion? I mean, who's to say that the math we've been doing could have been slightly incorrect?

A basic problem that is used as an example: What is 2+3*4? The answer is 14, but many people may say it's 20. Well why are we multiplying first? Is it because multiplying produces more than addition?

I'm just curious, because if the order were switched the other way, we'd really have a lot of equations to adjust.

Answer 1

Your friend has written out the formula 2 × 18 + 2 × 25 for the perimeter, in feet, of a rectangular lot, giving you the job of purchasing fencing to surround the lot.

But does your friend know about the order of operations? If he does, then you need to purchase 86 feet of fencing; if he does not, and he intends for you to simply perform the operations from left to right, then you need to purchase 950 feet of fencing.

Suppose, now, that you misread "18 ft" for "10 ft." Instead of the correct expression, 2 × 18 + 2 × 25, you calculate 2 × 10 + 2 × 25 as the perimeter of the lot. If YOU use the order of operations, you will purchase 70 feet of fencing; if you do not, you will purchase 550 feet of fencing.

As you can see, simply using the same system is not the idea behind the order of operations. If you both ignore the order of operations and simplify each expression from left to right, you will purchase 400 fewer feet of fencing than you need (off by a factor of about 1.7). If you both FOLLOW the order of operations, you will purchase 16 fewer feet of fencing than you need (off by a factor of about 1.2).

What happened was that when you used the order of operations, you multiplied the error (–8 ft) by only 2 (–16 ft); when you ignored the order of operations, you multiplied the error by 2 and then by 25, so your error was multiplied 50 times over (–400).

When you use the order of operations, you deal first with the operations that have the greatest impact on the outcome--exponents, multiplication and division--before you deal with those operations that have less of an impact, like addition and subtraction. If an error is made, using the order of operations helps to contain it.

Answer 2

That's because if you switch the 2+3*4 into 3*4+2, the answer with the correct order of operations would have the same answer...but if you do 2+3 first then multiply 4, the two answers would be different.

Answer 3

It isn't arbitrary. Math is a tool that is used to solve real world problems. If you don't do the math in the right order, you don't get the right answer. For example, suppose in a class of 20 students, each has 5 pencils and the teacher has 10 extra pencils in her desk. You could express the number of pencils in the room as
20 * 5 + 10 or as 10 + 20 * 5, right? However, if you actually count the pencils in the room, it is obvious that there are 110. The only way you get to that correct answer every time is to do the multiplication first.

It is NOT arbitrary and it is NOT just a convention.

Answer 4

It is just an agreed upon way of doing things to simplify an equation. If you did not have the order of operations as an accepted rule, then you would have to either rearrange the problem to get a good answer, or use a lot of parentheses.

In your example I could be forced to always rearrange the formula so that it read 3*4+2 to get the 14 answer or I could add parenthesis to get 2+(3*4), Either of these could be understood to get 14 without the order of operations rule. In larger formulas it is more convenient to just understand an order of operations. Keep working with it and it will come naturally to you in time.

Answer 5

A few of the above posters pointed out some sage examples of why this is not simply a convention, but is instead based on reason and logic. However, I would just like to take a moment to point out how shocking it is that our introductory math teachers are not explaining WHY mathematical operations work they way they do.

Answer 6

PEMDAS

Parenthes first, then exponents, the multimplication and division in the same stroke, then addition and subtraction in the same stroke, so it looks more like (P)(E)(MD)(AS).

And in answer to yer actual question, they tested it, using counters and other methods, physical representations and found the only by following to Order of Operations does math reflect nature.

Hehe those last few words make me think of something ironic, math always reflects nature, but nature never reflects math.

Answer 7

Thats why we have rules so everyone does it the same way. Also if you don't follow the rules with algebra your equation will not check. Lets say your answer is x=4 but if you did not follow the rules, 4 will not give you a true statement when you check the equation.

Answer 8

It's called a convention. A group of influential folks decided the order of operations would be My Dear Aunt Sally. They were influential enough for everyone to adopt this convention.

Another convention is the symbol "+" will stand for addition. There's no reason it has to be that way, but we all agreed it would be.

Answer 9

Basically you do the hard stuff first. This is to make it easier for you to answer the problem.Ya See adding and subtracting is easy stuff. Also when you multiply and divide first your answer tends to be bigger. That probably why.

Answer 10

It is just a rule of mathematics. This is the way it was originally defined and so this is the way it is taught.

PS : brackets (parentheses) have precedence