What makes non-PEMDAS answers invalid?
It seems to me that even the non-PEMDAS answer to an equation is logical since it fits together either way. If someone could show a non-PEMDAS answer being mathematically invalid then I’d appreciate it.
My teachers never really explained why, they just told us “This is how you do it” and never elaborated.
In: 834
Just like any language needs grammar, mathematics is a language that needs rules to be intelligible by everyone. If we resolved operations with any made up order two people would get different results for the same equation, and would write it differently to say the same thing, which is obviously not very practical. As such, everyone agreed to use this one made up order.
You can write words wrong but people will not understand what you are saying, so it is in the best interest of everyone to write words right. Right in this case means “As everyone else”. Same principle
Edit: By the way I had a similar problem the first time I started with technical drawings back in the day. I didn’t understand why one of the drawings was wrong, and it turns out that it was because I didn’t follow certain conventions. Which is vital, but at the time I didn’t understand the concept and the teacher just kept saying “that’s just how it is done”. Looking back it’s just that she was dumb as a rock, a teacher that can’t clearly explain to a kid something so simple yet so vital is a bad teacher
PEMDAS is like grammer for math. It’s not intrisicly right or wrong, but a set of rules for how to comunicate in a language. If everyone used different grammer maths would mean different things
Example
2*2+2
PEMDAS tells us to multiply then do addition 2*2+2 = 4+2 = 6
If you used your own order of operations SADMEP you would get 2*2+2 = 2*4 = 8
So we need to agree on a way to do the math to get the same results
In essence, it’s done for simplicity sake. It’s just something everyone can agree on, it could have easily been DEPSAM or some other abbreviation. However, think about it like this, exponentiation is repeated multiplication, multiplication is repeated addition, and addition is just counting up. It’s placed in the order of highest influence, but is still just done to create uniformity.
PEMDAS isn’t required.
What is required is that everyone agrees to the same order of operation.
Everyone needs to be on the same page in which order a term is processed.
If everyone agrees that we process the terms according to PEMDAS that works. If everyone agrees that we simply go left to right, that works too.
What doesn’t work is if some people want to read a term one way and some other people want to read it another way. That doesn’t work.
It is like finding a word written down and arguing whether reading it as a French word with French pronunciation and meaning or as an English word with English pronunciation and meaning is more correct.
One way of reading a word is not more correct than another, what is important is that everyone agrees on a single way to interpret the word in the context it is in otherwise it has no meaning at all.
For the same reason we require you to treat a + as “addition”.
Yes, the equation would still *look* fine and logical if you decided that a + now means “multiplicaiton” and that `*` means “subtraction”. You could also decide that the symbol “17” now means “two hundred and forty point three”. It would be mathematically valid, it just wouldn’t mean whatever the author *wanted* it to mean.
If I write `2 + 3 * 7`, my intent is for you to read it as “two plus the result of multiplying three by seven”. If we follow the same mathematical rules then you will be able to read it the way I intended it.
These conventions are communication tools. They allow us to write things down and have other people read them *and gain the same understanding*. If you don’t follow the same conventions as everyone else then you won’t understand what they meant by what they wrote, and they won’t understand what you mean with what you write. Then you’re no longer speaking the same language.
Think of a formula as communication.
When I write down a mathematical formula, I’m communicating the numbers involved, the operations that need to happen to those numbers and the order it should be done in.
There’s no universal need to do addition and subtraction after multiplication, but for any given formula, I’m trying to communicate a specific order for operations to be performed in. Having that order be standard means I can communicate in a simple compact way without needing to add in a lot of notes on the order.
The P in PEMDAS is the parentheses which IS more or less a note on how to order thing, and it provides a relatively simple tool to break and shuffle the order when needed.
But at the end of the day, PEMDAS is the grammar the person who wrote the formula is using, and so it’s the grammar needed to decode the formula they wanted to share with you. If you use a different grammar to read the formula, you won’t be reading the same formula they tried to give you, just like you won’t pronounce a word as it was intended if you read the letters out of order.
The order isn’t arbitrary.
Parentheses kinda need to go first to promote a calculation, or they rapidly get wildly complicated to use and you’d often have to use a bunch of them to get to the same result that a single parentheses can achieve with our current notation where it brings a part of the calculation to the front.
Similar with exponents – if you don’t have the exponent have priority over MDAS then you’d need to use parentheses almost every time you have an exponent to get the answer you need.
Multiplication and Division – these are next and equal in priority, because division is fundamentally multiplication by the inverse of the number. Eg 4/3 = = 4 x 1/3 = 4 x 0.333r. It makes sense to have MD come before AD because MD are essentially batched addition/subtractions, also again if you prioritize these after addition/subtraction you need a bunch of parentheses.
Fundamentally though – math isn’t a homework assignment, it’s a way to represent real quantities and things that are happening, so you can’t just evaluate everything in a different order and as long as everyone agrees on the order it’s all correct, because you’re not going to get to the real world answer you’re aiming for. It might be possible to write math using an alternative order of precedence, but in many cases you’re going to get a very convoluted equation vs the order we have, because you’re working against what the numbers are actually doing.
>If someone could show a non-PEMDAS answer being mathematically invalid then I’d appreciate it.
Try forming it as a word puzzle. If you have two lots of six apples, plus another two apples, what do you have? How do you write it? Well, there are a bunch of ways:
* (2 × 6) + 2
* 2 × 6 + 2
* (6 × 2) + 2
* 6 × 2 + 2
(There are others, but let’s just go with that for the moment.)
If we calculate those out using PEMDAS, we get:
* (2 × 6) + 2 = 14
* 2 × 6 + 2 = 14
* (6 × 2) + 2 = 14
* 6 × 2 + 2 = 14
If we calculate those same expressions out using a different system — for example, PESADM — we’d get:
* (2 × 6) + 2 = (12) + 2 = 14
* 2 × 6 + 2 = 2 × (8) = 16
* (6 × 2) + 2 = (12) + 2 = 14
* 6 × 2 + 2 = 6 × (4) = 24
But we’re talking about real, concrete things here: two packages of six apples, plus another two apples. You can take those apples out of the packages, line them up, and count them. There are 14 apples. That’s just a fact.
PEMDAS allows us to minimise the number of parentheses we need to use in order to get a consistent answer. (You’ll notice that in the last batch of answers, the two expressions that ‘worked’ both had parentheses right from the start.) Basically we use that order because it’s a way of both simplifying an expression and getting a consistent answer that everyone — if they follow the rules — can agree on.
Also something a lot of people forget/dont know, multiplication and division have the same priority so they could be swapped, same with addition and subtraction.
Edit: since there apparently is some confusion, by swapped I mean PEMDAS and PEDMSA are equivalent.
Some mathematical equations have some ambiguity in them. The simplest example I can think of is something like:
3 * 5 – 4 ÷ 2 + 9
There are many ways to solve this equation. Left to right (14.5). Right to left (-6.75). ASMD (16.5). MDAS (16). As you can see, each one will give you a different answer.
To remove those ambiguities, we need to have a convention to tell you the order of operations. The rules to ensure we all understand the math the same way and we all get the same answer.
That convention is PEMDAS.
Latest Answers