P E MD AS

What is PEMDAS?

PEMDAS is the order in which we do mathematical calculations. Each letter stands for the following. Parentheses, Exponents, Multiplication, Division, Addition and Subtraction.
These are arranged as follows.

P > E > MD > AS

P precedes E precedes M&D*  precedes A&S*

***NOTE: M&D are of equal importance and are to be resolved from left to right after Exponents.
***NOTE: A&S are of equal importance and are to be resolved from left to right after Multiplying and/or Dividing.

First is the P and that stands for Parentheses which have to be done first. e.g.

(1 + 3) x 2

First we do 1+3=4 and then 4 x 2 = 8. However whatever lies inside of the parentheses also have to follow the rules of PEMDAS e.g. (11 + (1 x 3 + 1)) x 2.*
Find the parentheses, within those parentheses look for more parentheses. When you are inside the innermost parentheses you may continue onto E which are the Exponents.

23 = 2x2x2 = 8.

Before we do any Multiplication we first have to figure out how many times we must multiply the same number by itself. For the benefit of this tutorial we are not going to deal with P and E. This will be a more advanced tutorial.  Right now we will just be dealing with MDAS.

***NOTE: Start with red, then add red to orange, add orange to yellow and multiply yellow by green

What is MDAS?

Well for a start let us first remove something which I find to be a little redundant.  The S on the end stands for subtraction which we can do completely without.  How can we do this? There really is no such thing as subtracting, this can be easily trumped by a different concept.

It is a lot better to add negative numbers instead of subtracting positive ones. Here is an example. With addition, it does not matter which order you add the numbers together you will get the same sum.  WIth subtraction, if you rearrange the numbers the difference changes based on which number you start with.  This is a problem. When we rearrange the numbers we are actually changing the value of the first number from a negative to a positive or vice versa.

6 - 2 = 4 2 - 6 = -4

2 is negative 2 is now positive

   6 is now negative

If we add negative numbers then we can rearrange the values however we want and still get the same answer because we are keeping the individual values constant

6 + (-2) = 4 (-2) + 6 = 4

2 is negative 2 remains negative

1 + 2 - 3 + 4 - 5 + 6 becomes 1 + 2 + (-3) + 4 + (-5) + 6 and equals (-5) + (-3) + 2 + 6 + 1 + 4

-8  -7  -6  -5  -4  -3  -2  -1  0  +1  +2  +3  +4  +5

If we were to use a number line to calculate the value of this sum, even if we rearrange the numbers to put the 5 as the first number, because 5 is actually negative (-5) we can make sure we start at the correct place on the number line and then continue to solve the sum and get the same answer no matter which order we do this in.

Another problem which subtraction gives us it when we include multiplication. Consider

9 - 4 x 2

If we do this from left to right then we would fail to recognize the 4 as (-4) and then when we multiply this number we will get the wrong value which will be wrong by 2 x absolute value. Multiplying negative numbers results in a negative answer unless both factors are negative.

9 + (-4) x 2 != 10 9 + (-4) x 2 = 1

9 + (-8) !=10 9 + (-8) = 1

Still if we continue to do this problem from left to right then we will still get a wrong answer which will be explained later

What is MD > A?

Now that we have been able to remove the concept of subtraction and replaced it with the concept of adding negative numbers we can now move onto the correct way to solve problems.
Imagine that there are real objects of a certain kind behind a curtain. You have no idea of knowing how many there are but you have a clue written on the outside of the curtain.  The clue was constructed using the number of objects as a starting point.  While we do not know the starting number of objects we can still use the clue and mathematics to derive the number of objects behind the curtain.

Lets start with 7.  There are seven chairs behind the curtain.  We can arrange those seven chairs a number of ways.  We can group them up into 3 groups of 2 which will leave 1 left over. Or we can group them into 2 groups of 3 with one left over or we can group them into 7 groups of 1.

Lets take the first example where we group them into 3 groups of 2 with one left over.  This would be represented as the following 3x2+1 now if we do this from left to right, we get the answer we started with which is 7.  What if we moved it around to make 2x3+1? well we get the same answer because like addition, it does not matter which order we arrange the multiplication numbers, we get the same product. Now here it gets tricky to explain.

Remember PEMDAS? we removed P and E and saved them for a later tutorial and we removed the S in llew of simply just adding negative numbers in order to keep the negative value rooted to the correct value. so now we are faced with MDA. really we need to look at this differently.

P > E > MD > AS

We arrange this in a way to remind us that we must Multiply and Divide (from left to right might I add) before we add numbers. Here is why

Lets take our previous example of 7 being represented by 3x2+1.  

3 x 2 + 1 = 7

Lets switch this around to make 1 + 3 x 2 and solve from left to right.

1 + 3 x 2 = ?

This should be valid right? I mean it does not matter which order we add things because we always get the same answer.  HOWEVER!! we do not know what 3x2 is yet.  What happens if we ignore the fact we have to do the multiplication first. Well we have 7 chairs behind the curtain and we represented with 3 x 2 + 1 so let’s flip the order in which we work with 1 + 3 x 2 and do this from left to right. well 1 + 3 is 4, times that by 2 and we get 8??  What happened there? We now have one more chair. Where did it come from? did someone sneak it in behind the curtain? Ok well if we pull back the curtain we will still see 7 chairs because the math is now wrong.  This ‘extra’ chair came into existence because we did addition before multiplication. when we added 1 to 3 we then created another chair to be added to the second group of 1+3 which is not what we started with.  We had 2 groups of 3 and then 1 more added to it. If we had 8 chairs then we would represent this by (1 + 3) x 2 because 8 chairs can be arranged into 2 groups of 4. And  4 chairs can be arranged into a single group of 1 plus a single group of 3.  So to represent 8 chairs we first must add 1 and 3, to get 4, before we define how many groups there are of 4 which in this case is 2. But we did not have 8 chairs, only 7 so to do addition before multiplication broke reality. Simple rule to remember is to never add to a factor.

In multiplication we take the two factors and multiply one by the other to get a product.
2 is a factor and 3 is the other factor required to get the product of 6

In the diagrams below you can see that 2 x 3 is represented by showing 2 groups of 3 TP rolls but you can also identify them as being represented by 3 groups of 2 TP rolls depending on whether you mentally group them horizontally or vertically.  Either way whether it is 3 x 2 or 2 x 3 they still produce 6. What we do notice is that we do not allow any part of the multiplication to cross the addition boundary. and vice versa unless we explicitly show that we want to do that with the use of parentheses as in the case of (1 + 3) x 2. This means that one of the factors of the multiplication has been represented by 1 + 3 instead of 4.  We are not adding to a factor, the addition is the factor.

A visual representation of 7 rolls of TP arranged into 2 groups of 3 with the addition of 1

A visual representation of 7 rolls of TP arranged into 1 with the addition of 2 groups of 3

A visual representation of what would happen if you added before multiplying with the same figures

Here are all the combinations using the same digits and operators and their answers.

1.)  1 + 2 x 3 = 7 2.)  (1 + 2) x 3 = 9 3.)  1 + 3 x 2 = 7 4.)  (1 + 3) x 2 = 8
5.)  2 x 3 + 1 = 7 6.)  3 x 2 + 1 = 7

Notice how the following equal example 4 (2 + 2) x 2 = 2 x 2 x 2 = 23 = 8
Where the following do not 2 + 2 x 2 = 2 x 3 = 6

Division

Although on the same tier as Multiplication in the hierarchy, Division is similar to Multiplication as it is the inverse of the way Multiplication operates. Where Multiplication takes a number of similar groups and combines them into a total, or product, Division takes groups and divides them into equal amounts. However; unlike Multiplication, where the factors can be in any order to produce the same product, Division has to have its dividend and divisor in explicit places.

4 ÷ 2 = 2 where 2 ÷ 4 = 0.5

Division is written differently in algebra and referred to differently.  Typically the phrase ‘divided by’ is replaced with ‘over’ and is written as the following

4 Dividend 2 Dividend 1
- = 2 - = 0.5 or - One half
2 Divisor 4 Divisor 2

The same applies when using Division around addition you must never add a number directly to either the Divisor or a Dividend in the same respect that you must never add a number directly to either factor of multiplication.

Consider the following

3 + 6 / 2 = 6 (3 + 6) / 2 = 4.5 3 + 2 / 6 = 3.3333~ (3 + 2) / 6 = 0.83333~

Which should be written as

     6
3 + - = 6
     2

And not as

3 + 6
------- = 4.5
  2

Now with all that said. The following problem which shows up on Facebook a lot is this

6 - 1 x 0 + 2 ÷ 2 = ?

Remember we first do the Multiplication & Division as we find them from left to right paying attention to the multiplication problem actually being -1x0 and not 1x0 even though in this case they both warrant the same answer. -1 x 0 = 0 and don’t forget that 6 - 1 x 0 is actually 6 + -1 x 0 because there is no actual subtraction but the addition of negative numbers.

6 - 1 x 0 + 2 ÷ 2

6 + (-1) x 0 + 2 ÷ 2

6 + 0 + 1

7

Even if we rearrange the problem to include the division first we will still get the same answer. Please note that when rearranging the problem the + and - operators were arranged also to correctly represent the values e.g. the dividend in 2 ÷ 2 as well as first factor in -1 x 0.

6 + 2 ÷ 2 - 1 x 0

6 + 2 ÷ 2 + (-1) x 0

6 + 1 + 0

7

In conclusion.
Here are a few simple rules to tackle simple math.

1. Do not add to a Factor, Divisor or Dividend. Only add to Products, Quotients and other actual values e.g. other additions.
2. Do not merely subtract but rather add negative numbers. (this helps keep track of whether values are negative or not)
3. When re-arranging a problem do not forget the polarity of the value, especially Factors, Divisors and Dividends. (e.g. make sure that when multiplying, if a factor is negative that it remains negative and vice versa.)

Fin