Order of Operations

The basics of arithmetic are the operations, the order of operations called PEMDAS, and the property laws. The acronym PEMDAS states the “order of operations”. Operations must be performed in a specific order or else answers will vary. The letters stand for 1. Parentheses, 2. Exponents, 3. Multiply and Divide, 4. Addition and Subtraction.

Parentheses are a grouping symbol. Operations inside parentheses are said to be "nested". Something inside parentheses should be done first. If there are parantheses nested inside parantheses you do the innermost ones first, then work your way out.

Example-1 parenthetical term: \(4 ⋅ ( 2 + 1 ) \)

\[ \begin{align} 4& ⋅ (2 + 1)\\ 4& ⋅ (3)\\ 4& ⋅ 3\\ 1&2 \end{align} \]

A parenthetical term is a "term" inside "parentheses". Because \( (2+1) \) is a "parenthetical term" it must be performed before "4 ⋅ ", even though multiplication comes before addition in PEMDAS.

Example-2 double nested terms: (4 ⋅ (2+1)) - 2

\[ \begin{align} (4 ⋅ (2 + 1)) - &2\\ ( 4 ⋅ 3 ) - &2\\ 12 - &2\\ 1&0 \end{align} \]

\((4 ⋅ (2 + 1))\) is the entire "nested term" or "parenthetical". \((2+1)\) is a "double nested term" because it is inside two sets of parentheses.

Tiers of Operations

There are three tiers of operations: ^, ⋅ , +.

The three tiers of operations are fully explained under "Rules Of Reciprocity". PEMDAS is an acrynym of 6 letters. The first letter stands for grouping symbols (which are not operators) and the other 5 letters are for operators/operations. As normally taught this creates fast learning but confusion later.

Where are radicals listed?

There are 6 operators but PEMDAS only lists 5. This is a problem. Where are radicals listed? Radicals are on the same tier as exponents because they are reciprocal operations. A reciprocal operation "undoes" the previous operation. For example subtraction undoes addition.

Why are radicals not listed in PEMDAS?

Radicals are complicated for beginners and usually taught way after PEMDAS. By then the rules of radicals can gently be introduced. Racidals themselves usually are down independently of other operation so it usually is not an issue. For completeness PEMDAS would add radicals and be written as PERMDAS: Parentheses, Exponents, Radicals, Multiply, Divide, Add, Subtract. This does not happen and no one is suggesting that it should. This just is to show how math and language do not always mutually agree with each other.

Of the six operations three increase value and 3 decrease value. Typically when your result is a Power, Product, or Total the operation moves right on the numberline to find the answer. This increases value. Therefore adding, multiplying, and powering are considered "positive operations". Their inverses rooting, dividing, and subtracting decrease value. Likewise they are considered "negative operations". Using this approach we can revise PEMDAS into a new framework that is logically more consistent called PEMA where we focus on 1 grouping convention and three tiers of operations.

PEMA lists the three tiers using only the positive operations ^, ⋅ , + but include the negative operations:

• Tier 1: ^ and √ (power or root symbol is " ^ √ ")
• Tier 2: ⋅ and ÷ (divide or times symbol is " ⋇ ")
• Tier 3: + and - (plus or minus symbol is " ± ")

Therefore P.E.M.A. stands for Parentheses, Exponents, Multiply, Add. It implies that negative operators are included at the explicitly stated tiers.

Some equations have only one, some, or all operator signs. Following the correct order means that two different people will get the same answer. Without rules for order of operation nobody would agree on the answer. These rules exist because mathematicians agreed on the order of numbers and the order of operations.

See how following a different order gives different answers? This is why it is important to follow the correct process or your answer will be different than everyone else's; like your friends or the answers in a teacher's book. Please follow the order so that we all are in agreement. After you master PEMA/PEMDAS you will learn times when breaking the rules of PEMA/PEMDAS can still guarantee same answers by following different rules called "properties". Addition and subtraction can always be done interchangeably as long as you use the correct sign for negative numbers. In rare cases so can multiplication and division:

Left to right:
1. Multiply first: \(2 ⋅ 3÷4\)
2. Divide second: \(6÷4 = 1.5\)

Right to left:
1. Divide first: \(2 ⋅ 3÷4\)
2. Multiply second: \(2 ⋅ ¾ = 6/4\)

The answers are both 1.5 because multiplication and division are on the same "tier". This is called "multiplication and division share reciprocity". Any operations that share reciprocity may be done in different order as long as it does not change final result.

Left to right:
1. Add first: \(5 + 3 – 1\)
2. Subtract second: \(8 – 1\) = 7

Right to left:
1. Subtract first: \(5 + 3 – 1\)
2. Add second: \(5 + 2\) = 7

Why does a different order work this time? Because adding and subtracting are reciprocal, meaning they share reciprocity. Changing this order does not matter because they are inverses which means they act like almost the same operation. We see that subtracting a positive is the same as adding a negative like \(3–1 = 3+(–1)\) and that dividing a whole number is the same as multiplying by a fraction such as \(10/2 = 10 ⋅ ½\). To be clear \(3–1\) does not equal \(1–3\). This will give additive inverses 2, –2. What is implied is \(3 – 1 = –1+3 = 2\). This is clarified further under Properties of Algebra.

Because + and – are inverse operations order doesn't matter. The are considered the same level. Because ⋅ and ÷ are inverse operations order doesn't matter. The are considered the same level. But we still have to do ⋅ and ÷ before + and – . We state this rule as “Multiplication does not share reciprocity with addition”. The problems occur with three or more terms and you alternate level operations. Consider add and subtract as level one and multiply divide as level two. If different levels of operations are involved then you must follow Pemdas left to right.

For cases like \(10 ÷ 2 = 10 ⋅ ½\) dividing by 2 is the same as multiplying by ½ because of fraction laws:

\(10÷2 = 5\)       \(10 ⋅ ½ = \frac{10}{1} \cdot \frac{1}{2}\) =       \( \frac{10 ⋅ 1}{1 ⋅ 2} \) =       \( \frac{10}{2} = 5 \)

We see both \(10 ÷ 2\) and \(10 ⋅ ½\) equal 5.

In algebra rewriting an equation is called algebraic manipulation. We do this almost every problem.
Four important points:

1. If a number is outside parentheses without an operator you multiply
2. If a number is in front of a radical you multiply
3. Rooting is done during “E” in Pemdas
4. If “ – ” is outside parentheses you flip all signs inside parentheses

Point 1: For exponents radicals are obvious when to perform because they act like a grouping symbol.

\(3\sqrt{9 − 5}\) \(3\sqrt{4}\) \(3 × 2 \) \(6\)

A radical is really a grouping symbol that nests operations under the vinculum (overline bar) similar to how a long division box does (although typically there are no nested operations in a division box only the dividend).

If the expressions was written as " \(3\sqrt{9} − 5\) " the minus operation is not under the radical (radical's vinculum) so it comes after rooting:

Point 2:

Numbers in front of grouping symbols invoke "implied multiplication". Taking the square root of 9 gave us 3. Since a 3 was in front of √9 we needed to write 3(9). Something like 3(2) means 3 ⋅ 2. If a number is outside parentheses without an operator you multiply: 3√4 = 3 ⋅ √4 = 3(2) = 3 ⋅ 2 = 6.

Point 3:

There are many rules for exponents. Although Pemdas does not explicitly state when to root it is done during exponents because rooting is the reciprocal operation of powering. In fact rooting uses exponents to determine which root to take be it the square, cube, or higher root. Here is one of the parentheses rules for roots:

3√4 3 = 3(2)3 = 3 ⋅ 43 = 3(2 ⋅ 2 ⋅ 2) = 3(8) = 3 ⋅ 8 = 24

First find the root. Then if an exponent is outside of the parentheses power the root before doing anything else.

Point 4:

–1 ⋅ any number = –that number.

This is true: – 1 ⋅ 4 = –4 Multiplying by a negative inverts the sign of the number
This is true: –1(7) = –7 The parentheses here imply multiplication
This is true: –(52) = –52 A sign outside parentheses inverts the sign of the number
This is true: –(2+3–4 ) = ( –2–3+4) The additive inverse of 2+3–4 is –2–3+4

Additive inverse means change all plus signs to minus signs and all minus signs to plus signs. We are inverting the operations and this means to reverse them so we are taking its inverse. The operations were additive so we call this type of inverse “additive inverse”. The slang term for taking an additive inverse is “flipping signs”.

Multiplying a number n by –1 creates the additive inverse for that n.
We write a rule, called a formula, that uses the script letter n be mean any number.
We write this rule as an equation and apply it by replacing n the actual number we want to take the inverse of. The formula for additive inverse is: n ⋅ –1 = –n.

Using letters to represent numbers is called variables. Variables are used to write rules called formulas. Formulas appear in algebra and every branch of math after that.