Algebra - Order of Operations

• Tale of Order of Operations
• Vignette
• What is the meaning of the word “Operations”?
• What is the meaning of “Order”? 
• Meaning of “Order” in BODMAS, “Indices” in BIDMAS, and “Exponent” in PEMDAS
• Examples for BODMAS (Brackets ⇒ Orders ⇒ Division ⇒ Multiplication ⇒ Addition ⇒ Subtraction)
• Examples for BIDMAS (Brackets ⇒ Indices ⇒ Division ⇒ Multiplication ⇒ Addition ⇒ Subtraction)
• Examples for PEMDAS (Parentheses ⇒ Exponents ⇒ Multiplication ⇒ Division ⇒ Addition ⇒ Subtraction)
• Why follow the order of operations?
• Practice Problems
• Frequently Asked Questions

Tale of Order of Operations

The order of operations dates back to the `16`th and `17`th centuries, when mathematicians Francois Viete, René Descartes, Augustin-Louis Cauchy, and Gottfried Leibniz developed the logic of performing operations. They then developed modern rules for power and root calculation, which resulted in the development of the order of operations.

Vignette

The order of operations is the sequence of operations used in mathematics to perform operations (addition, subtraction, division, and multiplication). The use of these operations guides everyone to get the same results, no matter who uses them. These are used in performing mathematical operations and solving mathematical equations.

What is the meaning of the word “Operations”?

In mathematics, operations are the actions undertaken with two numbers or variables for calculations. These operations include addition `[+]`, subtraction `[-]`, multiplication `[\times]`, and division `[÷]`. These symbols are used to represent any calculation between two numbers or variables.

What is the meaning of “Order”?

The order of operations is the sequence of performing mathematical operations in a calculation or equation to achieve the right answer. The internationally accepted sequence of operations is BODMAS/BIDMAS/PEMDAS. These abbreviations represent the order of operations in which the calculations are to be performed if multiple operations are given in an expression.

BODMAS/BIDMAS/PEMDAS are the acronyms to remember the order of operations in mathematics. The following table below shows the meaning of each letter with an example. BODMAS/BIDMAS/PEMDAS are to be followed from left to right for performing mathematical calculations.

Meaning of “Order” in BODMAS, “Indices” in BIDMAS and “Exponent” in PEMDAS

Brackets, parentheses, addition, subtraction, multiplication, and division are known operations. Order, Indices, and Exponents are the terms that can be understood in the following example. 

Take an expression `5+2^4` here, `4` in superscript of `2^4` is termed as Order, Indices, and Exponents, which are to be performed after brackets (in BODMAS and BIDMAS) or parenthesis (in PEMDAS) and before division (in BODMAS and BIDMAS) or multiplication (in PEMDAS).
 

Examples for BODMAS (Brackets ⇒ Orders ⇒ Division ⇒ Multiplication ⇒ Addition ⇒ Subtraction)

Consider the following examples.

1. `3\times (4+5)`
   Solve operations in Brackets first.
   `3\times (4+5)=3\times 9=27`
    
2. `3^2+5`
   Solve operations with Order (powers and roots), i.e. `3^2`
   `3^2+5=9+5=14`
    
3. `9\div 3\times 5`
   Rank Division and Multiplication equally and solve them from left to right as they appear. 
   `9\div 3\times 5=3\times 5=15`
    
4. `10-6+3`
   Rank Addition and Subtraction equally and solve them from left to right as they appear.
   `10-6+3=4+3=7`

Examples for BIDMAS (Brackets ⇒ Indices ⇒ Division ⇒ Multiplication ⇒ Addition ⇒ Subtraction)

Consider the following examples.

1. `4\times (3+5)`
   Solve operations in Brackets first, followed by other operations
   `4\times (3+5)=4\times 8=32`
    
2. `4^2+6`
   Solve operations with Indices (powers and roots), i.e. `4^2`
   `4^2+6=16+6=22`
    
3. `6\times 5\div 3`
   Rank Division and Multiplication equally and solve them from left to right as they appear. 
   `6\times 5\div 3=30\div 3=10`
    
4. `10+7-3`
   Rank Addition and Subtraction equally and solve them from left to right as they appear.
   `10+7-3=17-3=14`

Examples for PEMDAS (Parentheses ⇒ Exponents ⇒ Multiplication ⇒ Division ⇒ Addition ⇒ Subtraction)

Consider the following examples.

1. `(7+4)\times 3` 
   Solve operations in Parentheses first, followed by other operations
   `(7+4)\times 3=11\times 3=33`
    
2. `2^2+5`
   Solve operations with Exponents (powers and roots), i.e. `2^2`
   `2^2+5=4+5=9`
    
3. `9/3 \times 8`
   Rank Multiplication and Division equally and solve them from left to right as they appear.
   `9/3 \times 8=3\times 8=24`
    
4. `9-7+4`
   Rank Addition and Subtraction equally and solve them from left to right as they appear.
   `9-7+4=2+4=6`

Why follow the order of operations?

Practice Problems

1. Simplify using BODMAS

`56\div 8+(2\times 7)-11`

1. `10`
2. `45`
3. `37`
4. `27`

Solution:

`56\div 8+(2\times 7)-11`

Step 1: Solve Bracket operation `56\div 8+14-11`

Step 2: Solve Division operation `7+14-11`

Step 3: Solve Addition and Subtraction operations from left to right `10`.

2. Simplify using BIDMAS

`3^2+(8\div 4)-7\times 2`

1. `35`
2. `-3`
3. `3`
4. `7`

Solution:

Step 1: Solve Bracket operation `3^2+2-7\times 2`

Step 2: Solve Indices operation `9+2-7\times 2`

Step 3: Solve Multiplication operation `9+2-14`

Step 4: Solve Addition and Subtraction operations from left to right `-3`.

3. Simplify using PEMDAS

`15/3+4^2-(8+11)`

1. `5`
2. `4`
3. `3`
4. `2`

Solution:

Step 1: Solve Parentheses operation `15/3+4^2-19`

Step 2: Solve Exponential operation `15/3+16-19`

Step 3: Solve Division operation `5+16-19`

Step 4: Solve Addition and Subtraction operations from left to right `2`.

Frequently Asked Questions

Q`1`. Why are we calling it the order of operations?

In mathematics, to solve a problem certain rules are to be followed for getting a correct answer, and a standard is to be followed all over the globe. To follow common standards, these are called orders of operations and are valid all around the world.

Q`2`. Can we solve Multiplication before Division?

Yes, you can do that for multiplication and division as well as for addition and subtraction, but, you cannot solve multiplication/division and addition/subtraction before solving brackets/Parentheses and solving order/indices/exponential.

Q`3`. What is the difference between BODMAS/BIDMAS and PEMDAS? Which one should I follow?

There is no difference between all these. These are all acronyms, they are followed and called by different names in different countries. PEMDAS is followed in the U.S., whereas in regions like India, Australia, and the UK, BODMAS is followed. As the basic function is the same, people following any of the three will obtain the correct answer.