Bodmas vs Pemdas – Definition, Applications, and Examples

In the vast realm of mathematics, one of the foundational principles every student must grasp is the order of operations.  Two mnemonic devices, BODMAS and PEMDAS, stand as sentinels guiding this uniformity.

This article delves into the distinctions and similarities between BODMAS and PEMDAS, offering insights into their regional influences and core objectives.

What is the difference Between Bodmas vs Pemdas

The main difference between the Bodmas and Pemdas lies in their terminology and regional usage. BODMAS stands for Brackets, Orders (i.e., powers and square roots, etc.), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right).

Both BODMAS and PEMDAS are mnemonic devices used to remember the order of operations in mathematics, ensuring equations are solved in a consistent manner. It’s predominantly used in the UK and some other countries. PEMDAS, on the other hand, is an acronym for Parentheses, Exponents, Multiplication, Division (from left to right), and Addition and Subtraction (from left to right).

This mnemonic is more commonly used in the US. While the terms differ, the fundamental principle they represent remains consistent across both: a standard sequence for simplifying mathematical expressions.

Dissecting the Mnemonics

• BODMAS

  • Brackets: This includes both square brackets and round parentheses, i.e., $[\text{ and }(]$.
  • Orders: Refers to powers (exponents) and roots.
  • Division: The act of dividing one number by another.
  • Multiplication: The act of multiplying numbers.
  • Addition: Summing up numbers.
  • Subtraction: Taking one number away from another.

• PEMDAS

  • Parentheses: The American term for round brackets, i.e., $($.
  • Exponents: Powers or indices, such as squares and cubes.
  • Multiplication: The act of multiplying numbers.
  • Division: Dividing one number by another.
  • Addition: Summing up numbers.
  • Subtraction: Taking away one number from another.

Properties

let’s delve deeper into the properties of both BODMAS and PEMDAS. Understanding these mnemonics is crucial for ensuring mathematical operations are executed in a standard order, thereby yielding consistent results.

BODMAS

Brackets (B)

This property emphasizes that any arithmetic inside brackets must be resolved first. There are three types of brackets to consider:

• Round brackets or parentheses: ( )
• Square brackets: [ ]
• Curly brackets or braces: { }

The order to solve them is: Curly, Square, and then Round.

Orders (O)

Once bracketed operations are complete, the next focus is on ‘orders’. This term encompasses:

• • Powers (like squares, cubes, etc.)
  • Roots (like square root, cube root, etc.)

Division and Multiplication (D & M)

These are of equal precedence and are resolved from left to right. It means if both operations appear side by side, the one on the left is performed first.

Addition and Subtraction (A & S)

These too are of equal precedence, like multiplication and division. They are performed from left to right.

PEMDAS

Parentheses (P)

Similar to the brackets in BODMAS, any arithmetic inside parentheses must be resolved first. Though PEMDAS primarily references round parentheses, the rules for handling different brackets (like square or curly ones) remain consistent with BODMAS.

Exponents (E)

This step deals with powers and roots, just like ‘Orders’ in BODMAS.

Multiplication and Division (M & D)

As with BODMAS, multiplication, and division are equal in the order of operations. They are performed sequentially from left to right, depending on their appearance in the equation.

Addition and Subtraction (A & S)

Again, mirroring the rules of BODMAS, these operations are handled from left to right.

Shared Principles Between BODMAS and PEMDAS

Priority of Brackets/Parentheses

Both acronyms prioritize operations inside brackets or parentheses above all else.

Exponents/Orders

Both acronyms agree on the sequence, placing powers and roots after brackets but before standard arithmetic operations.

Left to Right Rule for Division & Multiplication

They are performed as they appear from left to right. Neither operation has precedence over the other.

Left to Right Rule for Addition & Subtraction

Similar to multiplication and division, addition and subtraction are performed as they appear from left to right.

Thus, regardless of which mnemonic you use, the result remains consistent.

Exercise 

Example 1

5 + 6 × 2

Solution

Using BODMAS/PEMDAS:

Multiplication before Addition:

= 5 + 12

= 17

Example 2 

$20-3($$2^2$$)+8÷4$

Solution

Using BODMAS/PEMDAS:

Exponents before others:

$($$2^2$$)$= 4 

= 20 − 3(4) + 8 ÷ 4

Multiplication and Division from left to right:

= 20 − 12 + 2

Subtraction and Addition from left to right:

= $0$

Example 3

$(5+3)\times 2-5\times 2$

Solution

Using BODMAS/PEMDAS:

Solve Parentheses/Brackets first:

= 8 × 2 − 5 × 2

Multiplication from left to right:

= 16 − 10

Subtraction:

= $6$

Example 4

$15-3(2+2)÷2$

Solution

Using BODMAS/PEMDAS:

Solve Parentheses/Brackets first:

= 15 − 3 (4) ÷ 2 

Multiplication then Division:

= 15 − 12 ÷ 2 

Division:

= 15 − 6

Subtraction:

= $9$

Applications 

The order of operations, as represented by mnemonics like BODMAS (or BIDMAS) and PEMDAS, isn’t just a classroom exercise. It’s a foundational principle in mathematics and is applied across many fields to ensure clarity, consistency, and correctness. Below are some applications of these rules across various fields:

• Engineering

  • Engineers routinely work with complex equations, whether designing bridges, circuits, or any machinery. Using the correct order of operations is essential to achieve the desired result and ensure safety.

• Computer Science

  • Algorithms and software codes often incorporate mathematical formulas. Following the correct order of operations ensures the computer returns accurate results. Programming languages inherently respect these rules, and developers must be keenly aware of them.

• Economics & Finance

  • Economists and financial analysts work with formulas to predict market trends, set prices, or assess financial risks. Using BODMAS or PEMDAS ensures they derive accurate projections and insights. For instance, calculating compound interest or depreciation involves multi-step equations where the order of operations is critical.

• Pharmacy & Medicine

  • Dosage calculations, interpreting medical statistics, or biomedical research often require mathematical computations. An error due to the incorrect order of operations can have serious implications in these fields.

• Architecture & Construction

  • Architects and construction managers use mathematics in designing buildings and infrastructures. Ensuring the correct sequence of operations can be the difference between a structure that stands robustly and one that’s flawed.

• Astronomy & Space Exploration

  • Space missions, satellite launches, and celestial calculations require precise mathematical computations. Following the correct order of operations is crucial to avoid catastrophic failures or misinterpretations.

• Cryptography

  • Modern cryptography, which is the art of encoding and decoding information, relies heavily on mathematics. Ensuring messages or data are securely encrypted (and can be decrypted) necessitates the correct application of mathematical operations.

• Graphics & Designing

  • Modern graphics software, used for animation or design, often involve mathematical transformations behind the scenes, like scaling, rotating, or translating objects. These transformations need to be done in a specific order to achieve the desired visual effect.

• Physics & Chemistry

  • Whether it’s calculating the trajectory of an object, the reaction rate of a chemical process, or the energy levels in quantum mechanics, the correct order of mathematical operations is pivotal.

• Daily Life

  • Even outside professional fields, we use the order of operations in daily activities, like budgeting, cooking (scaling recipes), or DIY projects.

Historical Background of Order of Operations (BODMAS vs. PEMDAS)

Pre-Formalized Order

Before we had formal rules for the order of operations, calculations were often performed from left to right, which could lead to ambiguous or varying results depending on the mathematician or the region. This lack of standardization posed problems, especially as mathematical and scientific inquiries became more advanced.

Origins

The need for a standardized order of operations can be traced back to the development of mathematical notation itself. As symbols for operations (like +, -, ×, ÷) and groupings (like parentheses) evolved, it became crucial to establish rules about which operations should be performed first to maintain consistency in results.

Parentheses

The usage of parentheses as a primary tool for grouping dates back centuries. The idea has always been straightforward: perform the calculations inside the brackets first. This was an essential concept as mathematics became more complex.

Renaissance Era

The use of exponents dates back to ancient times, but their formal notation and rules were developed during the Renaissance. Mathematicians of this era, such as René Descartes, made significant contributions to the way we notate and handle powers.

Modern Mnemonics

BODMAS

This mnemonic became popular, especially in British educational circles. The term “Orders” in BODMAS is a British term for exponents, which includes powers and root values.

PEMDAS

Predominantly used in the United States, this mnemonic emphasizes “Parentheses” and “Exponents” rather than “Brackets” and “Orders”.

Transition to Modern Education

As education became more standardized globally in the late 19th and early 20th centuries, it was necessary to teach a consistent method for the order of operations. Both BODMAS and PEMDAS, despite regional differences, aimed to provide clarity in an area that could otherwise lead to widespread mathematical confusion.

Technological Era

The advent of computers and calculators in the 20th century further necessitated a consistent order of operations. Computational devices are programmed with a set order of operations (typically following PEMDAS) to ensure that they return consistent results regardless of where or by whom they are used.