What Are BODMAS, BIDMAS, and PEMDAS? A Simple Explanation

In mathematics, accuracy in calculations is essential. To achieve this, specific rules guide us through the correct order of operations.

These rules are encapsulated in the acronyms BODMAS, BIDMAS, and PEMDAS. Each represents a sequence to follow when performing calculations to ensure consistency and precision.

This article breaks down what these acronyms stand for and how they are used in practice, making the world of math more manageable and less confusing.

What is BODMAS?

BODMAS is an acronym that helps remember the order of operations in mathematical calculations. It stands for:

• Brackets
• Orders (powers and square roots, etc.)
• Division
• Multiplication
• Addition
• Subtraction

These operations must be performed in the order specified to ensure accurate results.

Breaking Down BODMAS

Brackets (B): Solve expressions inside brackets first. This includes parentheses ( ), square brackets [ ], and curly braces { }.

• Example: (5+3)×2=8×2=16(5 + 3) \times 2 = 8 \times 2 = 16(5+3)×2=8×2=16

Orders (O): Next, calculate powers and roots.

• Example: 32+4=9+4=133^2 + 4 = 9 + 4 = 1332+4=9+4=13

Division (D) and Multiplication (M): These operations are performed from left to right, whichever comes first.

• Example: 12÷4×3=3×3=912 \div 4 \times 3 = 3 \times 3 = 912÷4×3=3×3=9

Addition (A) and Subtraction (S): Finally, perform addition and subtraction from left to right.

• Example: 10−2+4=8+4=1210 – 2 + 4 = 8 + 4 = 1210−2+4=8+4=12

Examples

Example 1: Calculate 6+2×56 + 2 \times 56+2×5.

Following BODMAS:

• Multiplication first: 2×5=102 \times 5 = 102×5=10
• Then addition: 6+10=166 + 10 = 166+10=16

Example 2: Calculate 3×(2+4)23 \times (2 + 4)^23×(2+4)2.

Following BODMAS:

• Brackets first: 2+4=62 + 4 = 62+4=6
• Then Orders (exponent): 62=366^2 = 3662=36
• Finally, Multiplication: 3×36=1083 \times 36 = 1083×36=108

Example 3: Calculate (8−3)÷(2+1)(8 – 3) \div (2 + 1)(8−3)÷(2+1).

Following BODMAS:

• Brackets first: 8−3=58 – 3 = 58−3=5 and 2+1=32 + 1 = 32+1=3
• Then Division: 5÷3=1.675 \div 3 = 1.675÷3=1.67

What is BIDMAS?

BIDMAS is another acronym used to remember the order of operations in mathematics, similar to BODMAS. It stands for:

• Brackets
• Indices (powers and roots)
• Division
• Multiplication
• Addition
• Subtraction

The order specified by BIDMAS ensures that calculations are performed correctly and consistently.

Breaking Down BIDMAS

Brackets (B): Solve expressions within brackets first, which can include parentheses ( ), square brackets [ ], and curly braces { }.

• Example: (4+2)×3=6×3=18(4 + 2) \times 3 = 6 \times 3 = 18(4+2)×3=6×3=18

Indices (I): Next, calculate powers and roots.

• Example: 23+1=8+1=92^3 + 1 = 8 + 1 = 923+1=8+1=9

Division (D) and Multiplication (M): These operations are performed from left to right, as they appear in the expression.

• Example: 16÷4×2=4×2=816 \div 4 \times 2 = 4 \times 2 = 816÷4×2=4×2=8

Addition (A) and Subtraction (S): Finally, perform addition and subtraction from left to right.

• Example: 15−5+3=10+3=1315 – 5 + 3 = 10 + 3 = 1315−5+3=10+3=13

Examples

Example 1: Calculate 7+3×227 + 3 \times 2^27+3×22.

Following BIDMAS:

• Indices first: 22=42^2 = 422=4
• Then Multiplication: 3×4=123 \times 4 = 123×4=12
• Finally Addition: 7+12=197 + 12 = 197+12=19

Example 2: Calculate (5+3)÷4×2(5 + 3) \div 4 \times 2(5+3)÷4×2.

Following BIDMAS:

• Brackets first: 5+3=85 + 3 = 85+3=8
• Then Division: 8÷4=28 \div 4 = 28÷4=2
• Finally Multiplication: 2×2=42 \times 2 = 42×2=4

Example 3: Calculate 10−32+610 – 3^2 + 610−32+6.

Following BIDMAS:

• Indices first: 32=93^2 = 932=9
• Then Subtraction: 10−9=110 – 9 = 110−9=1
• Finally Addition: 1+6=71 + 6 = 71+6=7

What is PEMDAS?

PEMDAS is an acronym widely used in the United States to remember the order of operations in mathematics. It stands for:

• Parentheses
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

PEMDAS ensures that calculations are carried out in a consistent and correct sequence.

Breaking Down PEMDAS

Parentheses (P): Solve any operations inside parentheses first. This can also include other types of brackets like square brackets [ ] and curly braces { }.

• Example: (3+5)×2=8×2=16(3 + 5) \times 2 = 8 \times 2 = 16(3+5)×2=8×2=16

Exponents (E): Next, calculate powers and roots.

• Example: 42+1=16+1=174^2 + 1 = 16 + 1 = 1742+1=16+1=17

Multiplication (M) and Division (D): These operations are performed from left to right, as they appear in the expression.

• Example: 20÷5×3=4×3=1220 \div 5 \times 3 = 4 \times 3 = 1220÷5×3=4×3=12

Addition (A) and Subtraction (S): Finally, perform addition and subtraction from left to right.

• Example: 18−6+4=12+4=1618 – 6 + 4 = 12 + 4 = 1618−6+4=12+4=16

Examples

Example 1: Calculate 8+2×328 + 2 \times 3^28+2×32.

Following PEMDAS:

• Exponents first: 32=93^2 = 932=9
• Then Multiplication: 2×9=182 \times 9 = 182×9=18
• Finally Addition: 8+18=268 + 18 = 268+18=26

Example 2: Calculate (7−2)×4+6(7 – 2) \times 4 + 6(7−2)×4+6.

Following PEMDAS:

• Parentheses first: 7−2=57 – 2 = 57−2=5
• Then Multiplication: 5×4=205 \times 4 = 205×4=20
• Finally Addition: 20+6=2620 + 6 = 2620+6=26

Example 3: Calculate 12÷32−112 \div 3^2 – 112÷32−1.

Following PEMDAS:

• Exponents first: 32=93^2 = 932=9
• Then Division: 12÷9=1.3312 \div 9 = 1.3312÷9=1.33
• Finally Subtraction: 1.33−1=0.331.33 – 1 = 0.331.33−1=0.33

Key Differences

        (2 + 3) × 4^2
        = 5 × 16
        = 80

        (3 + 4) × 2^3
        = 7 × 8
        = 56

        (1 + 2) × 3^2
        = 3 × 9
        = 27

Common Mistakes

When using BODMAS, BIDMAS, or PEMDAS, it’s easy to make mistakes that lead to incorrect results. Understanding these common pitfalls can help avoid errors and ensure accurate calculations.

Misinterpreting the Order of Operations

A frequent mistake is misunderstanding the hierarchy of operations. This often happens when learners assume operations should be performed from left to right without following the specific order.

• Example: In the expression 6+2×36 + 2 \times 36+2×3, adding first (6 + 2 = 8) and then multiplying (8 × 3 = 24) is incorrect. The correct approach is to multiply first (2 × 3 = 6), then add (6 + 6 = 12).

Ignoring Parentheses

Another common error is ignoring or misplacing parentheses, which changes the intended order of operations.

• Example: For 8÷(4−2)×38 \div (4 – 2) \times 38÷(4−2)×3, if the parentheses are ignored, you might divide first (8 ÷ 4 = 2) and then subtract (2 – 2 = 0) and finally multiply (0 × 3 = 0). The correct way is to solve the parentheses first (4 – 2 = 2), then divide (8 ÷ 2 = 4), and finally multiply (4 × 3 = 12).

Mixing Up Division and Multiplication

Learners sometimes incorrectly believe that division should always come before multiplication because it appears first in the acronym.

• Example: In 12÷3×212 \div 3 \times 212÷3×2, performing the division first (12 ÷ 3 = 4) and then the multiplication (4 × 2 = 8) is correct. However, doing it the other way (3 × 2 = 6, then 12 ÷ 6 = 2) is incorrect.

Misplacing the Importance of Addition and Subtraction

Similar to division and multiplication, addition and subtraction must be performed from left to right, not based on the order in the acronym.

• Example: For 10−5+210 – 5 + 210−5+2, the correct method is to subtract first (10 – 5 = 5) and then add (5 + 2 = 7). Performing addition first (5 + 2 = 7, then 10 – 7 = 3) is incorrect.

Misunderstanding Exponents

Errors often occur when students misplace the position of exponents in the order of operations.

• Example: In 3×22+13 \times 2^2 + 13×22+1, calculating the exponent first (2^2 = 4), then multiplying (3 × 4 = 12), and finally adding (12 + 1 = 13) is correct. Multiplying first (3 × 2 = 6, then 6^2 = 36, and adding 1 to get 37) is incorrect.

Why the Order of Operations Matters

The order of operations is a fundamental principle in mathematics that ensures calculations are performed correctly and consistently. This principle is crucial for several reasons:

Ensures Consistency

The order of operations provides a standard procedure for solving mathematical expressions. Without it, different people might interpret and solve the same expression differently, leading to multiple answers for the same problem. This consistency is vital in fields that rely heavily on precise calculations, such as engineering, physics, and computer science.

• Example: In the expression 8+2×58 + 2 \times 58+2×5, without a standard order, one might add first (8 + 2 = 10, then 10 × 5 = 50) while another might multiply first (2 × 5 = 10, then 8 + 10 = 18). The correct order of operations ensures everyone calculates it as 8+10=188 + 10 = 188+10=18.

Avoids Ambiguity

Mathematical expressions often involve multiple operations. The order of operations removes ambiguity by specifying which operations to perform first. This clarity is essential for both simple arithmetic and complex algebraic expressions.

• Example: For the expression 3+6÷23 + 6 \div 23+6÷2, the correct interpretation is to divide first (6 ÷ 2 = 3), then add (3 + 3 = 6). Without the order of operations, one might add first (3 + 6 = 9) and then divide (9 ÷ 2 = 4.5), resulting in a different and incorrect answer.

Facilitates Complex Calculations

In advanced mathematics, calculations often involve nested operations and multiple levels of complexity. The order of operations allows mathematicians to break down complex problems into manageable steps, ensuring accuracy and coherence in their work.

• Example: In the expression (5+3)×22(5 + 3) \times 2^2(5+3)×22, following the order of operations, you first solve the parentheses (5 + 3 = 8), then the exponent (2^2 = 4), and finally the multiplication (8 × 4 = 32). This step-by-step approach simplifies solving complex expressions.

Essential for Technology and Programming

Computers and calculators use the order of operations to process mathematical instructions correctly. In programming, algorithms rely on these rules to execute commands in the proper sequence, ensuring that software functions as intended.

• Example: In programming, an expression like a+b×ca + b \times ca+b×c would be interpreted by the computer according to the order of operations, ensuring b×cb \times cb×c is calculated before adding aaa. Misinterpreting this order could lead to software bugs and incorrect outputs.

Conclusion

Understanding BODMAS, BIDMAS, and PEMDAS is essential for ensuring accurate and consistent mathematical calculations. These acronyms provide a clear framework for solving expressions by specifying the correct order of operations: Brackets/Parentheses first, followed by Orders/Exponents, then Division and Multiplication from left to right, and finally Addition and Subtraction from left to right.

By adhering to these rules, one can avoid common mistakes, reduce ambiguity, and achieve precise results in both simple and complex calculations.