Basic Arithmetic - BODMAS / PEDMAS Rule
Arithmetic is the foundation of math, focusing on how we manipulate numbers. Here are the four core operations:
- Addition (+): Combining two or more amounts to get a total (the sum).
- Example: 5 + 3 = 8
- Subtraction (−): Finding the difference between two numbers or taking an amount away.
- Example: 10 − 4 = 6
- Multiplication (×): Repeated addition. It’s like adding the same number to itself a specific number of times.
- Example: 4 × 3 is the same as 4 + 4 + 4 = 12
- Division (÷): Splitting a number into equal parts or seeing how many times one number fits into another.
- Example: 15 ÷ 3 = 5
Key Concepts to Remember:
- Order of Operations (PEMDAS/BODMAS): When a problem has multiple steps, you follow a specific order: Parentheses, Exponents, Multiplication/Division (left to right), and Addition/Subtraction (left to right).
- Number Types: Arithmetic usually starts with integers (whole numbers), but it also applies to fractions, decimals, and percentages.
- Properties:
- Commutative: For addition and multiplication, the order doesn't matter (5 + 2 is the same as 2 + 5).
- Identity: Adding 0 or multiplying by 1 doesn't change the original number.
Arithmetic is the foundation of mathematics, dealing with basic operations like addition, subtraction, multiplication, and division. It also includes concepts like prime numbers, numeral systems, and progressions, forming the building blocks for all higher-level math.
🔢 Core Operations in Arithmetic
- Addition (+) → Combining numbers (e.g., (3 + 5 = 8))
- Subtraction (−) → Finding the difference (e.g., (9 − 4 = 5))
- Multiplication (×) → Repeated addition (e.g., (6 × 3 = 18))
- Division (÷) → Splitting into equal parts (e.g., (20 ÷ 4 = 5))
These four are the elementary arithmetic operations taught first in schools.
🔢 Numeral Systems
- Decimal (base-10) → Uses digits 0–9 (most common system worldwide).
- Other systems → Examples include binary (base-2, used in computers) and vigesimal (base-20, used in some indigenous languages).
🔢 Fundamental Theorem of Arithmetic
- Every integer greater than 1 is either a prime number or can be expressed as a unique product of prime numbers.
- Example:
- (12 = 2 × 2 × 3)
- (17) is prime.
This theorem is crucial for number theory and factorization.
🔢 Arithmetic Progressions (AP)
- A sequence of numbers where the difference between consecutive terms is constant.
- Examples:
- Natural numbers: (1, 2, 3, 4, …) (difference = 1)
- Even numbers: (2, 4, 6, 8, …) (difference = 2)
- Multiples of 3: (3, 6, 9, 12, …) (difference = 3)
📊 Quick Comparison Table
| Concept | Definition | Example |
|---|---|---|
| Addition | Combine values | (7 + 2 = 9) |
| Subtraction | Find difference | (10 − 6 = 4) |
| Multiplication | Repeated addition | (4 × 5 = 20) |
| Division | Equal sharing | (15 ÷ 3 = 5) |
| Prime Numbers | Divisible only by 1 & itself | 2, 3, 5, 7 |
| Arithmetic Progression | Sequence with constant difference | (5, 10, 15, 20, …) |
✅ Key Takeaway
Arithmetic is the gateway to mathematics, teaching how numbers interact through operations, patterns, and structures. Mastery of these basics—especially the BODMAS rule, prime factorization, and progressions—sets the stage for algebra, geometry, and advanced math.
Would you like me to prepare a step-by-step practice set of arithmetic problems (with solutions) so you can strengthen your basics interactively?
Here’s your diagrammatic infographic of the BODMAS Rule in mathematics — a visual breakdown of the order of operations used in arithmetic.
✅ It follows this hierarchy:
- B → Brackets (solve inside first)
- O → Orders (powers, roots)
- D/M → Division & Multiplication (left to right)
- A/S → Addition & Subtraction (left to right)
This layout helps you remember the correct sequence when solving expressions.
https://copilot.microsoft.com/th/id/BCO.2fc98992-ab5e-411c-897a-dfda39b41509.png
The BODMAS rule is the standard order of operations in arithmetic: Brackets → Orders (powers/roots) → Division → Multiplication → Addition → Subtraction. Following this sequence ensures accurate results when solving expressions.
🔢 What BODMAS Stands For
- B → Brackets (solve inside parentheses first)
- O → Orders (exponents, powers, square roots, etc.)
- D → Division (from left to right)
- M → Multiplication (from left to right)
- A → Addition (from left to right)
- S → Subtraction (from left to right)
📘 Step-by-Step Examples
Example 1:
Expression: (7 + (6 \times 5^2 + 3))
- Brackets: (6 \times 5^2 + 3)
- Orders: (5^2 = 25) → (6 \times 25 + 3 = 150 + 3 = 153)
- Add: (7 + 153 = 160)
Answer: 160
Example 2:
Expression: (30 \div 5 \times 3)
- Division first: (30 \div 5 = 6)
- Multiplication: (6 \times 3 = 18)
Answer: 18
Example 3:
Expression: (2 + 5 \times 3)
- Multiplication first: (5 \times 3 = 15)
- Addition: (2 + 15 = 17)
Answer: 17
⚠️ Common Mistakes
- Doing operations in the wrong order leads to errors.
- Example: (4 \times (5+3)) should be (4 \times 8 = 32), not ((4 \times 5) + 3 = 23).
- Forgetting that division and multiplication are done left to right, not one before the other. Same applies to addition and subtraction.
📊 Quick Reference Table
| Step | Operation | Example | Correct Result |
|---|---|---|---|
| Brackets | Inside parentheses | (4 \times (5+3)) | 32 |
| Orders | Powers/roots | (5 \times 2^2) | 20 |
| Division | Left to right | (30 \div 5 \times 3) | 18 |
| Multiplication | Left to right | (2 + 5 \times 3) | 17 |
| Addition | Left to right | (10 + 4 - 2) | 12 |
| Subtraction | Left to right | (20 - 5 + 3) | 18 |
✅ Key Takeaway
Always apply BODMAS strictly to avoid confusion. Think of it as a hierarchy: brackets first, then powers, then division/multiplication, and finally addition/subtraction. This rule is universal in mathematics and ensures consistency in solving problems.
Would you like me to create a practice quiz with BODMAS-based problems so you can test your understanding interactively?
https://copilot.microsoft.com/shares/BsomX7QCiM6wfYoaBUvk4
Here’s the diagrammatic infographic of the PEMDAS Rule in mathematics you requested — it visually lays out the hierarchy of operations step by step.
✅ The flowchart shows:
- P → Parentheses (solve inside first)
- E → Exponents (powers, roots)
- M/D → Multiplication & Division (left to right)
- A/S → Addition & Subtraction (left to right)
This makes it easy to remember the correct order when solving expressions.
https://copilot.microsoft.com/th/id/BCO.98de9658-6fe3-4815-b717-3bab5bf54961.png
The PEMDAS rule is the American version of the order of operations in arithmetic. It’s essentially the same as BODMAS, but the acronym is slightly different:
🔢 What PEMDAS Stands For
- P → Parentheses (solve inside brackets first)
- E → Exponents (powers, roots, etc.)
- M → Multiplication (left to right)
- D → Division (left to right)
- A → Addition (left to right)
- S → Subtraction (left to right)
👉 Notice that multiplication/division are handled together (left to right), and addition/subtraction are also handled together (left to right).
📘 Step-by-Step Examples
Example 1
Expression: (8 + (3 \times 2^2))
- Parentheses: (3 \times 2^2)
- Exponents: (2^2 = 4) → (3 \times 4 = 12)
- Addition: (8 + 12 = 20)
Answer: 20
Example 2
Expression: (40 \div 5 \times 2)
- Division first: (40 \div 5 = 8)
- Multiplication: (8 \times 2 = 16)
Answer: 16
Example 3
Expression: (10 - 2 + 4)
- Subtraction first (left to right): (10 - 2 = 8)
- Addition: (8 + 4 = 12)
Answer: 12
⚠️ Common Pitfalls
- Thinking multiplication always comes before division → Wrong! They are equal priority, solved left to right.
- Same with addition and subtraction → Always left to right.
- Forgetting to handle exponents before multiplication/division.
📊 Quick Comparison: PEMDAS vs BODMAS
| Rule | Acronym | Key Terms |
|---|---|---|
| PEMDAS | Parentheses, Exponents, Multiplication, Division, Addition, Subtraction | Used mainly in the US |
| BODMAS | Brackets, Orders, Division, Multiplication, Addition, Subtraction | Used in UK, India, and elsewhere |
They are the same rule, just different wording.
✅ Key Takeaway
PEMDAS ensures consistency in solving math expressions. Always follow the hierarchy: Parentheses → Exponents → Multiplication/Division → Addition/Subtraction.
Would you like me to create a visual flowchart of PEMDAS steps so you can see the hierarchy clearly?
https://copilot.microsoft.com/shares/y5fghDRxsHAHLvnQne2C6
🔢 Core Properties of Arithmetic
1. Commutative Property
- Applies to: Addition and Multiplication
- Definition: Changing the order of numbers does not change the result.
- Examples:
- Addition: (a + b = b + a) → (3 + 5 = 5 + 3 = 8)
- Multiplication: (a \times b = b \times a) → (4 \times 2 = 2 \times 4 = 8)
- Does NOT apply to: Subtraction or Division
2. Associative Property
- Applies to: Addition and Multiplication
- Definition: Grouping of numbers does not affect the result.
- Examples:
- Addition: ((a + b) + c = a + (b + c)) → ((2 + 3) + 4 = 2 + (3 + 4) = 9)
- Multiplication: ((a \times b) \times c = a \times (b \times c))
- Does NOT apply to: Subtraction or Division
3. Distributive Property
- Applies to: Multiplication over Addition/Subtraction
- Definition: Multiplying a number by a group of numbers added together is the same as doing each multiplication separately.
- Formula: (a \times (b + c) = a \times b + a \times c)
- Example: (2 \times (3 + 4) = 2 \times 3 + 2 \times 4 = 6 + 8 = 14)
4. Identity Property
- Addition Identity: (a + 0 = a)
- Multiplication Identity: (a \times 1 = a)
- These identities help simplify expressions and maintain values.
5. Zero Property of Multiplication
- Definition: Any number multiplied by zero equals zero.
- Example: (7 \times 0 = 0)
📊 Summary Table
| Property | Applies To | Example | Result |
|---|---|---|---|
| Commutative | Addition, Multiplication | (5 + 2 = 2 + 5) | 7 |
| Associative | Addition, Multiplication | ((2 + 3) + 4 = 2 + (3 + 4)) | 9 |
| Distributive | Multiplication over Addition | (2 \times (3 + 4)) | 14 |
| Identity | Addition, Multiplication | (a + 0 = a), (a \times 1 = a) | — |
| Zero Property | Multiplication | (9 \times 0 = 0) | 0 |
🧠 Why These Properties Matter
- They simplify complex expressions, especially in algebra.
- They ensure accuracy and consistency in calculations.
- They are used in mental math, programming, and problem-solving.
https://copilot.microsoft.com/shares/EVE6Aq2vVFCebpQok8Z4Y
Here’s a diagrammatic infographic of the key properties of basic arithmetic — a visual guide to the rules that govern addition, subtraction, multiplication, and division.
✅ It highlights:
- Commutative Property → Order doesn’t matter (for + and ×)
- Associative Property → Grouping doesn’t matter (for + and ×)
- Distributive Property → Multiplication distributes over addition/subtraction
- Identity Property → Adding 0 or multiplying by 1 keeps the number unchanged
- Zero Property → Any number × 0 = 0
- Inverse Property → Adding opposites gives 0, multiplying reciprocals gives 1
This infographic makes it easy to remember and apply these properties in problem-solving.
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