Number Operations and Concepts

TL;DR

This note covers basic number operations like addition, subtraction, multiplication, and division, along with key number concepts such as factors, multiples, prime numbers, and order of operations. Mastering these fundamentals is crucial for all math problems. You'll learn how to apply these concepts systematically.

1. The Mental Model

Think of numbers as building blocks and operations as tools to combine or break them apart. Understanding number concepts helps you classify and relate these blocks, making complex problems easier to tackle by breaking them into smaller, manageable steps.

2. The Core Material

2.1 Basic Operations

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These are the four fundamental ways to combine or separate numbers:

• Addition (+): Combining two or more numbers. (e.g., 5 + 3 = 8)
• Subtraction (-): Finding the difference between two numbers. (e.g., 8 - 3 = 5)
• Multiplication (* or x): Repeated addition. (e.g., 5 * 3 = 15 means 5 + 5 + 5)
• Division (/ or ÷): Splitting a number into equal parts. (e.g., 15 / 3 = 5)

2.2 Number Concepts

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• Factors: Numbers that divide exactly into another number without leaving a remainder.
  • Example: Factors of 12 are 1, 2, 3, 4, 6, 12.
• Multiples: Numbers you get when you multiply a given number by an integer.
  • Example: Multiples of 3 are 3, 6, 9, 12, 15...
• Prime Numbers: A whole number greater than 1 that has only two factors: 1 and itself.
  • Example: 2, 3, 5, 7, 11, 13... (2 is the only even prime number).
• Composite Numbers: A whole number greater than 1 that has more than two factors.
  • Example: 4 (factors: 1, 2, 4), 6 (factors: 1, 2, 3, 6).

2.3 Order of Operations (BIDMAS/PEMDAS)

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When an expression has multiple operations, you must follow a specific order to get the correct answer.

graph TD
    B["Brackets (Parentheses)"] --> I["Indices (Exponents)"]
    I --> D["Division"]
    D --> M["Multiplication"]
    M --> A["Addition"]
    A --> S["Subtraction"]

    subgraph Operations Handled Left-to-Right if at Same Level
        D -- "then" --> M
        A -- "then" --> S
    end

Remember: Division and Multiplication are at the same level of priority, as are Addition and Subtraction. When you have operations at the same level (e.g., multiplication and division), you work from left to right.

3. Worked Example

Let's evaluate: 25 - (3 * 4) + 10 / 2

1. Brackets first: (3 * 4) = 12
   Expression becomes: 25 - 12 + 10 / 2
2. Division next: 10 / 2 = 5
   Expression becomes: 25 - 12 + 5
3. Now, Addition and Subtraction from left to right:
   25 - 12 = 13
   Expression becomes: 13 + 5
4. Finally, Addition: 13 + 5 = 18

So, 25 - (3 * 4) + 10 / 2 = 18.

4. Key Takeaways

• Understand the definitions of factors, multiples, and prime numbers to categorize numbers effectively.
• Always apply the BIDMAS/PEMDAS rule to solve expressions with multiple operations consistently.
• Remember that multiplication is repeated addition, and division is sharing into equal groups.
• Prime numbers are fundamental building blocks for all other whole numbers through multiplication.
• Practice identifying factors by thinking about which numbers can divide evenly into another.

Common Mistakes to Avoid:
* Ignoring the order of operations, especially when brackets are present.
* Confusing factors with multiples (e.g., thinking 6 is a factor of 3).
* Forgetting that 1 is neither prime nor composite.
* Thinking all odd numbers are prime (e.g., 9 is odd but not prime because 3 * 3 = 9).

5. Now Try It

Evaluate the expression: 4 * (12 - 3) / 6 + 7 and list all the factors of 18.

What success looks like: You should arrive at a single correct numerical answer for the expression and correctly list all numbers that divide evenly into 18.