Understanding Numbers

1. Base Ten Numeration

Base ten numeration, also known as the decimal system, is a positional numeral system that uses ten symbols (0-9) to represent numbers. Each position in a number represents a power of ten.

2. Place Value of a Digit in a Number

Each digit in a number has a place value depending on its position. 

Example:

In the number 5827:

• The digit 5 is in the thousands place: 5 × 1000 = 5000,
• The digit 8 is in the hundreds place, 8 × 100 = 800,
• The digit 2 is in the tens place, 2 × 10 = 20,
• The digit 7 is in the ones place, 7 × 1 = 7.

3. Total Value

The total value of a number is the sum of the values of its digits based on their place values.

Example:

For 5827, the total value is:  5000 + 800 + 20 + 7 = 5827.

4. Writing Numbers in Words and Numerals

Numbers can be expressed in two forms: 

Example: The numeral 245 can be written in words as "two hundred forty-five."

5. Natural and Whole Numbers

Natural Numbers: The positive integers starting from 1 (i.e., 1, 2, 3, ...).

Whole Numbers: The set of natural numbers including zero (i.e., 0, 1, 2, 3, ...).

6. Even, Odd, and Prime Numbers

Even Numbers: Numbers that are divisible by 2 (e.g., 0, 2, 4, 6, ...).

Odd Numbers: Numbers that are not divisible by 2 (e.g., 1, 3, 5, ...).

Prime Numbers: Numbers greater than 1 that have no divisors other than 1 and themselves (e.g., 2, 3, 5, 7, 11).

Examples:

- Even: 4, 16

- Odd: 3, 27

- Prime: 11, 13

7. Operations on Whole Numbers

The main operations on whole numbers are addition, subtraction, multiplication, and division.

Examples:

- Addition: 7 + 3 = 10

- Subtraction: 10 - 4 = 6

- Multiplication: 5 × 6 = 30

- Division: 20 ÷ 4 = 5

8. BODMAS

BODMAS stands for Brackets, Orders (i.e., powers and square roots, etc.), Division and Multiplication (from left to right), Addition and Subtraction (from left to right). It dictates the order of operations in calculations.

Example:

For the expression 3 + 6 x (5 + 4) ÷ 3 - 7:

  1. Brackets: 5 + 4 = 9

  2. Then it becomes 3 + 6 x 9 ÷ 3 - 7

  3. Multiplication/Division: 6 x 9 = 54 and 54 ÷ 3 = 18

  4. Now, 3 + 18 - 7 = 14.

9. Factors and Multiples of Numbers

    Factors: Numbers that divide another number exactly with no remainder.

     Example:

     Factors of 12: 1, 2, 3, 4, 6, 12.

  

     Multiples: The product of a number and an integer.

     Example:

     Multiples of 5: 5, 10, 15, 20, ...

10. Operations on Integers

Integers include whole numbers and their negative counterparts. Operations with integers must consider the sign.

Examples:

Addition: 5 + (-3) = 2

Subtraction: 7 - 10 = -3

Multiplication: (-4) x 3 = -12

Division: (-8) ÷ 4 = -2