Number - Operations with Integers (Addition, Subtraction, Multiplication, Division)

Integers on a Number Line: Integers consist of zero, positive whole numbers, and negative whole numbers. Visually, imagine a horizontal line where $0$ is the center; positive integers like $1, 2, 3$ extend infinitely to the right, and negative integers like $-1, -2, -3$ extend infinitely to the left.

Adding Integers with the Same Sign: When adding two integers that have the same sign, add their absolute values and keep the common sign. On a number line, this is like starting at a point and moving further in the same direction (e.g., $+2 + (+3)$ means starting at $2$ and moving $3$ units further right to $5$).

Adding Integers with Different Signs: When adding integers with different signs, find the difference between their absolute values and use the sign of the number with the larger absolute value. Visually, this is like moving right for a positive number and then moving left for a negative number (e.g., $-5 + 3$ means starting at $-5$ and moving $3$ units right to end at $-2$).

Subtraction as Adding the Inverse: To subtract an integer, add its opposite (additive inverse). The expression $a - b$ is rewritten as $a + (-b)$. On a number line, subtracting a negative number like $4 - (-2)$ is visualised as a 'double negative' that results in moving to the right, effectively becoming $4 + 2 = 6$.

Multiplication and Division Signs: When multiplying or dividing two integers, if the signs are the same (both positive or both negative), the result is positive. If the signs are different (one positive and one negative), the result is negative. This can be remembered using a sign triangle or a grid where $(-1) \\times (-1) = 1$ and $(-1) \\times 1 = -1$.

The Role of Zero: Adding or subtracting zero does not change an integer's value. Multiplying any integer by zero results in zero ($a \\times 0 = 0$). Zero divided by any non-zero integer is zero ($\frac{0}{a} = 0$), but dividing any integer by zero is undefined.

Order of Operations (BODMAS/BIDMAS): When a calculation involves multiple operations, they must be performed in the order: Brackets, Orders (Indices), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). This ensures consistent results for complex expressions.