Number - Operations with Integers

Integers are a set of whole numbers that include positive numbers, negative numbers, and zero: $\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$. Visually, they are represented on a horizontal number line where zero is the origin, positive integers extend infinitely to the right, and negative integers extend infinitely to the left.

The Absolute Value of an integer, denoted as $|x|$, is its distance from zero on a number line. Because it represents distance, it is always non-negative. For example, $|-5| = 5$ and $|5| = 5$, meaning both numbers are exactly $5$ units away from zero.

Adding Integers with the Same Signs: To add integers with the same sign, add their absolute values and keep the common sign. Visually, adding two negative numbers like $-2 + (-3)$ is like starting at $-2$ on the number line and moving $3$ units further to the left to land on $-5$.

Adding Integers with Different Signs: Subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value. Visually, this can be seen as 'Zero Pairs'; if you have $4$ positive counters and $6$ negative counters, $4$ pairs cancel out to zero, leaving $2$ negative counters, so $4 + (-6) = -2$.

Subtracting Integers: To subtract an integer, add its additive inverse (its opposite). The rule is $a - b = a + (-b)$. Visually, subtracting a negative number like $2 - (-3)$ is the same as $2 + 3$, moving $3$ units to the right on the number line to reach $5$.

Multiplying and Dividing Integers with Same Signs: When multiplying or dividing two integers with the same sign (both positive or both negative), the result is always positive. For example, $(-4) \times (-5) = 20$ and $(-10) \div (-2) = 5$.

Multiplying and Dividing Integers with Different Signs: When multiplying or dividing two integers with different signs (one positive and one negative), the result is always negative. For example, $8 \times (-3) = -24$ and $(-15) \div 3 = -5$.

Order of Operations (BIDMAS/BODMAS): When an expression contains multiple operations, follow the order: Brackets, Indices, Division and Multiplication (from left to right), and then Addition and Subtraction (from left to right). This ensures consistent results when working with complex integer expressions.