## Integers ### Definition Integers are all whole numbers including positive numbers, negative numbers, and zero. They are represented by the symbol $\mathbb{Z}$. ### Identify integers Consider a number line below

Positive integers lie to the right of zero and increase in value as we move right. Negative integers lie to the left of zero and decrease in value as we move left. To compare integers on a number line: - The number to the right is greater. - The number to the left is smaller. - $a > b$ means a is greater than b. - $a < b$ means a is less than b. - $a \ne b$ means a is not equal to b. --- **Example 1:** Represent integers $\mathbb{Z}$ such that: $0 > \mathbb{Z} > -4$ **Step:** We are looking for integers less than 0 but greater than -4. These integers are: $\mathbb{Z} = \{-3, -2, -1\}$ Number line: ``` ← −5 −4 −3 −2 −1 0 1 2 → ``` --- **Example 2:** Represent integers $\mathbb{Z}$ such that: $-2 < \mathbb{Z} \leq 1$ **Step:** We are looking for integers greater than -2 and less than or equal to 1. These integers are: $\mathbb{Z} = \{-1, 0, 1\}$ Number line: ``` ← −3 −2 −1 0 1 2 3 → ```

--- **Example:** Put the correct sign to make a true statement: $3 \, \_ \, 5 \quad \Rightarrow \quad 3 < 5$ (because 3 is less than 5) $-4 \, \_ \, -2 \quad \Rightarrow \quad -4 < -2$ (because -4 is further left on the number line) $7 \, \_ \, 7 \quad \Rightarrow \quad 7 = 7$ (because they are equal) --- ## Addition of Integers **Rule 1:** To add a negative integer, you move to the left on the number line. **Example:** $-2 + (-5)$ Start at -2 and move 5 steps to the left: $-2 + (-5) = -7$ Number line movement: ``` ← −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 → ↑ ↑ Result Starting Point ``` **Alternative:** Start at 0, move 2 steps left to reach -2, then 5 more steps left: ``` ← −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 → ↑ ↑ Result After -2 from 0 ``` **Therefore,** $-2 + (-5) = -7$ **Rule 1:** To add a negative integer you move to the left on the number line. **Example:** $$ -2 + (-5) = -7 $$ Start at -2, move 5 steps to the left or Start at 0, move 2 steps to the left, then 5 more steps to the left.

**Example:** Evaluate: $-4 + (-6)$ **Step:** Start at -4 on the number line, then move 6 steps to the left. This gives: $$ -4 + (-6) = -10 $$ **Number line illustration:**

When drawing integers on a number line, the arrows for the positive numbers goes to the right while the arrows for the negative numbers goes to the left. ## Subtraction of Integers

## Multiplication of Integers

**Multiplication of Integers** **Example:** $2 \times 6 = 2 + 2 + 2 + 2 + 2 + 2 = 12$ This can be shown on a number line by taking 2 and adding it six times to get 12. **Note:** Multiplication of a negative integer by a negative integer cannot be shown on a number line, but the result is a positive integer. **Multiplication:** $$ (+) \times (+) = (+) \\ (-) \times (-) = (+) \\ (+) \times (-) = (-) \\ (-) \times (+) = (-) $$ ### Division of Integers **Example:** $6 \div 3 = 2$ Since $2 \times 3 = 6$, division is the opposite of multiplication. If two integers have the same sign (both positive or both negative), the quotient is positive. If the signs are different, the quotient is negative. **Division:** $$ (+) \div (+) = (+) \\ (-) \div (-) = (+) \\ (+) \div (-) = (-) \\ (-) \div (+) = (-) $$ ### Mixed Operations on Integers When solving problems with multiple operations, follow the BODMAS rule: **B**rackets → **O**rders → **D**ivision → **M**ultiplication → **A**ddition → **S**ubtraction **Example:** $$ 9 \div 3 + 3 \times 2 - 1 $$ Step 1: Do division and multiplication first: $$ = 3 + 6 - 1 $$ Step 2: Add and subtract: $$ = 8 $$ **Example:** $$ (12 \div 4 - 2) + 4 - 7 $$ Step 1: Inside the brackets — divide then subtract: $$ = (3 - 2) + 4 - 7 $$ Step 2: Simplify inside brackets: $$ = 1 + 4 - 7 $$ Step 3: Add and subtract: $$ = 5 - 7 = -2 $$