Integers
Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.
First Principles
Objective: Add, subtract, multiply, and divide positive and negative integers using number line models and algebraic sign rules.
Concrete Scenario: Picture Mama Akinyi writing entries in her duka ledger in Kisumu. A profit of KES 500 is written as \(+500\), while taking goods on credit or suffering a loss of KES 300 is recorded as \(-300\). Integers are simply these signed whole numbers: \(\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}\). They extend infinitely in both directions from zero.
Geometric Insight: Place integers on a continuous number line. Zero is the origin. Positive integers lie to the right; negative integers lie to the left. Adding a positive integer moves you to the right. Adding a negative integer (or subtracting a positive) moves you to the left. Subtracting a negative integer reverses direction, moving you to the right.
Algebraic Sign Rules: Multiplication is repeated addition or scaling. When signs are identical, the product or quotient is positive (\((-) \times (-) = (+)\) and \((+) \times (+) = (+)\)). When signs differ, the product or quotient is negative (\((-) \times (+) = (-)\)).
Key Formulas
1. Addition & Subtraction Equivalences
\[ a + (-b) = a - b \]\[ a - (-b) = a + b \]Adding a negative reduces value (moves left). Subtracting a negative adds value (moves right).
2. Multiplication & Division Sign Laws
\[ (\text{positive}) \times (\text{positive}) = \text{positive} \]\[ (\text{negative}) \times (\text{negative}) = \text{positive} \]\[ (\text{positive}) \times (\text{negative}) = \text{negative} \]\[ (\text{negative}) \div (\text{positive}) = \text{negative} \]\[ (\text{negative}) \div (\text{negative}) = \text{positive} \]3. Absolute Value (Magnitude)
\[ |a| = \begin{cases} a, & \text{if } a \ge 0 \\ -a, & \text{if } a < 0 \end{cases} \]\(|a|\) represents the undirected geometric distance from 0 on the number line; thus, \(|-7| = 7\) and \(|7| = 7\).
Worked Examples
Evaluate \((-3) + 7\).
- Identify signs: We are adding a negative number (\(-3\)) and a positive number (\(7\)).
- Compare magnitudes: \(|7| = 7\) and \(|-3| = 3\). The magnitudes differ, so subtract the smaller from the larger: \[ 7 - 3 = 4 \]
- Determine sign: The number with the larger magnitude is \(7\) (positive), so the result is positive.
Answer: \( 4 \)
Evaluate \((-5) - (-2) + 4\).
- Simplify double negative: Subtracting \(-2\) is equivalent to adding \(+2\): \[ (-5) - (-2) + 4 = (-5) + 2 + 4 \]
- Combine positive terms: \[ 2 + 4 = 6 \]
- Combine remaining terms: \[ (-5) + 6 = 6 - 5 = 1 \]
Answer: \( 1 \)
Calculate: \[ (-6) \times (-3) + (-12) \div 4 - | -8 | \]
- Perform multiplication and division first (BODMAS/PEMDAS):\[ (-6) \times (-3) = +18 \quad \text{(same signs give positive)} \]\[ (-12) \div 4 = -3 \quad \text{(different signs give negative)} \]
- Evaluate absolute value: \[ |-8| = 8 \]
- Substitute and combine: \[ 18 + (-3) - 8 \]\[ = 18 - 3 - 8 \]\[ = 15 - 8 = 7 \]
Answer: \( 7 \)
Common Mistakes
Mistake: Believing that two negatives always make a positive during addition; e.g., writing \((-4) + (-2) = +6\).
Correction: \((-4) + (-2) = -6\).
Why it feels right: Learners confuse the multiplication sign rule with addition. When combining two debts, your total debt increases; you are moving further left on the number line.
Mistake: Subtracting larger numbers incorrectly by swapping them, e.g., writing \(3 - 8 = 5\).
Correction: \(3 - 8 = -5\).
Why it feels right: The brain intuitively subtracts \(8 - 3 = 5\) to avoid negative numbers. Subtraction is not commutative: \(a - b \neq b - a\).
Mistake: Thinking absolute value means "make the sign negative", e.g., \(|-9| = -9\).
Correction: \(|-9| = 9\).
Why it feels right: Learners mistake the absolute value symbol \(|\cdot|\) for brackets or an inversion operator. Absolute value represents physical distance from zero, which can never be negative.
Real World
Signed integers are essential across everyday economic, meteorological, and scientific activities in Africa:
1. M-Pesa Overdraft (Fuliza)
If an account has KES 200 and a merchant is paid KES 700 via Fuliza, the account balance is \(200 - 700 = -500\) KES. Depositing KES 800 leaves \(-500 + 800 = +300\) KES.
2. Elevation & Diving
A scuba diver exploring coral reefs in Mombasa at \(-18\text{ m}\) ascends \(7\text{ m}\). Her new position is \(-18 + 7 = -11\text{ m}\) (11 metres below sea level).
3. Cold Fronts on Mt. Kenya
At Point Lenana on Mt. Kenya, the nighttime temperature is \(-6^\circ\text{C}\). By noon, it warms up by \(11^\circ\text{C}\). The new temperature is \(-6 + 11 = +5^\circ\text{C}\).
Practice