Integers and Operations on Integers

Directed numbers on the number line, addition, subtraction, multiplication, and sign rules.

Full Detailed Notes Video Lesson Worked Examples (1)Take Diagnostic Quiz

Curated Video Lesson

Visual explanation and practical step-by-step walk-through

Watch on YouTube

Comprehensive Study Notes

Aligned with Ghana NaCCA & WAEC BECE syllabus standards

Topic Introduction & Real-World Context:

Integers extend our number system into negative territory, allowing us to quantify temperatures below freezing, financial debts, depths below sea level, and deficits in sports. On the number line, negative numbers mirror positive numbers across the central zero. In the BECE exam, questions testing operations on directed numbers (+ and - signs) appear in both objective questions and algebraic equation steps.

What You Will Master in This Lesson (NaCCA Objectives):

Represent positive and negative integers on a horizontal number line.

Compare and arrange integers in ascending and descending order.

Add and subtract directed integers with and without a number line.

Multiply and divide positive and negative integers using sign laws.

Apply the BODMAS rule to multi-step expressions involving integers.

1. The Number Line and Integer Order

The set of Integers (ℤ) is written as {..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...}. • Zero (0) is neutral: it is neither positive nor negative. • As you move to the RIGHT on the number line, numbers INCREASE in value: -1 > -5. • As you move to the LEFT, numbers DECREASE in value. Rule for Negative Numbers: The larger the numeral behind the negative sign, the smaller its actual value! (e.g. -100 is much smaller than -2).

2. Addition and Subtraction of Integers

Rules for Addition: • Adding numbers with the SAME sign: Add their numerical values and keep the common sign. (+4) + (+6) = +10 (-5) + (-3) = -8 • Adding numbers with DIFFERENT signs: Subtract the smaller numerical value from the larger, and take the sign of the larger number. (-9) + (+4) = -5 (9 - 4 = 5; 9 is negative) (+12) + (-7) = +5 Rules for Subtraction: • Subtracting an integer is identical to adding its opposite (additive inverse)! a - (+b) = a - b a - (-b) = a + b (Double negative becomes positive!) Example: 7 - (-5) = 7 + 5 = 12. Example: -4 - (-9) = -4 + 9 = +5.

3. Multiplication and Division of Integers

When multiplying or dividing two integers: • Same signs produce a POSITIVE result: (+) × (+) = (+) (-) × (-) = (+) (+) ÷ (+) = (+) (-) ÷ (-) = (+) • Different signs produce a NEGATIVE result: (+) × (-) = (-) (-) × (+) = (-) (+) ÷ (-) = (-) (-) ÷ (+) = (-) Examples: (-6) × (-4) = +24 (-18) ÷ (+3) = -6 (-5) × (+7) = -35

Common Mistakes Students Make in BECE Examinations:

⚠️Thinking that two negative numbers added together make a positive: (-4) + (-3) = +7 is WRONG! (-4) + (-3) = -8. Only multiplication and division of two negatives make positive!

⚠️Sign errors with double negatives: Writing 8 - (-3) = 5 instead of 8 + 3 = 11.

⚠️Arranging negative numbers backwards: Writing -1, -2, -3 as ascending order instead of descending.

Teacher's BECE Exam Pro-Tips:

⭐Remember the phrase: "Minus of a minus is plus" when removing brackets: -(-x) = +x.

⭐Think of positive as money you HAVE, and negative as money you OWE.

⭐Apply BODMAS strictly when simplifying expressions with multiple integer operations.

Quick Revision Summary Checklist:

Negative numbers to the left of 0; -1 is greater than -10.

Double negative: -(-b) = +b.

Multiplication: (-) × (-) = (+); (-) × (+) = (-).

Addition: (-a) + (-b) = -(a + b).

Step-by-Step Worked Examples (1)

Real BECE exam-standard problems with complete solution steps

Example 1: Subtracting Negative Integers

Problem StatementEvaluate: (i) 8 - (-6) (ii) -15 - (-9)

Step-by-Step Solution:

Step 1: For 8 - (-6), double negative becomes positive: 8 + 6 = 14.

Step 2: For -15 - (-9), rewrite as: -15 + 9.

Step 3: Since signs differ, subtract 9 from 15 (gives 6) and keep the negative sign: -6.

💡

Key Takeaway / Exam Rule: Subtracting a negative number is equivalent to adding its positive counterpart.

Mastery Diagnostic Quiz

Ready to test your knowledge on Integers and Operations on Integers?

Timed computerized test with instant feedback, scoring, and comprehensive question rationales.

Start Diagnostic Quiz

Previous TopicReal Number System and Place Value

Next TopicNumber Bases and Binary System (Base 2)