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JAC • Class 6 • Mathematics • Ch 3

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Integers

In JAC Board Class 6 Mathematics, "Integers" extends the number line into negative territory: $\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$. Students explore real-world representations (profit/loss, above/below sea level, temperature), number line ordering, absolute value, and integer addition and subtraction rules with and without number lines.

View All Chapters JCERT Hindi Textbook

Overview Core Concepts Formulas & Identities Solved Examples Common Traps Summary Self Check Take MCQ Test

❄️ Have You Ever Wondered?

If you are on the ground floor (0) and take an elevator down 2 basement levels, which floor are you on?

Negative numbers were historically called "debt numbers" in ancient Indian mathematics! Discover how integers make subtraction universally possible without boundaries.

Why This Chapter Matters

Before You Begin (Prerequisites)

Whole numbers and number line
Basic addition and subtraction

What You Will Learn (Core Objectives)

Represent real-world opposite situations using positive and negative integers.
Locate and compare integers on the number line ($a < b$ if $a$ lies to the left of $b$).
Perform addition of integers using sign rules.
Perform subtraction of integers by adding the additive inverse: $a - b = a + (-b)$.

Chapter Roadmap & Progression

1 Syllabus and book alignment

2 1. Need for Negative Numbers & The...

3 2. Addition and Subtraction Rules f...

Complete Concept Guide (100% Curriculum Coverage)

Syllabus and book alignment

JCERT 2026–27: Regular schools (Non-SOE) — Chapter 3 · Integers. Syllabus pages 42–69. These notes follow this topic. Official syllabus. Official JCERT Hindi textbook. Some source PDFs are damaged. The supplementary reference uses different chapter numbering.

1. Need for Negative Numbers & The Integer Line

Whole numbers cannot represent quantities below zero (e.g. sub-zero Himalayan winter temperatures of $-15^\circ\text{C}$, deep sea depths, or financial bank debts). The set of Integers is:$$\mathbb{Z} = \{\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots\}$$

Positive integers: $1, 2, 3, \dots$ (to the right of 0).
Negative integers: $-1, -2, -3, \dots$ (to the left of 0).
$0$ is an integer that is neither positive nor negative.
Ordering Rule: Every positive integer is greater than zero and greater than every negative integer. For any two numbers on the number line, the number to the right is always greater: $-1 > -100$!

2. Addition and Subtraction Rules for Integers

Addition Rules

Same Signs: Add the absolute values and keep the common sign:
$(+5) + (+3) = +8$
$(-5) + (-3) = -(5 + 3) = -8$
Different Signs: Subtract the smaller numerical value from the larger and attach the sign of the larger value:
$(-8) + (+5) = -(8 - 5) = -3$
$(+8) + (-5) = +(8 - 5) = +3$

Subtraction via Additive Inverse

Golden Subtraction Rule: To subtract an integer $b$ from $a$, add the additive inverse (opposite) of $b$:$$a - b = a + (-b)$$

Examples: $7 - (-3) = 7 + (+3) = 10$; $(-8) - (-5) = -8 + (+5) = -3$.

Key Formulas, Identities & Theorems

Additive Inverse Law

a + (-a) = 0

-a is the additive inverse of a.

Integer Subtraction Rule

$$a - b = a + (-b), \quad a - (-b) = a + b$$

Subtracting a negative is adding a positive.

Conceptual Solved Examples & Case Studies

Example 1

Find the value of: (-13) + 32 - 8 - 1.

Step-by-Step Solution:

Group positive and negative numbers:
Positive: $32$
Negative: $(-13) + (-8) + (-1) = -(13 + 8 + 1) = -22$
Combining: $32 + (-22) = 32 - 22 = \mathbf{10}$.

Example 2

Subtract -4 from -10.

Step-by-Step Solution:

$(-10) - (-4) = -10 + 4 = \mathbf{-6}$.

Common Misconceptions & Examiner Traps

Common Misconception

Thinking -10 is greater than -2.

Scientific Reality & Correction

On the number line, -10 is further to the left than -2, so $-10 < -2$! Think of debt: owing $10 is worse than owing $2.

Visual Learning & Conceptual Map

The Integer Number Line

Negative Integers <-- 0 --> Positive Integers

-4 -3 -2 -1 0 +1 +2 +3 +4

Value INCREASES going rightwards | Value DECREASES going leftwards

Chapter Summary & 10 Key Takeaways

Takeaway 1

Integers: $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$.

Takeaway 2

0 is neither positive nor negative.

Takeaway 3

Numbers to the right on the number line are larger.

Takeaway 4

Subtracting an integer is equivalent to adding its additive inverse.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

A submarine was at 350 m below sea level. It ascended 120 m. What is its new depth?

Reveal Answer & Explanation

Answer: Initial depth $= -350\text{ m}$. Ascending means adding $+120\text{ m}$. New depth $= -350 + 120 = \mathbf{-230\text{ m}}$ (230 m below sea level).
Ascending adds positive height.

Chapter practice

CBT tests for this chapter are not published yet. Solve the self-check questions above and compare your answers.

All Class 6 Mathematics Chapters

Ch 1: Knowing Our Numbers Ch 2: Whole Numbers Ch 3: Integers Ch 4: Playing with Numbers Ch 5: Fractions Ch 6: Decimals Ch 7: Ratio and Proportion Ch 8: Algebra Ch 9: Line, Line Segment and Ray Ch 10: Pairs of Lines Ch 11: Angles Ch 12: Polygons Ch 13: Circle Ch 14: Three-Dimensional Shapes Ch 15: Symmetry Ch 16: Practical Geometry Ch 17: Perimeter and Area Ch 18: Data Handling

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