Operations with Integers

Chapter Overview

Integers help us describe quantities on both sides of zero, such as temperature, profit and loss, elevation and game scores.

1. Addition and Subtraction

When adding on a number line:

• add a positive integer → move right;
• add a negative integer → move left.

For example:

Subtraction means adding the opposite.

2. Multiplication of Integers

Same signs → positive
Different signs → negative

3. Division of Integers

Division follows the same sign rule.

4. Useful Properties

5. Subtraction as Adding the Opposite

Subtraction of integers becomes easier when it is rewritten as addition of the additive inverse.

For example:

and:

6. Why Multiplication Sign Rules Work

Repeated addition explains part of integer multiplication. For example:

The full sign pattern is:

The same sign pattern applies to division, provided the divisor is not zero.

7. Zero in Integer Operations

Zero behaves in important ways:

But division by zero is not defined.

8. Evaluating Integer Expressions

When several operations appear, deal with brackets and multiplication/division before addition/subtraction.

9. Real-life Interpretation

Integers are useful whenever a reference point separates two directions or states:

• temperatures above and below $0^\circ$C;
• floors above and below ground level;
• profit and loss;
• credits and debits;
• heights above and below sea level;
• game points gained and lost.

10. Complete Integer Notes

Integers on the Number Line

Positive integers lie to the right of zero and negative integers lie to the left. Moving right increases value; moving left decreases value.

Therefore:

because $-2$ lies to the right of $-5$.

Adding Integers

When signs are the same, add the magnitudes and keep the common sign:

When signs differ, subtract the smaller magnitude from the larger magnitude and keep the sign of the number with the larger magnitude:

Additive Inverses

Two numbers whose sum is zero are additive inverses:

The additive inverse of $-13$ is 13.

Subtracting Integers

Rewrite subtraction as addition of the opposite:

Multiplying Several Integers

For a product of several non-zero integers:

• an even number of negative factors gives a positive product;
• an odd number of negative factors gives a negative product.

Example:

There are three negative factors, so the product is negative.

Division and Sign

Division follows the same sign pattern as multiplication:

The divisor can never be zero.

Properties

Addition and multiplication of integers are commutative and associative:

Multiplication distributes over addition:

Subtraction and division are not commutative.

Word-problem Signs

Choose zero as a reference point. Examples:

• ₹200 profit → $+200$
• ₹75 loss → $-75$
• 6 m above sea level → $+6$
• 12 m below sea level → $-12$

A final answer should include its meaning, not only the signed number.

11. More Worked Examples

Integer Properties to Remember

For integers $a,b,c$:

But subtraction and division cannot generally be regrouped or reordered in the same way.

For example:

and:

Comparing Negative Values

Think of a temperature scale. $-2^\circ$C is warmer than $-7^\circ$C, so:

Similarly, a debt of ₹20 is better than a debt of ₹100, so $-20>-100$.

Integer Word Problems

A diver is 12 m below sea level and rises 7 m:

The diver is now 5 m below sea level.

A team loses 4 points in each of 3 rounds:

The total change is $-12$ points.

Common Mistakes

Quick Revision

Practice Questions

Additional Practice

Chapter Summary