Complete Guide to Integers

Understanding Integers and Their Operations

What are Integers?

Integers are whole numbers that include positive numbers, negative numbers, and zero. They can be represented on a number line extending infinitely in both directions.

The set of integers (Z) includes:

{..., -3, -2, -1, 0, 1, 2, 3, ...}

Properties of Integers

Closure Property

The sum, difference, and product of any two integers is always an integer.

Example: 5 + (-3) = 2, which is an integer.

Commutative Property

Changing the order of integers in addition or multiplication doesn't change the result.

Examples: a + b = b + a and a × b = b × a

Associative Property

Changing the grouping of integers in addition or multiplication doesn't change the result.

Examples: (a + b) + c = a + (b + c)

Distributive Property

Multiplication distributes over addition.

Example: a × (b + c) = a × b + a × c

Operations with Integers

Addition of Integers

Rules:

When adding integers with the same sign, add their absolute values and keep the sign.
When adding integers with different signs, subtract the smaller absolute value from the larger absolute value, and use the sign of the number with the larger absolute value.

Example 1: Adding positive integers

5 + 3 = 8

Example 2: Adding negative integers

(-5) + (-3) = -8

Example 3: Mixed signs, larger positive

8 + (-3) = 5

Example 4: Mixed signs, larger negative

(-8) + 3 = -5

Subtraction of Integers

Rules:

To subtract an integer, add its additive inverse (opposite).
a - b = a + (-b)

Example 1: Subtracting positive from positive

8 - 3 = 8 + (-3) = 5

Example 2: Subtracting negative from positive

8 - (-3) = 8 + 3 = 11

Example 3: Subtracting positive from negative

(-8) - 3 = (-8) + (-3) = -11

Example 4: Subtracting negative from negative

(-8) - (-3) = (-8) + 3 = -5

Multiplication of Integers

Rules:

When multiplying integers with the same sign, the result is positive.
When multiplying integers with different signs, the result is negative.

Example 1: Positive × Positive

5 × 3 = 15

Example 2: Negative × Negative

(-5) × (-3) = 15

Example 3: Positive × Negative

5 × (-3) = -15

Example 4: Negative × Positive

(-5) × 3 = -15

Division of Integers

Rules:

When dividing integers with the same sign, the result is positive.
When dividing integers with different signs, the result is negative.
Division by zero is undefined.

Example 1: Positive ÷ Positive

15 ÷ 3 = 5

Example 2: Negative ÷ Negative

(-15) ÷ (-3) = 5

Example 3: Positive ÷ Negative

15 ÷ (-3) = -5

Example 4: Negative ÷ Positive

(-15) ÷ 3 = -5

Real-World Applications of Integers

Temperature

Temperature can be above zero (positive) or below zero (negative).

Example: Temperature changed from -3°C to 5°C, representing an increase of 8°C.

Banking

Deposits (positive) and withdrawals (negative) in a bank account.

Example: Starting balance $100, withdrawal of $25, deposit of $50 gives $100 - $25 + $50 = $125.

Altitude

Heights above sea level (positive) and depths below sea level (negative).

Example: A submarine diving from the surface to -200m and then rising 75m would be at -125m.

Sports

Gains (positive) and losses (negative) in yards or points.

Example: A football team gaining 15 yards, then losing 7 yards has a net gain of 8 yards.

Problem-Solving Techniques

Number Line Method

Use a number line to visualize integer operations.

For addition:

Start at the first number on the number line, then move right for positive numbers and left for negative numbers.

Example: To calculate -3 + 5, start at -3 and move 5 units to the right to reach 2.

Counters Method

Use positive and negative counters (or chips) to model integer operations.

For addition and subtraction:

Use positive counters for positive integers and negative counters for negative integers. Paired positive and negative counters cancel each other out (zero pairs).

Example: To calculate -3 + 5, place 3 negative counters and 5 positive counters. After pairing, you're left with 2 positive counters, so the answer is 2.