Integers Lesson - Definition, Rules, Properties and Examples

Lesson Overview

• What Are Integers?
• Types of Integers
• Set of Integers
• Integers on a Number Line
• Integer Operations

Whole numbers show amounts we can count. But sometimes, we need numbers below zero. Integers solve this problem. 

Integers are a set of numbers that include whole numbers (0, 1, 2, 3, ...) and their negative counterparts (-1, -2, -3, ...). They extend the number line to both positive and negative directions, providing a more comprehensive representation of numerical values.

They provide a framework for solving equations, analyzing data, and modeling real-world phenomena. By mastering integers, one can enhance their problem-solving skills and develop a deeper understanding of mathematical concepts.

What Are Integers?

Integers are whole numbers (including negatives) that represent quantities above and below a reference point on a number line. They don't include fractions or decimals, such as:

• Temperature: -5°C (below zero) or 20°C (above zero)
• Elevation: -100 meters (below sea level) or 5000 meters (above sea level)
• Financial balance: -$50 (debt) or $100 (credit)
  

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Types of Integers

There are three types of integers that form a set of integer numbers - 

• Positive Integers are numbers greater than zero. They represent quantities greater than a reference point.

Example: 5 represents a gain of 5 points, a temperature of 5°C above freezing, or a location 5 meters above sea level.

• Negative Integers are numbers less than zero. They represent quantities less than a reference point.

Example: -3 represents a loss of 3 points, a temperature of 3°C below freezing, or a location 3 meters below sea level.

• Zero is neither positive nor negative. It represents a neutral point or the origin on a number line.

Example: 0 represents no change, the freezing point of water (0°C), or sea level.

Set of Integers

Integers include all whole numbers (no fractions or decimals) and stretch endlessly in both positive and negative directions. They are represented by the symbol ℤ and can be written as:

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

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Integers on a Number Line

A number line visually represents integers as a horizontal line with evenly spaced points, each marking an integer.

• Zero (0) marks the origin, the central point of the number line.
• Positive integers extend to the right of zero.
• Negative integers extend to the left of zero.

The number line demonstrates the order and magnitude of integers.

Integer Operations

Integers follow specific rules for arithmetic operations. Understanding these rules is crucial for solving problems involving quantities with direction.

Addition of Integers

When adding integers with the same sign, you're essentially moving in the same direction on the number line. When adding integers with different signs, you're moving in opposite directions on the number line.

• Adding integers with the same sign: Add their absolute values (the distance from zero) and keep the common sign.
  

Example 1: 5 + 3 = 8 (Both positive, move right on the number line, resulting in a larger positive number)

(Alt text- Addition of positive integers on number line)

Example 2: -5 + (-3) = -8 (Both negative, move left on the number line, resulting in a smaller negative number)

• Adding integers with different signs:
  
  • Find the absolute value of each integer.
  • Subtract the smaller absolute value from the larger absolute value.
  • The sum has the sign of the integer with the larger absolute value.

Example 1: -5 + 3 = -2 (Move 5 units left, then 3 units right)

Example 2: 5 + (-3) = 2 (Move 5 units right, then 3 units left.)

Subtraction of Integers

Subtracting a positive number is the same as adding a negative number (moving left on the number line). While subtracting a negative number is the same as adding a positive number (moving right).

• Subtraction is the same as adding the opposite.
  • Change the subtraction sign to an addition sign.
  • Change the sign of the integer being subtracted.
  • Follow the rules for addition.

Example 1: 8 - (-2) = 8 + 2 = 10 (Start at 8, move 2 units right)

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Multiplication of Integers

Adding a positive number repeatedly gives a positive result. Repeatedly adding a negative number also gives a positive result.

• Multiplying integers with the same sign: The product is always positive.

Example 1: 5 x 3 = 15 (5 added to itself 3 times)

Example 2: -5 x (-3) = 15 (Subtracting -5 three times is the same as adding 5 three times)

• Multiplying integers with different signs: The product is always negative.

Example 1: 5 x (-3) = -15 (5 added to itself -3 times, which means subtracting 5 three times)

Example 2: -5 x 3 = -15 (-5 added to itself 3 times)

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Integers: Positive and Negative Numbers

Division of Integers

Dividing by a positive number is simple. Dividing two negative numbers gives a positive result. Dividing a positive number by a negative, or the reverse, moves in opposite directions on the number line, giving a negative result.

• Dividing integers with the same sign: The quotient is always positive.

Example 1: 15 / 3 = 5 

Example 2: -15 / (-3) = 5

• Dividing integers with different signs: The quotient is always negative.

Example 1: 15 / (-3) = -5

Example 2: -15 / 3 = -5

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