The Four Operations, Integers and Number Lines

Q: What is an integer?

An integer is any whole number that can be positive, negative, or zero, excluding fractions and decimals.

Q: How is numerical value represented by position and direction on a horizontal number line?

Moving to the right of zero represents positive, increasing values, while moving to the left represents negative, decreasing values.

Q: How do you determine which of two integers is greater using a number line?

Any number located further to the right on a horizontal number line is strictly greater than any number to its left.

Q: How is distance between numbers measured on a number line?

Distance is always a non-negative quantity representing the total number of units between two points, calculated as $b - a$ where $b \ge a$.

Q: What direction do you move on a number line when adding positive and negative integers?

Adding a positive integer means moving to the right, whereas adding a negative integer means moving to the left.

Q: What is the rule for subtracting an integer, particularly when subtracting a negative number?

Subtracting a number is equivalent to adding its opposite: $a - b = a + (-b)$ and $a - (-b) = a + b$.

Q: How does the zero pair method explain the addition of integers with different signs?

One positive unit and one negative unit cancel each other out to make zero: $(+1) + (-1) = 0$.

Q: What are the sign rules for multiplying and dividing two integers?

Multiplying or dividing two numbers with the same sign gives a positive result; numbers with different signs give a negative result.

Q: Why is dividing by zero impossible in mathematics, and how does it differ from dividing zero by a non-zero integer?

Division by zero is undefined, whereas zero divided by any non-zero integer is always zero ($\frac{0}{a} = 0$ for $a \neq 0$).

Q: In what order must mathematical operations be performed when evaluating an expression?

Follow BEDMAS/PEMDAS: Brackets first, then Exponents, followed by Division and Multiplication (left-to-right), and finally Addition and Subtraction (left-to-right).

Practice

Mathematics · IB Middle Years Programme (MYP) · 10 min read · Updated: 23 Oct 2026

Welcome to The Four Operations, Integers and Number Lines

Have you ever checked the weather forecast in the middle of winter and seen a temperature like $-5^\circ\text{C}$? Or perhaps you have played a board game where you lost points or moved backward? In everyday life, numbers do not just count upward from zero; they move in both directions! In this chapter, we explore integers, how to visualize them clearly on a number line, and how to master the four operations (addition, subtraction, multiplication, and division) with positive and negative values.

Note: For a broader look at different number classifications such as rational or real numbers, see the chapter on Forms of Numbers and Number Systems.

1. Understanding Integers and the Number Line

What is an Integer?

An integer is any whole number that can be positive, negative, or zero. Integers do not include fractions or decimals.

We can think of integers in three distinct groups:

• Positive integers: $\{1, 2, 3, 4, \dots\}$ (values greater than zero)
• Zero: $0$ (neither positive nor negative; it acts as the neutral origin)
• Negative integers: $\{-1, -2, -3, -4, \dots\}$ (values less than zero)

Visualizing the Number Line

A number line is a straight horizontal line where numbers are placed at equal intervals. It provides a visual map of numerical value and direction:

• To the right of $0$: Numbers are positive and increase in value ($\dots \to +\infty$).
• To the left of $0$: Numbers are negative and decrease in value ($\dots \to -\infty$).
• Origin: The point marked $0$.

Analogy: Imagine standing on a ladder where ground level is $0$. Climbing up step $1, 2, 3$ takes you into positive heights. Stepping down into a basement cellar at levels $-1, -2, -3$ takes you into negative heights.

Comparing and Ordering Integers

On a standard horizontal number line, any number located further to the right is always greater than any number to its left.

We use inequality symbols to compare integers:

• $>$ means strictly greater than (e.g., $4 > -2$)
• $<$ means strictly less than (e.g., $-7 < -3$)
• $=$ means equal to (e.g., $-5 = -5$)

Important Tip: Be careful with negative values! Even though $7$ is bigger than $3$, $-7$ is colder or lower down than $-3$, which means $-7 < -3$.

Distance and Absolute Value on the Number Line

The distance between any number and zero on the number line is always a positive quantity or zero. For instance, both $+4$ and $-4$ are exactly $4$ units away from $0$. When calculating the distance between two values $a$ and $b$ on a number line, we count the number of units between them: the distance between $-3$ and $5$ is $5 - (-3) = 8$ units.

Section Takeaway: Moving right on a number line means increasing in value; moving left means decreasing in value. A number further to the left is always smaller.

2. Adding and Subtracting Integers

Addition: Moving Along the Line

Think of addition as taking steps along the number line:

• Adding a positive number means moving to the right.
• Adding a negative number means moving to the left.

Let's look at two examples:

1. Calculate $-2 + 5$:
Start at $-2$. Move $5$ units to the right. You arrive at $3$. Therefore, $-2 + 5 = 3$.

2. Calculate $1 + (-4)$:
Start at $1$. Adding a negative means walking left. Move $4$ units to the left. You arrive at $-3$. Therefore, $1 + (-4) = -3$.

Subtraction: The "Opposite Direction" Rule

Subtracting a number is defined as adding its opposite. When you subtract, you reverse direction:

• Subtracting a positive number means moving to the left (just like adding a negative):
$a - b = a + (-b)$
Example: $3 - 5 = -2$

• Subtracting a negative number turns into moving to the right (two negatives make a positive):
$a - (-b) = a + b$
Example: $4 - (-3) = 4 + 3 = 7$

Everyday Analogy: Think of a financial ledger. If someone removes (subtracts) a debt (a negative value) from your account, you end up wealthier (positive change)!

The "Zero Pair" Method

Another helpful way to understand integer addition is the concept of zero pairs:

• One positive counter $(+1)$ and one negative counter $(-1)$ cancel each other out to make zero: $(+1) + (-1) = 0$.
• To solve $-5 + 3$: You have $5$ negatives and $3$ positives. Pair up $3$ positives with $3$ negatives to make $0$. You are left with $2$ negatives. Hence, $-5 + 3 = -2$.

Section Takeaway: Adding a negative is identical to subtracting a positive. Subtracting a negative is identical to adding a positive.

3. Multiplying and Dividing Integers

The Sign Rules

Multiplication and division follow consistent rules based on whether the signs of the two numbers are the same or different.

Rule 1: Same Signs Give a Positive Result

• $\text{positive} \times \text{positive} = \text{positive}$ | $6 \times 3 = 18$
• $\text{negative} \times \text{negative} = \text{positive}$ | $(-6) \times (-3) = 18$
• $\text{positive} \div \text{positive} = \text{positive}$ | $18 \div 3 = 6$
• $\text{negative} \div \text{negative} = \text{positive}$ | $(-18) \div (-3) = 6$

Rule 2: Different Signs Give a Negative Result

• $\text{positive} \times \text{negative} = \text{negative}$ | $6 \times (-3) = -18$
• $\text{negative} \times \text{positive} = \text{negative}$ | $(-6) \times 3 = -18$
• $\text{positive} \div \text{negative} = \text{negative}$ | $18 \div (-3) = -6$
• $\text{negative} \div \text{positive} = \text{negative}$ | $(-18) \div 3 = -6$

Why Does Negative Times Negative Equal Positive?

Don't worry if this feels counterintuitive at first! Consider the following descending multiplication pattern:

$3 \times (-4) = -12$
$2 \times (-4) = -8$ (increases by $+4$)
$1 \times (-4) = -4$ (increases by $+4$)
$0 \times (-4) = 0$ (increases by $+4$)
$-1 \times (-4) = 4$ (must continue increasing by $+4$)
$-2 \times (-4) = 8$

To preserve logical consistency in mathematics, the pattern demands that $(-1) \times (-4) = +4$.

Division by Zero

In mathematics, division by zero is undefined. For example, $\frac{7}{0}$ has no valid numerical value. However, zero divided by any non-zero integer is always zero: $\frac{0}{-5} = 0$.

Section Takeaway: Same signs multiply or divide to yield a positive; opposite signs multiply or divide to yield a negative.

4. The Four Operations and Order of Operations

The Standard Order of Operations

When an expression contains multiple operations, we must follow a globally recognized hierarchy to ensure consistent, correct answers. A common mnemonic is BEDMAS or PEMDAS:

1. Brackets / Parentheses: Work inside the innermost grouping symbols first.
2. Exponents / Indices: Evaluate powers and square roots next.
3. Division and Multiplication: Perform these from left to right as they appear.
4. Addition and Subtraction: Perform these from left to right as they appear.

Worked Example: Step-by-Step Calculation

Evaluate the expression: $14 - 3 \times (-2) + (-8) \div 4$

Step 1: Identify operations. We have subtraction, multiplication, addition, and division.
Step 2: Carry out multiplication and division first, moving left to right:
• $3 \times (-2) = -6$
• $(-8) \div 4 = -2$
Rewrite the expression: $14 - (-6) + (-2)$
Step 3: Carry out addition and subtraction from left to right:
• $14 - (-6) = 14 + 6 = 20$
• $20 + (-2) = 20 - 2 = 18$
Final Answer: $18$

Working with Nested Brackets

When expressions feature brackets inside brackets (such as round brackets $( )$ inside square brackets $[ ]$), always resolve the inner set first:

Evaluate: $5 \times [12 - (3 - 7)]$
• Inner bracket: $3 - 7 = -4$
• Substitute back: $5 \times [12 - (-4)]$
• Simplify inside square brackets: $12 - (-4) = 12 + 4 = 16$
• Final multiplication: $5 \times 16 = 80$

Section Takeaway: Always follow the order of operations. Never perform addition or subtraction before multiplication or division unless grouping brackets instruct you to do so.

5. Real-World Applications (Criterion D Contexts)

Integers and number lines frequently model authentic physical and financial scenarios in IB MYP tasks:

1. Temperature Variations

Scenario: At midnight in a mountain village, the temperature is $-8^\circ\text{C}$. By midday, the temperature rises by $13^\circ\text{C}$. By 8:00 PM, a cold front causes it to fall by $9^\circ\text{C}$. What is the temperature at 8:00 PM?

• Starting temperature: $-8^\circ\text{C}$
• After the midday rise: $-8 + 13 = 5^\circ\text{C}$
• After the evening drop: $5 - 9 = -4^\circ\text{C}$
• Conclusion: The temperature at 8:00 PM is $-4^\circ\text{C}$.

2. Elevations Above and Below Sea Level

Scenario: A submarine cruises at an elevation of $-140\text{ m}$ (relative to sea level, which is $0\text{ m}$). It rises $65\text{ m}$ to avoid an underwater ridge, and then dives $30\text{ m}$. What is its new depth?

• Mathematical model: $-140 + 65 - 30$
• Step 1: $-140 + 65 = -75\text{ m}$
• Step 2: $-75 - 30 = -105\text{ m}$
• Conclusion: The submarine is at an elevation of $-105\text{ m}$, which represents a depth of $105\text{ m}$ below sea level.

3. Financial Accounting

Scenario: A school club bank account has a starting balance of $\$45$. They purchase supplies costing $\$80$ (represented as $-\$80$) and later receive a donation of $\$60$. What is their closing balance?

• $45 - 80 + 60 = -35 + 60 = 25$
• Conclusion: The club has a positive closing balance of $\$25$.

Section Takeaway: When applying mathematics to real-world contexts, identify key values, assign the correct positive or negative sign, show clear working, and state final units (e.g., $^\circ\text{C}$, $\text{m}$, $\$$).

6. Communicating Mathematical Reasoning (Criterion C)

In the MYP on-screen eAssessment, Criterion C: Communicating evaluates how well you explain your methods and present your solutions. To earn top marks:

• Show full working step-by-step: Avoid writing only the final answer. Write down each intermediate step clearly so examiners can follow your logic.
• Use proper mathematical notation: Write $(-4) \times (-3) = 12$ rather than informal abbreviations.
• Use brackets around negative numbers when operating: Writing $5 \times -2$ looks confusing; write $5 \times (-2) = -10$ to maintain clarity.
• Label number line sketches: When drawing or interpreting a number line, always include the origin ($0$), consistent scale markings, and an arrow indicating direction.

7. Common Pitfalls and Mistakes to Avoid

• Confusing the rules for addition with the rules for multiplication:
Incorrect: $-3 - 5 = +8$ (thinking "two negatives make a positive").
Correct: $-3 - 5 = -8$ (you start at $-3$ and move $5$ units to the left). The "two negatives make a positive" rule applies only when two signs sit directly together, such as $-(-5) = +5$, or when multiplying/dividing $(-3) \times (-5) = +15$.

• Ignoring the order of operations:
Incorrect: $10 - 2 \times 3 = 8 \times 3 = 24$.
Correct: Multiplication comes first: $2 \times 3 = 6$, then $10 - 6 = 4$.

• Misinterpreting inequality signs with negatives:
Incorrect: $-12 > -4$.
Correct: $-12 < -4$, because $-12$ is further to the left on the number line than $-4$.

• Dropping negative signs during multi-step equations: Keep track of negative signs at every line of working by wrapping negative numbers in parentheses.

8. Quick Review Summary

• Integers: All positive whole numbers, negative whole numbers, and zero: $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.
• Number Line: Moving right is positive / increasing; moving left is negative / decreasing.
• Adding & Subtracting: Adding a negative is subtraction ($a + (-b) = a - b$); subtracting a negative is addition ($a - (-b) = a + b$).
• Multiplying & Dividing: Same signs yield a positive answer; different signs yield a negative answer.
• Hierarchy: Follow BEDMAS/PEMDAS (Brackets, Exponents, Division & Multiplication, Addition & Subtraction).

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is an integer?

An integer is any whole number that can be positive, negative, or zero, excluding fractions and decimals.

How is numerical value represented by position and direction on a horizontal number line?

Moving to the right of zero represents positive, increasing values, while moving to the left represents negative, decreasing values.

How do you determine which of two integers is greater using a number line?

Any number located further to the right on a horizontal number line is strictly greater than any number to its left.

How is distance between numbers measured on a number line?

Distance is always a non-negative quantity representing the total number of units between two points, calculated as $b - a$ where $b \ge a$.

What direction do you move on a number line when adding positive and negative integers?

Adding a positive integer means moving to the right, whereas adding a negative integer means moving to the left.

What is the rule for subtracting an integer, particularly when subtracting a negative number?

Subtracting a number is equivalent to adding its opposite: $a - b = a + (-b)$ and $a - (-b) = a + b$.

How does the zero pair method explain the addition of integers with different signs?

One positive unit and one negative unit cancel each other out to make zero: $(+1) + (-1) = 0$.

What are the sign rules for multiplying and dividing two integers?

Multiplying or dividing two numbers with the same sign gives a positive result; numbers with different signs give a negative result.

Why is dividing by zero impossible in mathematics, and how does it differ from dividing zero by a non-zero integer?

Division by zero is undefined, whereas zero divided by any non-zero integer is always zero ($\frac{0}{a} = 0$ for $a \neq 0$).

In what order must mathematical operations be performed when evaluating an expression?

Follow BEDMAS/PEMDAS: Brackets first, then Exponents, followed by Division and Multiplication (left-to-right), and finally Addition and Subtraction (left-to-right).

What is the common pitfall when subtracting a positive number, such as $-3 - 5$?

Thinking "two negatives make a positive" is incorrect here; $-3 - 5 = -8$ because you start at $-3$ and move $5$ units to the left.

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