Integers and the Number Line

If you have ever checked the weather and seen a temperature like “-5 degrees,” or looked at a bank statement showing you are “$50 in the red,” then you have already encountered integers in the wild. Negative numbers might seem strange at first - after all, how can you have “less than nothing”? But here is the good news: you already understand more about integers than you think. Every time you talk about owing money, going below sea level, or counting down from ten, you are working with the ideas behind integers.

In this chapter, we are going to expand your number toolkit beyond the counting numbers you have known since childhood. By the end, you will be able to work confidently with positive numbers, negative numbers, and zero - and you will see that the rules governing them are simpler than they first appear.

Core Concepts

What Are Integers?

An integer is any whole number - positive, negative, or zero. That is it. No fractions, no decimals, just whole numbers that can go in either direction from zero.

The integers include:

• Positive integers: $1, 2, 3, 4, 5, …$ (the counting numbers you already know)
• Zero: $0$ (the starting point, neither positive nor negative)
• Negative integers: $-1, -2, -3, -4, -5, …$ (the “mirror image” of the positives)

Written as a set, the integers look like this:

$${…, -3, -2, -1, 0, 1, 2, 3, …}$$

The three dots (called an ellipsis) mean the pattern continues forever in both directions.

The Number Line

The number line is your visual map for understanding integers. Picture a horizontal line with zero in the middle:

$$\leftarrow ;-5 \quad -4 \quad -3 \quad -2 \quad -1 \quad 0 \quad 1 \quad 2 \quad 3 \quad 4 \quad 5; \rightarrow$$

A few key observations:

• Zero sits in the center - it is the dividing point between positive and negative
• Positive numbers are to the right of zero, getting larger as you move right
• Negative numbers are to the left of zero, getting “more negative” as you move left
• Numbers further right are always greater than numbers to their left

This means $-2$ is greater than $-5$ (even though 2 is less than 5 as positive numbers). Why? Because $-2$ is to the right of $-5$ on the number line. Think of it like temperature: -2 degrees is warmer than -5 degrees.

Absolute Value

Absolute value measures how far a number is from zero, regardless of direction. It answers the question: “How many steps away from zero is this number?”

We write absolute value using vertical bars: $|x|$

Some examples:

• $|5| = 5$ (five is 5 steps from zero)
• $|-5| = 5$ (negative five is also 5 steps from zero)
• $|0| = 0$ (zero is 0 steps from itself)

Think of absolute value as the “distance” version of a number. Distance cannot be negative - you cannot walk negative three miles. So absolute value is always zero or positive.

In plain English: Absolute value strips away the negative sign (if there is one) and tells you the magnitude of the number.

Notation and Terminology

Term	Symbol	Meaning
Positive integer	$+n$ or just $n$	A whole number greater than zero
Negative integer	$-n$	A whole number less than zero
Absolute value	$|n|$	Distance from zero (always non-negative)
Opposite	$-n$	The number on the other side of zero, same distance away
Greater than	$>$	Further to the right on the number line
Less than	$<$	Further to the left on the number line

A note on writing positive numbers: You can write positive 5 as $+5$ or simply $5$. Most of the time, we drop the plus sign because it is understood. However, when working with both positives and negatives, writing $+5$ can help you keep track of signs.

Examples

Key Properties and Rules

Rules for Addition

Situation	Rule	Example
Same signs	Add absolute values, keep the sign	$(-4) + (-5) = -9$
Different signs	Subtract absolute values, use sign of larger absolute value	$(-7) + 3 = -4$

Rule for Subtraction

Subtracting is adding the opposite:

$$a - b = a + (-b)$$

This single rule transforms every subtraction problem into an addition problem, which you can then solve using the addition rules above.

Rules for Multiplication and Division

Signs	Result	Memory Aid
$(+) \times (+)$	$+$	Positive times positive is positive
$(-) \times (-)$	$+$	Two negatives make a positive
$(+) \times (-)$	$-$	Mixed signs give negative
$(-) \times (+)$	$-$	Mixed signs give negative

The same rules apply to division:

$$\frac{+}{+} = + \qquad \frac{-}{-} = + \qquad \frac{+}{-} = - \qquad \frac{-}{+} = -$$

Quick memory trick: Count the negative signs. An even number of negatives gives a positive result; an odd number gives a negative result.

Absolute Value Properties

• $|a| \geq 0$ for all numbers $a$ (absolute value is never negative)
• $|a| = |-a|$ (a number and its opposite have the same absolute value)
• $|a \times b| = |a| \times |b|$ (absolute value of a product equals product of absolute values)

Real-World Applications

Temperature

Temperature is perhaps the most common everyday use of negative numbers. When meteorologists say it is -15 degrees Fahrenheit, they mean 15 degrees below zero.

Situation: The temperature at noon was $12°F$. By midnight, it had dropped $20$ degrees. What was the midnight temperature?

$$12 + (-20) = 12 - 20 = -8°F$$

The temperature at midnight was 8 degrees below zero.

Elevation and Depth

Sea level is defined as zero elevation. Places above sea level have positive elevation; places below have negative elevation.

Situation: A submarine is at -200 feet (200 feet below sea level). It descends another 150 feet. What is its new depth?

$$(-200) + (-150) = -350 \text{ feet}$$

The submarine is now 350 feet below sea level.

Money: Debt and Credit

Your bank balance can be thought of as positive (you have money) or negative (you are overdrawn or in debt).

Situation: Your account balance is -$75 (you owe the bank $75). You deposit $120. What is your new balance?

$$(-75) + 120 = 45$$

Your new balance is $45.

Football Yardage

In football, teams can gain yards (positive) or lose yards (negative).

Situation: A team runs three plays: a gain of 8 yards, a loss of 3 yards, and a loss of 6 yards. What is their total yardage?

$$8 + (-3) + (-6) = 8 - 3 - 6 = -1$$

They had a net loss of 1 yard.

Golf Scores

In golf, scores are measured relative to par. Under par is negative (good), over par is positive (not as good).

Situation: A golfer shoots -3 on the first round and +2 on the second round. What is their total score relative to par?

$$(-3) + 2 = -1$$

They are 1 under par overall.

Self-Test Problems

Summary

Here is what you need to remember about integers:

• Integers are whole numbers that can be positive, negative, or zero: ${…, -3, -2, -1, 0, 1, 2, 3, …}$

• The number line helps you visualize integers, with negatives to the left of zero and positives to the right. Numbers further right are always greater.

• Absolute value $|n|$ measures distance from zero and is always non-negative.

• Adding integers:

  • Same signs: Add absolute values, keep the sign
  • Different signs: Subtract absolute values, take the sign of the larger absolute value

• Subtracting integers: Change to addition by adding the opposite: $a - b = a + (-b)$

• Multiplying and dividing integers:

  • Same signs give a positive result
  • Different signs give a negative result

• Integers appear everywhere: temperature, elevation, money, sports statistics, and countless other real-world contexts.

The most important thing to remember is that negative numbers are not mysterious or “unnatural” - they simply extend our number system to describe situations where we need to count below zero. Once you get comfortable moving in both directions on the number line, working with integers becomes second nature.