Lesson 4.4: Negative Numbers

New Concept

This lesson introduces negative numbers, expanding our number system to values less than zero. You will learn how to represent quantities like debt or temperatures below freezing, graph them on a number line, and compare their values.

What’s next

Now that you have the basic idea, you'll work through interactive examples, practice graphing on a number line, and solve problems involving real-world change.

Property

A negative number is a number less than zero. The symbol $-$ in front of a number means negative or opposite of.

Examples

• A debt of 100 dollars is represented by $-100$ dollars.
• A location 50 feet below sea level has an elevation of $-50$ feet.
• A loss of 5 points in a game is recorded as $-5$ points.

Explanation

Negative numbers represent values less than zero, like debt, temperatures below freezing, or depths below sea level. They are the opposites of positive numbers, existing on the other side of zero.

Property

When we deal with positive and negative numbers together, we call them signed numbers. When we work with signed numbers, we use the symbol $+$ to indicate a positive number. For example, $+3$ means positive 3. However, unless some special distinction is needed, we usually write positive 3 as just 3.

Examples

• A gain of 8 yards in football is written as $+8$ yards.
• A company profit of 20,000 dollars is represented by $+20,000$ dollars.
• A temperature increase of $5^{\circ}$ is shown as $+5^{\circ}$.

Explanation

Signed numbers tell us two things: a value and a direction (positive or negative). A plus sign means an increase or gain, while a minus sign means a decrease or loss. No sign usually means positive!

Property

The positive whole numbers, the negative whole numbers, and zero make up a set of numbers called the integers. Numbers increase as we move from left to right on the number line, so every negative number is less than every positive number. We use inequality symbols to compare numbers:
$<$ means 'is less than'
$>$ means 'is greater than'

Examples

• The integers between $-4$ and $1$ are $-3, -2, -1, 0$.
• To compare $-9$ and $-2$, notice $-9$ is to the left of $-2$ on a number line, so $-9 < -2$.
• The statement '$-3$ is greater than $-10$' is written with an inequality symbol as $-3 > -10$.

Explanation

Integers are all the whole numbers and their opposites, with zero in the middle. They don't include fractions or decimals. A number line helps us see that numbers to the right are always greater.

Property

To show a particular number on the number line, we place a dot at its position. This is called the graph of the number. Fractions, such as $2\frac{3}{4}$ or $4.8$, are located between the integers on a number line.

Examples

• To graph $-1.5$, you place a dot exactly halfway between $-1$ and $-2$.
• The number $3\frac{1}{2}$ is graphed by placing a dot halfway between $3$ and $4$.
• The graph of $-4\frac{3}{4}$ is located between $-4$ and $-5$, closer to $-5$.

Explanation

Graphing a number means pinning its exact location on a number line. This visual map helps you compare numbers easily. Negative fractions and decimals live between the negative integers, just like positive ones.

Property

The opposite of a negative number is a positive number. We write the opposite of $-6$ as $-(-6)$, so

Examples

• The opposite of $-25$ is written as $-(-25)$, which simplifies to $25$.
• If you cancel a debt of 30 dollars, your financial change is $-(-30)$ dollars, which is a gain of $30$ dollars.
• On a number line, the opposite of $-8.2$ is found by reflecting it over zero, resulting in $8.2$.

Explanation

Taking the 'opposite' of a number flips it across zero on the number line. If you flip a negative number, it lands on the positive side. So, a double negative like $-(-6)$ becomes a positive $6$.

Property

We can use negative numbers to indicate a decrease or a loss, and positive numbers to indicate an increase or a gain. An increase or decrease is called a net change.

Examples

• If the temperature was $8^{\circ}$ and changed by $-12^{\circ}$, the new temperature is $-4^{\circ}$.
• A stock was worth 25 dollars per share and is now worth 18 dollars. The net change in value is $-7$ dollars.
• Corinne's bank account had 150 dollars, and she wrote a check for 200 dollars. Her account's value changed by $-200$ dollars, resulting in a balance of $-50$ dollars.

Explanation

Signed numbers are perfect for showing change. A positive number shows a gain or increase, like the temperature rising. A negative number shows a loss or decrease, like a stock price dropping. This is the 'net change.'