Understanding Negative Numbers

Negative numbers are numbers that have a value less than zero. You can think of them as opposites to the positive numbers you already know. We use a minus sign ($-$) to show that a number is negative.

Real-World Examples of Negative Numbers

Negative numbers aren't just for math class; they show up in the real world all the time to represent situations that go "below" a certain starting point (zero).

• Sea Level: Sea level is considered $0$. If a mountain is $1,000$ feet above sea level, it is $+1000$. If a submarine is $50$ feet below sea level, it is represented by the integer $-50$.
• Temperature: When the temperature drops below freezing, we use negative numbers. A temperature of 5 degrees below zero is written as $-5^\circ$.
• Money: If you have \10$in your bank account, your balance is$10$. If you spend$$15$, you now owe$$5$, which means your balance is$-5$.

The Number Line

A number line is the best tool for visualizing negative numbers.

• The number $0$ is in the middle. It is neither positive nor negative.
• Positive numbers ($1, 2, 3, \dots$) go to the right of zero.
• Negative numbers ($-1, -2, -3, \dots$) go to the left of zero.

Imagine you need to place the numbers $-3$, $2$, $-1$, $0$, and $4$ on a number line. You would start at $0$, move to the right to place $2$ and $4$, and move to the left to place $-1$ and $-3$.

$\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots$

Comparing and Ordering Negative Numbers

When comparing numbers on a number line, the rule is simple: Any number to the right is greater than any number to the left.

This can be tricky with negative numbers because the digits look backwards. For example, $8$ is bigger than $2$, but $-8$ is less than $-2$. Why? Because $-8$ is further to the left on the number line than $-2$.

Let's order this set of numbers from least to greatest: $-5$, $3$, $-2$, $7$, $-8$.

1. First, identify the negative numbers: $-5, -2, -8$. The one furthest to the left is $-8$, so it is the smallest. Next is $-5$, then $-2$.
2. Then, look at the positive numbers: $3$ and $7$. We know $3$ is less than $7$.
3. Put them all together from left to right:

Answer: $-8, -5, -2, 3, 7$

Remember: The larger the negative number looks, the smaller its actual value is, because it is further away from zero in the negative direction!