Negative Numbers on the Number Line | Multiplying and Dividing Negative Numbers |

Absolute Value | Summary Table of Negative Number Operations |

Adding and Subtracting Negative Numbers | More Practice |

## Negative Numbers on the Number Line

Negative numbers seem a little scary at first, but they really aren’t that bad. Let’s first re-introduce our** number line**:

Notice how the negative integers (the ones with the minus in front of them) are the same distance from zero ($ 0$) as the positive numbers — but they are to the left of the $ 0$. That’s all negative numbers are; they just go backward the same way that positive numbers go forward.

## Absolute Value

The **absolute value** of a number (also called the **modulus**) is the distance from $ 0$, so it is always a positive number. It is written with two lines around the number, and it is simply the **positive value** of what’s inside the lines, whether the number is positive or negative.

It can get a little more complicated in algebra when we work with variables, or unknowns, but for now, here are examples to show how really simple the concept is:

$ \left| {-5} \right|=5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| 5 \right|=5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\left| {-5} \right|=-5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| 0 \right|=0$

Two numbers are called **opposites** if they are the same distance from $ 0$. For example, $ 2$ and $ -2$ are opposites.

Remember that numbers with a larger absolute value can actually be smaller when the numbers are negative – for example, $ -6<-5$, and, in the case of fractions, $ \displaystyle -\frac{3}{4}<-\frac{1}{2}$. So if we’re comparing negative numbers, it’s actually backwards compared to what we’re used to.

Let’s think of an example of how a negative might exist in real life. Let’s say your mom is having a little sister and the new baby is supposed to be born in $ 5$ months. In a weird way, your baby sister is $ =5$ months old. Next month, she will be $ -4$ months old, and so on. On the day she is born, she is $ 0$, and then she’ll start being positive months after that. Again, negative numbers are the same “**distance**” (distance is always positive) away from **0**, but just in the opposite direction. If we wanted to know how long it is until she’s $ 9$ months old, we’d add the $ 5$ months **before** she’s $ 0$ to the $ 9$ months **after** she’s $ 0$ to get $ 14$ months. This is actually a subtraction problem, when you think about it: $ 9-\left( {-5} \right)=14$ – weird!. Can you guess that when we have two negatives together, it becomes a positive?

More advanced topics on **Absolute Value **are found in the **Solving Absolute Value Equations and Inequalities **section.

## Adding and Subtracting Negative Numbers

Let’s talk about **adding and subtracting** negative numbers. For the number $ -2$, this means that we are two places to the left of the $ 0$ and four places to the left of the $ 2$. Remember that when we **add**, we count to the **right**, and when we **subtract**, or add a negative number, we count to the **left**. Thus, to add $ -2$ and $ 4$, we’d get $ 2$. There are some rules about adding and subtracting negative numbers that we’ll talk about shortly.

Let’s do some more addition and add some subtraction. As just mentioned, **adding means moving to the right**, and **subtracting means moving to the left**, as in the following graphic:

**Remember that **$ \boldsymbol {-3+5}$** is the same as **$ \boldsymbol {5-3}$** and **$ \boldsymbol {5+-6}$** is the same as **$ \boldsymbol {5-6}$; **it’s OK to memorize this! Also note that the sign of a number comes before it and many times there is an invisible plus sign before numbers, like when they are at the beginning.**

Here are some rules for **adding** **and** **subtracting** negative numbers:

- Adding two positive numbers yields a positive number. For example, $ 4+4=8$.
- Adding two negative numbers yields a negative number. Add the two numbers and put negative in front of it. For example, $ -4+-4=-8$ . This is the same as $ -4-4=-8$.
- Adding a negative number is the same thing as subtracting that number. For example, $ \displaystyle -4+-4\text{ }=-4-4=-8$.
- When adding a positive number and negative number,
**subtract**the absolute values of the two numbers (larger – smaller) and make the sign whichever has**largest**original number without the signs (in other words, the largest absolute value). Some examples:- For $ -4+10$, since one is positive and the other one is negative, subtract $ 4$ from $ 10$ (the absolute values of the numbers), to get $ 6$. Since $ \left| {10} \right|>\left| {-4} \right|$ and the $ 10$ is positive, the answer is a positive $ 6$.
- For $ -15+8$, subtract $ 8$ from $ 15$, to get $ 7$. Since $ \left| {-15} \right|>\left| 8 \right|$ and
**15**has the – sign before it, add a “–” to the**7**, to get**–7**.

- If you have two minuses in a row, turn those into pluses. For example, if you have $ 4-\left( {-8} \right)=4- -8$, it’s the same as $ 4++\,8$, which is $ 4+8$, which is
**12**.

Don’t worry; once you do a lot of these, they will become second nature!

## Multiplying and Dividing Negative Numbers

Now let’s talk about **multiplying** and **dividing** with negative numbers. This is actually easier, as there as fewer rules:

- Multiplying or dividing two positive numbers results in a positive number. For example, $ 5\times 5=25$.
- Multiplying or dividing two negative numbers results in a positive number. For example, $ \displaystyle -5\times -5=25$, and $ \displaystyle \frac{{-5}}{{-5}}=1$.
- Multiplying or dividing a positive number with a negative number is always negative. For example, $ -5\times 5=-25$.

## Summary Table of Negative Number Operations

Here is a table that sums all this up. Interesting how multiplication with negative numbers is easier than addition!

You really need to practice with negative numbers, because you will use them a lot all throughout your mathematical life; maybe not so much in your “real” life, but when you take math classes through Calculus in high school, you’ll still be using them. And, as always, be careful with fractions!

**Learn these rules and practice, practice, practice!**

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