Negative Numbers and Absolute Value

This section covers:

Negative Numbers

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

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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 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.

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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.

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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).

For more advanced topics on Absolute Value, see Solving Absolute Value Equations and Inequalities.

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.  So 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.

Now 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:

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Remember that “3 + 5” is the same as “5 3” and “5 + 6” is the same as “5 6” — just memorize this!!! And 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.

So 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. Just add the two numbers and put negative in front of it. For example, $$-4+-4=-8$$ . This is the same as $$-4-4=-8$$.
• Again, adding a negative number is the same thing as subtracting that number. For example, $$\displaystyle -4-4=-4+-4\text{ }=-8$$.
• When you add a positive number and negative number, you’ll want to subtract the absolute values of the two numbers (larger – smaller) and make the sign of the difference 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, you’d subtract 4 from 10 (the absolute values of the numbers) and get 6. Now since $$\left| {10} \right|>\left| {-4} \right|$$ and the 10 is positive, the answer is a positive 6.
• For  $$-15+8$$, you would subtract 8 from 15, and get 7. Since $$\left| {-15} \right|>\left| 8 \right|$$ and 15 has the – sign before it, you would 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 $$4-\left( {-8} \right)$$, it’s the same as $$4++\,8$$, which is $$4+8$$, which is 12.

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 2 positive numbers results in a positive number. For example, $$5\times 5=25$$.
• Multiplying or dividing 2 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, $$-5\times 5=-25$$.

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

 Multiplication and Division Addition and Subtraction $$\begin{array}{c}\text{Positive }\times \,\,\text{Positive}=\text{Positive}\\4\times 3=12\end{array}$$ $$\begin{array}{c}\text{Positive }+\,\,\text{Positive}=\text{Positive}\\4+4=8\end{array}$$ $$\begin{array}{c}\text{Negative }\times \,\,\text{Negative}=\text{Positive}\\-5\times -5=25\end{array}$$ $$\begin{array}{c}\text{Negative }+\,\,\text{Negative}=\text{Negative}\\-4+-5=-9\end{array}$$ $$\begin{array}{c}\text{Negative }\times \,\,\text{Positive}=\text{Negative}\\-4\times 4=-16\end{array}$$ $$\displaystyle \text{Negative }+\,\,\text{Positive }=\,\,\text{Depends}\,\,\text{:)}$$ When you add a positive number and negative number, you’ll want to subtract the absolute values of the two numbers (larger – smaller) and make the sign the same sign of the largest absolute value of the numbers: $$-15+8=$$ Subtract 8 from 15 to get 7; since $$\left| {-15} \right|>\left| 8 \right|$$ and 15 has the – sign before it, you would add a “–” to the 7 to get  –7. $$-8+15=$$ Subtract 8 from 15 to get 7; since $$\left| {15} \right|>\left| {-8} \right|$$ and 15 is positive, you get 7.   NOTE:  Subtracting a negative number is the same thing as adding numbers. $$5–10=5++10=5+10=15$$

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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On to Powers, Exponents, Radicals and Scientific Notation – you are ready!!

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