Negative Numbers
and
Greater Than/Less Than


Image



Up until now, some of you may only be familiar with the number line shown above. It starts at zero and goes all the way up to "positive infinity." What is positive infinity? Positive infinity represents the largest number that's impossible to ever reach. Why can't we reach it? Think for a minute. Let's say that you think the largest number is something like 1,000,000,000,000,000,000. What happens when you add one more to that number? You get 1,000,000,000,000,000,001. You see, no matter how large the number is, you can always add one more to it. That's why we have to use positive infinity to represent that number. Infinity goes on and on forever. Since numbers go on and on forever also, we say that positive infinity represents the largest number we'll never be able to reach.


Image


Let's discuss greater than and less than. Many of you may already be familiar with these signs. If you're not, please take a close look at the picture above. When you see a statement like: 5 > 1, you can tell that it says: 5 "is greater than" 1, because the sign begins with the wide side. The wide side always faces the larger number and the narrow side always points to the smaller number. In a statement like: 1 < 5, you can tell that it says: 1 "is less than" 5, because the sign begins with the narrow side. Again, the narrow side is always going to point toward the smaller number, and the wide side will always face the larger number. It's easy to remember the difference in the signs if you think of it like this: narrow side first = less than, and wide side first = greater than. Study the picture closely to see what I mean. Even though we used two different examples, both times the narrow side pointed to the smaller number and the wide side faced the larger number.


Image


Now we'll discuss "real numbers." "Real numbers" are numbers that can fall anywhere on the number line shown above. Why are they called "real numbers?" Were the numbers on the other number line "fake numbers?" No, no, no... It's nothing like that. Before, you were only working with real numbers that started at zero and went to positive infinity. Now we're going to look at the real numbers on the other side of zero. These numbers and the numbers that you are already familiar with make up the entire set of real numbers. As you go right on the number line, from any point, the numbers grow larger and larger. As you go left, from any point, the numbers get smaller and smaller.

The smallest number that we'll never be able to reach is known as "negative infinity." Just like with positive infinity, you'll always be able to subtract one more from any number. For that reason, negative numbers also never end. The yellow area in the picture represents the negative numbers. I should point out that 0 is not considered to be either negative or positive. Zero is the point where positive numbers end and negative numbers begin, or vice-versa. Negative numbers are always smaller than zero. Numbers further left on the number line are always smaller than those to their right. Take a look below to see what I mean.

1 > 0 (says: 1 is greater than 0)
(but)
-1 < 0 (says: -1 is less than 0)

5 > 1 (says: 5 is greater than 1)
(but)
-5 < -1 (says: -5 is less than -1)

How can -5 be less than -1? It's less because it's further left on the number line. A good way to think about negative numbers, and one you may already be familiar with without knowing it, is a thermometer. 0 degrees Celsius is the point where water freezes. Anything below 0 degrees is negative. -5 degrees is colder than -1 degrees. You can also think about money. If you are broke and don't have any money, you could say that you have 0 dollars. If you somehow manage to talk someone into loaning you a dollar, you're going to owe them 1 dollar. Now you're in debt 1 dollar and are going to have to pay that back before you'll be at 0 dollars again. Until you pay it back, your money is actually -1 dollars. The more you borrow, the worse it gets. I guess this proves that being broke is much better than owing lots of money. Let's take a look at the real number line again and try doing some adding and subtracting.


Image


If you think about it, you can count up the number line to add or down the line to subtract. What I mean is that 1+3=4, because starting at 1 on the number line and counting up 3 places lands on 4. The same is true for subtraction, except that you have to count down. 4-3=1, because starting at 4 and counting down 3 places lands on 1. Now here's the cool part! The same thing can be done using negative numbers! 1-2=-1, because starting at 1 on the number line and counting down 2 places lands on -1. -3+6=3, because starting at -3 on the number line and counting up 6 places lands on 3. Look at the problems below, and count up or down the number line to see how I got the answers.

3-7=-4
-4+7=3
-1-7=-8
-8+7=-1
-10<-1
-1>-10

Will we always have to use a number line in order to add or subtract negative numbers? No way! There's a much easier way to do it, but first we need to talk about "absolute value." The absolute value of a number is its value without the negative sign. For instance, the absolute value of 3 is 3 because it doesn't even have a negative sign to begin with. The absolute value of -3 is also 3, because this time it did have a negative sign, but the absolute value of a number is its value without the negative sign. In other words, finding the absolute value of any number is the same as making the number positive. In math problems, if you see something like: |-5|=5, those two lines around the -5 are saying: "get the absolute value of -5." With that said, let's look at some examples of absolute values.

The absolute value of 100 is 100
The absolute value of -100 is 100
The absolute value of 23 is 23
The absolute value of -23 is 23

In order to add 2 positive numbers, we just add them. In order to add 2 negative numbers, we just add them also, but we have to remember to put the negative sign on the answer. In other words, -2+-2=-4. It's sometimes easier to use parenthesis around negative numbers so that we don't make things look confusing. In the previous problem, it probably would have been better for me to have written it like this: -2+(-2)=-4. Both problems are exactly the same; it's just that the last one, using parenthesis, is easier to read.

So far we haven't needed to use the absolute value. What happens when we try to add a negative and a positive number together? That's when we need to use the absolute value! First we write down the number with the largest absolute value. If it's the negative number, write the negative sign down in front of where your answer is going to go, because the answer is going to be negative also. Next write the number with the smaller absolute value below or to the right of the first number. Subtract the two and you have your answer! Remember that if the number with the larger absolute value was negative, you have to put the negative sign on your answer. If the number with the larger absolute value was positive, the answer will also be positive. Below are two examples.

Problem: -10+5
The absolute value of -10 is 10 so this is our larger number.
(Remember the sign since the larger number was negative)
Now subtract the two: 10-5=5
(Since our larger absolute value was negative, add the "-")
Our final answer is: -5
So: -10+5=-5

Problem: 10+(-5)
The absolute value of -5 is 5 so this is our smaller number.
(Forget the sign since the larger number was positive)
Now subtract the two: 10-5=5
Our final answer is: 5 (since we didn't need the sign)
So: 10+(-5)=5

Subtraction gets a little trickier, but we'll use our own tricks in order to make it easy! When you're subtracting, and there's a negative number involved, change the sign of what's being subtracted, then change the subtraction sign to addition. From there just follow the same rules that we used to add. Check out the examples below to see what I'm talking about.

Problem: 5-10
The number being subtracted is 10 so change the sign.
5-(-10)
Next change the subtraction sign to addition.
5+(-10)
Now we can follow the rules for addition.
The absolute value of -10 is 10 so this is our larger number.
(Remember the sign since the larger number was negative)
Now subtract the two: 10-5=5
(Since our larger absolute value was negative, add the "-")
Our final answer is: -5
So: 5-10=-5

Problem: -10-(-5)
The number being subtracted is -5 so change the sign.
-10-(5)
Next change the subtraction sign to addition.
-10+5
Now we can follow the rules for addition.
The absolute value of -10 is 10 so this is our larger number.
(Remember the sign since the larger number was negative)
Now subtract the two: 10-5=5
(Since our larger absolute value was negative, add the "-")
Our final answer is: -5
So: -10-(-5)=-5

Let's look at something that you might have already noticed. When you subtract a negative number from something, it's the same as adding the absolute value of that negative number. For instance: 10-(-5) is the same as: 10+5. You might have also noticed that when you add a negative number to something, it's the same as subtracting the absolute value of that negative number. For instance: 10+(-5) is the same as: 10-5. Don't worry too much about this right now. It's just something I thought I'd point out.

Hopefully this negative business hasn't gotten too confusing yet. If it has, guess what? The worst of it is over. What I'm going to do now is to tell you how to multiply and divide using negative numbers. You might not believe it, but it's actually a whole lot easier than adding and subtracting with them. When you multiply or divide, using negative numbers, you simply use the absolute values of the numbers. For the answer, if both numbers had the same signs, the answer will be positive. If the numbers had different signs, the answer will be negative. It's that simple! I'll go over it again. Use the absolute values of the numbers to multiply or divide. If the numbers are both positive or both negative, the answer will be positive. If one was negative and the other was positive, the answer will be negative. Let's work a few problems, and I'll show you how easy it is!

Problem: -10x-5
(Use the absolute values)
10x5=50
(Both numbers were negative so the answer is positive)
Answer: -10x-5=50

Problem: 10/5
(Numbers are positive so we already have the absolute values)
10/5=2
(Both numbers were positive so the answer is positive)
Answer: 10/5=2

Problem: 10x-5
(Use the absolute values)
10x5=50
(One number was positive and the other was negative)
(The answer will be negative)
Answer: 10x-5=-50

Problem: -10/5
(Use the absolute values)
10/5=2
(One number was positive and the other was negative)
(The answer will be negative)
Answer: -10/5=-2

Please go over everything we've talked about, and make sure that you're comfortable with it. If there is anything that you don't feel comfortable with, scroll back up and go over it again. It's ok to use a calculator for some of this stuff, but you won't always have a calculator handy. For that reason, it's best to know how to do things on paper. When you feel you're ready, I've included a 10 question quiz to see how much you've learned. Let's review what we've learned before taking the quiz.

1. Positive infinity represents the largest number that's impossible to ever reach.

2. Negative infinity represents the smallest number that's impossible to ever reach.

3. The greater than sign begins with the wide side.

4. The less than sign begins with the narrow side.

5. With greater than/less than the narrow side always points to the smaller number.

6. With greater than/less than the wide side always faces the larger number.

7. Numbers further left on the number line are always smaller than those to their right.

8. Numbers further right on the number line are always larger than those to their left.

9. The real numbers are numbers that lie anywhere on the number line between positive and negative infinity.

10. The absolute value of a number is its value without the negative sign.

11. |-5| means to get the absolute value of -5, which would be 5.

12. Parenthesis are often used around negative numbers in order to make them easier to read.

13. To add 2 negative numbers, we just add them and remember to put the negative sign on the answer.

14. To add a negative and positive number we write down the number with the largest absolute value. If it's the negative number, the answer will also be negative. Subtract the number with the smaller absolute value from the larger one to get the answer.

15. To subtract when negative numbers are involved, change the sign of what's being subtracted, then change the subtraction sign to addition. From there, just follow the rules of addition.

16. To multiply or divide using negative numbers, just use the absolute values of the numbers. If both signs were the same, the answer will be positive. If the signs were different, the answer will be negative.

17. 10-(-5) is the same as 10+5.

18. 10+(-5) is the same as 10-5.


Image


QUIZ

Question 1:

-5 < 0

Answer:
A:  True
B:  False

Question 2:

-10 > 5

Answer:
A:  True
B:  False

Question 3:

-10 > -5

Answer:
A:  True
B:  False

Question 4:

1 < -5

Answer:
A:  True
B:  False

Question 5:

-2 + 1 = _____

Answer:
A:  -3
B:  3
C:  -1
D:  1

Question 6:

-1 - 2 = _____

Answer:
A:  3
B:  -3
C:  1
D:  -1

Question 7:

5 - 7 = _____

Answer:
A:  -2
B:  2
C:  -12
D:  12

Question 8:

-2 + (-4) = _____

Answer:
A:  2
B:  -2
C:  6
D:  -6

Question 9:

-5 x 2 = _____

Answer:
A:  -10
B:  10
C:  -7
D:  None of the above

Question 10:

-3 / -1 = _____

Answer:
A:  -4
B:  -2
C:  3
D:  -3




~ Other Links ~

PurpleMath - Absolute Value

PurpleMath - Negative Numbers

Math League - Positive and negative numbers

MathsIsFun - Negative Numbers

Infinity and Its Symbol

FunBrain - Line Jumper

Harcourt E-Lab

Harcourt Math Glossary


Back to Learn