Adding and Subtracting Real Numbers

Let's discuss how to add and subtract real numbers.

You can always use a number line to add and subtract real numbers. For instance, if you take a look at the number line below, you can use it to add or subtract numbers. number-line -10 to 10.jpg

If you are given a simple problem such as -3 + 5.

1. Graph -3 on the number line

2. Add 5 means to move the point 5 places to right.

3. After moving 5 places you will see that the answer is 2. number-line -10 to 10 example.jpg

It works the same with subtraction too, but you need to move to the left instead of right.

Example 1: Solve using a number line: -1 - 6

number-line -10 to 10 example 1 lesson 5.jpg

You will see that you end up at -7.

That is a fairly simple procedure to follow:

Plot the first point on the number line
Move your point to the right for addition and to the left for subtraction.

Just be aware that if the second number is negative, you have to reverse direction. To add a negative number you will move left, and to subtract a negative number you will move right.

Example 2: Solve using a number line: $(- 4) + (- 2)$

Since the second number is negative the problem really becomes $- 4 - 2$ .

That means plot -4 and move 2 places to the left.

number-line -10 to 10 example 2 lesson 5.jpg

So, you can see the answer is -6.

Practice problem 1: Solve using a number line:

Click the question prompt below to reveal the answer.

$- 2 - (- 6)$

number-line -10 to 10 PP1 lesson 5.jpg
$- 2 - (- 6) = 4$

The Absolute Value of a number is the distance of the number from zero on a number line. Since it is a distance, it must always be positive because there is no such thing as negative distance. It is impossible to go a negative distance. Even if you walk backwards, you still walk a positive distance.

Now, there is always the argument of a football player that gets tackled behind the line of scrimmage.

football player.jpg

That looks like it hurt and he may well have lost yards on the play, but the number of yards he lost is a positive number. It is amazing what students come up with when it comes to this argument, but what it comes down to is that THERE IS NO SUCH THING AS NEGATIVE DISTANCE. Try to draw a line with a length less than zero. I'll wait...

So to restate: Absolute Value is the distance on a number line from zero to the number, no matter the direction.

Absolute value is written as this: $| x |$ which means the absolute value of $x$ .

If you say that $x = 9$ , then $| 9 | = 0$

If you say that $x = - 9$ , then $| - 9 | = 0$

Adding numbers with the same sign: To add two numbers that have the same sign, keep the sign and add the numbers together.

$4 + 8 = 12$

$- 4 + (- 8) = - 12$

Now that appears pretty straight forward. But what if they do not have the same sign?

Adding numbers with different signs: To add two numbers together with different signs. Use the sign of the number that is larger and then subtract the smaller number from the bigger number. Consider

$- 4 + 8$

8 is larger, so the answer is positive. $8 - 4 = 4$ . Therefore the answer is positive 4. Now consider

$4 + (- 8)$

8 is larger, so the answer is negative. $8 - 4 = 4$ . Remember we keep the sign of the larger number, therefore the answer is $- 4$ .

This works for all real number including decimals, fractions and whole numbers.

Practice problem 2: Find the answer using the addition concepts discussed above:

Click the question prompt below to reveal the answer.

$- 10 + 3$

$- 7$

If two different numbers are the same distance from zero on a number line, they are said to be opposites.

number-line -10 to 10 opposites.jpg

Since 7 and -7 are equidistant from 0 on the number line, they are called opposite numbers.

A number and its opposite number are called additive inverses. To find the sum of a number and its opposite number, you can use the inverse property of addition.

The Inverse Property of Addition: For every real number $x$ , there is an additive inverse $- x$ such that

$x + (- x) = - x + x = 0$

For example: $- 12 + 12 = 0$ and $12 + (- 12) = 0$

Don't let the fancy terminology fool you - this is pretty simple. Important, but simple. All the times in previous math classes when you thought you were subtracting, you were actually adding inverses!

Subtracting Real Numbers: To subtract a real number, add its opposite: $x - y = x + (- y)$

Example 3: Subtract using the inverse property of addition: $2 - 8$

$2 - 8 = 2 + (- 8) = - 6$

Example 4: Subtract using the inverse property of addition: $9 - (- 7)$

$9 - (- 7) = 9 + 7 = 16$

The opposite of -7 is 7. So instead of subtracting -7, we add the opposite. That is, we add 7.

If either example above is unclear, try doing them on a number line.

Practice problem 3:

Click the question prompt below to reveal the answer.

You want to go SCUBA diving. You start at the surface and descend 26 feet. You need to rise 11 feet to go over a reef and then descend again another 52 feet. Finally you rise 31 feet to pet a manta ray that is swimming over your head. What is your depth when you are petting the manta ray?

$0 - 26 = 0 + (- 26) = - 26$

$- 26 + 11 = - 15$

$- 15 - 52 = - 15 + (- 52) = - 67$

$- 67 + 31 = - 36$

you are at a depth of 36 feet