GED Mathematical Reasoning: Number Line and Signed Numbers I – Adding

 

Adding with signed numbers

  • When adding numbers with the same sign, add the number parts and use the common sign.
  • When adding numbers with different signs, take the difference of the number parts and use the sign of the number having the larger number part.

 

Example 1

Add:  3 + 1

 

It probably took you just a split second to calculate the answer to our first example when you saw it appear on the screen. Three plus one is equal to four.

Let’s take a more visual approach to this problem, which will help as we dive into the world of signed numbers. Recall that all real numbers, signed numbers included, can be represented visually on what we call a number line. A number line is a line – made up of points – that stretches forever in both directions. And every signed number has a place on that line.

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On a basic number line, zero makes a good “center” value. To the right of zero are the positive numbers and to the left of zero are the negative numbers. Zero is considered to be neither positive nor negative.

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Notice that each negative number is preceded by what looks like a subtraction sign. We call that sign a “negative” sign. All negative numbers have a negative sign in front. Sometimes positive numbers will be preceded by a “plus” sign, which we call a “positive” sign, but most of the time a positive number will have no sign in front. If a number has no sign in front, we assume it is positive.

Notice that each positive number labeled on the number line has a negative counter-part. For this reason, we sometimes refer to the negative concept as “opposite.” That will be very important when we discuss how to subtract with signed numbers.

algeb2A signed number gives us two pieces of information.

The sign tells us the direction from zero – a negative number implies a left-ward direction whereas a positive number implies a right-ward direction. And the number part tells us the distance from zero.

Consider the number ‘negative three.’  The negative sign tells us that this signed number lies to the LEFT of zero. So the number -3 lies three units left of zero.

For another example, consider the number ‘positive 6.’ This number lies six units to the right of zero.

Let’s go back to our example. To add ‘positive 3’ plus ‘positive 1’, start at zero on the number line and move three units to the right – in the positive direction. Then move one more unit to the right – in the positive direction. Notice where we land: on the answer, positive 4.

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If you’re thinking that using a number line isn’t all that efficient for adding and subtracting signed numbers, you’re right. Especially when the numbers are large or there are more than two numbers in the calculation, using a number line won’t be the best strategy to use for finding the answer. So let’s discuss another approach for adding and subtracting signed numbers.

When we added numbers with the SAME sign, the answer was the same as if we added the number parts and used the common sign.

3 + 1 = 4

3 + 1 is 4 and the answer is positive

 

Example 2

Add:  (-5) + 1

 

Sometimes we use parenthesis around signed numbers to emphasize that the number is either positive or negative. In example two, the parenthesis around the ‘negative five’ is to emphasize that the 5 is negative.

Visually, using a number line, this problem tells us to start at zero and move 5 units to the LEFT – in the negative direction. Then, we’ll turn around and move one unit to the RIGHT – in the positive direction. We land on the answer of -4.

algeb4When we added numbers with DIFFERENT signs, the answer was the same as if we found the difference between the numbers and used the sign of the number with the larger number part.

(-5) + 1 = -4

difference between 5 and 1 is 4 and we used the sign of 5

 

Example 3

Add:  (-2) + (-4)

 

Visually, using a number line, this problem tells us to start at zero and move 2 units to the LEFT – in the negative direction. Then, we’ll continue moving 4 more units to the left – in the negative direction. We land on the answer of -6.

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Both -2 and -4 have the SAME sign, the answer was the same as if we added the number parts and used the common sign.

(-2) + (-4) = -6

2 + 4 is 6 and the answer is negative

 

Example 4

Add:  12 + (-15)

 

We are adding two numbers with different signs, so we will first take the difference between 12 and 15, which is 3. And since 15 is the larger number part and it’s negative, our final answer will be negative.

12 + (-15) = -3

As a side note, some people find it helpful to think of money going in and out of a bank account when adding with signed numbers, where a positive number represents a deposit and a negative number represents a withdrawal.

In this example, consider a bank account with a zero balance. If $12 is deposited and then $15 is withdrawn, the balance would be negative $3.

 

Example 5

Add:  -23 + (-25)

 

Since both numbers are negative, we will add the number parts and use the common negative sign. 23 plus 25 is equal to 48, and since both numbers are negative, our final answer is ‘negative 48.’

-23 + (-25) = -48

To relate this example to money in a bank account, it’s as if we start with a zero balance and withdraw $23. If we withdraw an additional $25, we are even more in debt making our balance:  -$48

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