 In this lesson, we will learn about adding integers or signed numbers. Using the number line and idea of absolute value, we will see that the rules used to add them are simply an extension of the rules of basic arithmetic.

Absolute Value and the Number Line

First, we need to understand absolute value. Think about a walk from your house in a straight line without regard for the direction you are walking.

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Then, measure how far you walked. If you think of your house as being the point zero, then the measurement would be the absolute value of the distance you walked. Absolute value is the distance from zero without regard to sign or direction. Since the absolute value is distance, the absolute value of a number is always positive.In mathematical notation, we represent the absolute value of a number with two vertical bars on either side of the number. For example, the absolute value of 5 is written like this | 5|.To understand absolute value, a number line is often helpful. Integers With Like Signs

To add two integers with the same sign, let’s start with what we already know. What is the value of 3 + 2? We know that 3 + 2 = 5. Now, let’s see how this would be visualized on the number line. Basically, the two arrows point in the same direction because both 3 and 2 have a positive sign. The final arrow ends up at the sum of the integers.Now, let’s look at a problem with two negative integers. What is -3 + -2? Similar to the sum of the positive integers, the arrows for the sum of the negative integers both point in the same direction. Now, however, since they both have a negative sign, they both point in the negative direction.

The final arrow shows the sum of the integers to be -5.Notice that both the (absolute value) of | 5 | = 5 and the (absolute value) of | -5 | = 5. Why? The answer is that even though the sums were on different sides of zero, they were both the same distance from zero. This idea is behind the first rule we will learn for adding integers.

• To add two integers with the same sign, add their absolute values. The answer will have the same sign as the integers.

We do this every day to add positive integers. For example, we know that 4 + 7 = 11. We didn’t learn to go through these steps:

1. | +4 | = 4 and |+ 7| = 7
2. +4 +7 = +11

Yes, that is actually what is happening behind the scenes to get the final answer.Let’s try a problem with two negative integers.

What is the value for = -4+ -2?

1. First, find the absolute value of each integer. | -4 | = 4 and | -2 | = 2
2. Second, add the absolute values. 4 + 2 = 6
3. Third, give the answer the common sign of the integers you added. -4 + -2 = -6. The answer is -6.

While some examples can be demonstrated with number lines, some cannot.

You should not depend on number lines to add integers. Number lines are handy for visualizing the process of adding integers. Once you understand the general idea behind adding integers using the number line, you should rely on the rules to complete the calculation.

Here is another example. What is -52 + -63?

1. Find the absolute value for each integer: | -52 | = 52 and | -63 | = 63
2. Add the absolute values: 52 + 63 = 115
3. Give the answer the common sign of the integers you added: So, -52 + – 63 = -115

To add integers with different signs, let’s again start with a problem we already know how to do. For example, we know that 5 – 3 = 2.

You might say, ‘This is not adding, this is subtracting!’ In mathematics, subtraction can also be seen as adding a negative. If I were to write the problem as 5 + (- 3) = 2, we can now see that this is really the same as adding a positive integer and a negative integer. It also shows that -3 really is the same as + ( -3).Showing this problem on the number line may be helpful. On the number line, one arrow points to five in the positive direction. The second arrow takes 3 away from five by pointing back in the negative direction. The final answer is 2.What would -5 + 3 look like on the number line? One arrow points to five in the negative direction.

The second arrow points in the positive direction 3 units, taking the final answer back towards positive. The final answer is -2.Notice that the answer to -5 + 3 was almost the same as 5 + ( – 3) except for the sign of the final answer. We know from arithmetic that 5 – 3 is a difference. The following rule will demonstrate that the addition of numbers of unlike signs always involves finding a difference.

• To find the sum of two numbers with unlike signs, subtract their absolute values. Give the difference the sign of the number with larger absolute value.

To demonstrate using the problem 5 + (-3), the steps are as follows:

1. First, | 5 | = 5 and | -3 | = 3
2. Next, 5 – 3 = 2, meaning the difference is 2
3. Finally, since +5 has a larger absolute value, the difference will be positive. Thus, 5 – 3 = +2

To demonstrate using the problem -5 + 3,

1. | -5 |= 5 and | 3 | = 3
2. 5 -3 = 2
3. Since -5 has larger absolute value, the difference will be negative. Thus, -5 + 3 = -2

The order of the numbers in the problem will not affect the answer, either. This means, for example, that 5 + ( -3 ) has the same answer as -3 + 5.

In both cases, the 5 is positive and the 3 is negative.Let’s look at another example. What is 45 + ( -61) ?

1. | 45 | = 45 and | -61 | = 61
2. 61 – 45 = 16
3. Since -61 has larger absolute value, the answer is negative.

45 + (-61) = -16.

Lesson Summary

Let’s review! Absolute value plays an important role in adding integers. The number line is useful for visualizing the addition of integers but cannot take the place of the rules for adding integers.

To add integers with like signs we add absolute their values, then determine the sign of the answer. To add integers with unlike signs we subtract their absolute values, then determine the sign of the answer. Although we learn these rules for integers, they also apply to every day arithmetic addition.

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