Add and Subtract Positive and Negative Integers
A strong foundation in Number Properties is essential for understanding various mathematical concepts. It helps us predict outcomes, comprehend equations, and verify our work. In this Student Guide, we will focus on Positive and Negative Numbers. We will begin by explaining the number line and what integers are and then dive into adding and subtracting integers.
Positive and negative numbers play a significant role in our daily lives. While positive numbers are more commonly used while counting or performing operations like addition, subtraction, multiplication, and division, negative numbers are equally important. They frequently appear in functions, other algebraic concepts, and even in situations outside school.
Before we proceed, let's clearly define the word integer. In mathematics, an integer refers to a positive or negative whole number, including zero. For this guide, we will only discuss integers.
Some other important terms we need to know for this lesson are opposite values and absolute values. Opposite values have the same number but with opposite signs. For example, the opposite value for (-5) is +5. A number’s absolute value is the distance from zero (0) and is represented as a positive value. For example, the absolute value of (-5) is 5, and the absolute value of 5 is also 5.
The Number Line
Now, let's explore how integers are represented on a number line. Here is a visual representation of a number line where different values are defined.
As we move left on the number line, the numbers decrease in value, while moving right indicates an increase. It's important to note that zero (0) is neither positive nor negative. Hence, whenever we traverse the number line, we must count zero before transitioning from negative to positive or vice versa. Take a look at Example 1 and Example 2 to see how we can determine whether a number is greater than or less than another number.
Once we are comfortable with the number line and comparing different numbers, we can move on to understanding how these numbers interact with each other. These interactions occur through various mathematical operations.
To aid us in solving addition and subtraction problems, we can utilize the number line.
Let's now look at specific scenarios and the rules for adding and subtracting positive and negative numbers.
Adding Integers
Examples 3 and 4
Positive Number Plus a Positive Number (Adding the Same Sign)
To add two positive numbers, we start at zero and move to the first number on the number line. Then, we advance to the right by the number of units indicated by the second number.
Using Example 3 shown and the number line, we started from zero (0) and moved to the right 5 units. Then to add 7, we moved 7 more units to the right to end at the number 12.
Negative Number Plus a Negative Number (adding The Same Sign)
To add a negative number, we move left on the number line. Thus, we start at zero, move to the first number, and then proceed left by the number of units specified by the second number.
Using Example 4 shown and the number line, we started from zero (0) and moved to the left 5 units. Then to add (-7), we moved 7 more units to the left to end at the number (-12).
To solve this mathematically, we can add the absolute values of the numbers and include a negative sign in the final answer.
Positive Number Plus a Negative Number (Adding Different Signs)
Examples 5 and 6
When adding numbers with different signs, the sum will be their difference.
To visualize this on the number line, we move towards the first number indicated in the problem and then shift left by the number of units specified by the second number.
Using Example 5 shown and the number line, we started from zero (0) and moved to the right 5 units. Then to add (-7), we move 7 units to the left to end at the number (-2).
Mathematically, when we add numbers with different signs, we find the difference between the two numbers, and the answer's sign is the same as the number with the greater absolute value.
Negative Number Plus a Positive Number (Adding Different Signs)
Adding numbers with different signs follows the same principle as explained above.
On the number line, we move towards the first number and then proceed right by the number of units specified by the second number.
Using Example 6 shown and the number line, we started from zero (0) and moved to the left 5 units. Then to add 7, we move 7 units to the right to end at the number 2.
Mathematically, when we add numbers with different signs, we find the difference between the two numbers, and the answer's sign is the same as the number with the greater absolute value.
Now, let's shift our focus to subtracting integers.
Subtracting Integers
Any Number Minus a Positive Number
Examples 7 and 8
When subtracting two numbers, we have to pay attention to the order of the problem.
On the number line, we move from zero toward the first number. Then to subtract, we move to the left that many units.
Using Example 7 shown and the number line, we started from zero (0) and moved to the right 7 units. Then to subtract 5, we move 5 units to the left to end at the number 2.
Mathematically, to subtract a number, we keep the first number, change the minus sign to a plus sign, and change the sign of the second number (add the opposite).
For example, 7 – 5 changes to 7 + (-5), and then we can use the addition rules we learned above.
Any Number Minus a Negative Number
Subtracting negative numbers requires careful attention to operations and calculations, and an important rule comes into play: when we subtract a negative number, we change it to an addition problem.
Subtracting integers is equivalent to adding the opposite.
Using Example 8 shown and the number line, we started from zero (0) and moved to the left 5 units. Then to subtract (-7), we move 7 units to the right to end at the number 2.
It's crucial to remember that the order of the numbers affects the result. For example, changing the order in the problem -7 - (-5) results in -7 + 5, which equals (-2).
Conclusion
By incorporating these rules and concepts, you can confidently add and subtract integers. If you are feeling ready, head to our next post about the rules for multiplying and dividing integers.
