Problem 33
Question
Add. Do not use the number line except as a check. \(19+(-19)\)
Step-by-Step Solution
Verified Answer
The result of adding 19 and -19 is 0.
1Step 1 - Identify the Numbers
Identify the numbers given in the problem. In this case, the numbers are 19 and -19.
2Step 2 - Understand the Operation
Notice that the operation is addition, but one of the numbers is negative. Specifically, we have to add 19 and -19.
3Step 3 - Apply the Rule for Adding Positive and Negative Numbers
When adding a positive number to its negative counterpart, the sum is 0 because they cancel each other out. Mathematically, this can be shown as: \(19 + (-19) = 0\).
4Step 4 - Verify the Solution
Optionally, use a number line to check the correctness of the solution. Starting at 19, moving 19 steps to the left (since we add -19) will land us at 0.
Key Concepts
additionpositive and negative numberscancellation rule
addition
Addition is one of the basic arithmetic operations. It involves combining two or more numbers to get a total. Understanding how to add both positive and negative numbers is essential for solving many types of math problems.
When adding, it's important to consider the signs of the numbers involved. If both numbers are positive, you simply combine their values to get the total. For example, \(5 + 3 = 8\).
If both numbers are negative, you still combine their absolute values, but the result will also be negative. For example, \( -2 + (-3) = -5\).
However, things get slightly more complex when dealing with one positive number and one negative number. This brings us to our next core concept.
When adding, it's important to consider the signs of the numbers involved. If both numbers are positive, you simply combine their values to get the total. For example, \(5 + 3 = 8\).
If both numbers are negative, you still combine their absolute values, but the result will also be negative. For example, \( -2 + (-3) = -5\).
However, things get slightly more complex when dealing with one positive number and one negative number. This brings us to our next core concept.
positive and negative numbers
Working with positive and negative numbers is a crucial skill in mathematics. Positive numbers are greater than zero and are often written without a sign. Negative numbers, on the other hand, are less than zero and are indicated with a minus sign (-).
When adding a positive number to a negative number, you should think about it like taking steps forward and backward. For example, consider the problem \(19 + (-19)\). Here, the positive and negative numbers are equal in magnitude.
Simply put, they balance each other out.
This concept leads directly to our next important rule in arithmetic.
When adding a positive number to a negative number, you should think about it like taking steps forward and backward. For example, consider the problem \(19 + (-19)\). Here, the positive and negative numbers are equal in magnitude.
Simply put, they balance each other out.
This concept leads directly to our next important rule in arithmetic.
cancellation rule
The cancellation rule is a handy trick for simplifying addition problems involving both positive and negative numbers. When a positive number and its negative counterpart are added, they cancel each other out. This means their sum is zero.
For example, in the problem \(19 + (-19)\), the positive 19 and the negative 19 are opposites. They cancel each other out, leaving you with zero.
Put simply, \( a + (-a) = 0 \).
This rule helps keep calculations straightforward and enables you to solve problems quickly, without always needing a number line or other tools.
It’s a powerful and simple rule to remember for your future math problems!
For example, in the problem \(19 + (-19)\), the positive 19 and the negative 19 are opposites. They cancel each other out, leaving you with zero.
Put simply, \( a + (-a) = 0 \).
This rule helps keep calculations straightforward and enables you to solve problems quickly, without always needing a number line or other tools.
It’s a powerful and simple rule to remember for your future math problems!
Other exercises in this chapter
Problem 33
Simplify. $$ 9 \div 3+16 \div 8 $$
View solution Problem 33
Change the sign. (Find the opposite.) $$ -1 $$
View solution Problem 33
Multiply. $$ -\frac{3}{8} \cdot\left(-\frac{2}{9}\right) $$
View solution Problem 33
Write decimal notation for each number. $$ -\frac{1}{2} $$
View solution