Problem 30 - Fundamentals of Mathematics

Question

Find the sums in the following 27 problems. If possible, use a calculator to check each result. $$ -22+(-1) $$

Step-by-Step Solution

Answer

The sum of \(-22 + (-1)\) is \(-23\).

Step 1: Identify the components

The expression we need to evaluate is \(-22 + (-1)\). Here, \(-22\) is the first number and \(-1\) is the second number that needs to be added.

Step 2: Understand the operation

We need to add two negative numbers: \(-22\) and \(-1\). When adding negative numbers, you add their absolute values and keep the negative sign in the result.

Step 3: Calculate the sum

Add the absolute values of the numbers together. The absolute value of \(-22\) is 22, and the absolute value of \(-1\) is 1. The sum is 22 + 1 = 23.

Step 4: Apply the sign

Since both numbers were negative, the sum will also be negative. Thus the answer is \(-23\).

Step 5: Verification with a calculator

Use a calculator to check: enter \(-22 + (-1)\) and verify that the result is indeed \(-23\).

Key Concepts

Absolute Values

Absolute values refer to the non-negative value of a number without considering its sign. It indicates how far a number is from zero on a number line, irrespective of its direction. To denote absolute values, we use vertical bars: for a number \(-22\), its absolute value is written as \(|-22| = 22\). Similarly, for \(-1\), its absolute value is \(|-1| = 1\).

When adding negative numbers, as in the expression \(-22 + (-1)\), the first step is to consider their absolute values.

• The absolute value of \(-22\) is 22.
• The absolute value of \(-1\) is 1.

After determining the absolute values, we sum them to get 23. It's useful to keep in mind that the absolute value only helps us compute the magnitude of the numbers during addition, and we apply the appropriate sign based on the original numbers, which in this case were both negative.

Negative Integers

Negative integers are numbers that are less than zero. On the number line, they are found to the left of zero. Common examples, such as \(-1, -22\), exist in everyday situations like temperature below freezing or when indicating a loss, as in debt.

When adding negative integers like \(-22\) and \(-1\), the operation differs slightly from adding positive numbers:

• Adding two negative integers involves summing their absolute values.
• Once added, apply a negative sign to the result as indicated by their shared negative nature.

In our example \(-22 + (-1)\), after summing the absolute values (22 and 1) to get 23, we apply a negative sign to this result. This results in an answer of \(-23\). Understanding how to handle operations with negative integers is crucial for solving problems accurately.

Calculator Verification

After performing arithmetic operations manually, especially with negative numbers, it's a good practice to verify your results using a calculator. Modern calculators are equipped to handle both positive and negative integers seamlessly.

In our example of adding \(-22 + (-1)\), here's how you can verify the result:

• Enter the first negative number, \(-22\).
• Add the second negative number by inputting it as \(-1\).
• Hit the equals button to receive the calculation.

The calculator should display \(-23\). This corroborates your manual computation, confirming that both processes agree. Note that using the calculator not only reassures you of the accuracy but also helps reinforce your understanding of the concept of adding negative integers.