Problem 43 - Elementary Algebra

Question

Find the sums. \(-1+(-1)\)

Step-by-Step Solution

Answer

Answer: The sum of -1 and -1 is -2.

Step 1: Identify the numbers

We are given two numbers to find the sum: \(-1\) and \(-1\).

Step 2: Add the absolute values of the numbers

Ignore the negative signs and add the absolute values of the numbers: \(1 + 1 = 2\).

Step 3: Place a negative sign in front of the result

As both numbers are negative, place a negative sign in front of the sum obtained in Step 2: \(-2\). The sum of \(-1\) and \(-1\) is \(-2\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers that are less than zero. They are typically represented with a minus sign in front of them, like \(-1\), \(-2\), and so on. Negative numbers are often used to describe values below a certain point, such as below sea level or temperatures below freezing.
When working with negative numbers, it's essential to remember:

• They are the opposite of positive numbers.
• A negative number is always less than zero.
• When you add or subtract them, the rules for arithmetic slightly differ from positive numbers.

In our example, \(-1\) is a negative number because it is less than zero. Adding two negative numbers requires careful attention to their absolute values and signs.

Exploring Absolute Value

The absolute value of a number represents its distance from zero on the number line, regardless of direction. It is always a positive number or zero.
For example, the absolute value of both \(+3\) and \(-3\) is \(3\) because each is three units away from zero. The absolute value helps us understand the size of the number without considering whether it is positive or negative.
When calculating the sum of negative numbers, we initially focus on their absolute values:

• Ignore the sign to find the absolute value.
• Add the absolute values together.
• In our case, the absolute values of \(-1\) and \(-1\) are each \(1\), so we added them to get \(2\).

Understanding absolute value simplifies operations involving integers, especially negative numbers.

Mastering Integer Arithmetic

Integer arithmetic involves performing calculations using whole numbers, which can be positive, negative, or zero. When adding integers, it's crucial to keep these rules in mind:

• Same signs: Add the absolute values and keep the common sign.
• Different signs: Subtract the smaller absolute value from the larger one, and use the sign of the larger absolute value.
• Zero: Adding or subtracting zero does not change the number.

In our example \(-1 + (-1)\), both numbers have the same sign (negative). We add their absolute values \(1 + 1\) to get \(2\) and apply the negative sign back, yielding \(-2\).
Working with integer arithmetic effectively requires practice and understanding these rules to manage positive and negative values smoothly.