Problem 72 - Algebra 1

Question

Find the sum. $$5+(-3)+(-5)$$

Step-by-Step Solution

Answer

The sum of the numbers \(5, -3, -5\) is \(-3\).

Step 1: Evaluate the sum

First step is to add the three given numbers. This is done by knowing that adding a negative number is the same as subtracting its positive counterpart. So 5 + (-3) is the same as 5 - 3.

Step 2: Continue Adding Numbers

Then continue the calculation by adding the result in step 1 with the next number as mentioned in the exercise. So the result of step 1 i.e. 2, added to -5 is the same as 2 - 5.

Step 3: Final solution

Now, obtain the final answer by subtracting 5 from 2 i.e. \(2 - 5 = -3\).

Key Concepts

Integer Addition

Integer addition is a fundamental concept in arithmetic where two or more integers are combined to produce a sum. Integers can be both positive and negative, and the rules for adding them are straightforward, but may require practice to become second nature. For example, when we have both positive and negative numbers, like in the equation
\[5 + (-3) + (-5)\],
we approach the addition step by step. We can start by adding any two integers. If we add 5 and (-3), which is same as subtracting 3 from 5, we get 2. The process is known as integer addition because we are adding whole numbers that have no fractional or decimal components.

Subtracting Positive Counterparts

When it comes to adding negative numbers, one effective strategy is to think of it as subtracting the positive counterpart. The 'positive counterpart' is the positive version of any negative number. In our given problem, we have to add (-3) to 5, which can be interpreted as subtracting 3 from 5.

In mathematical terms, this is written as:
\[5 + (-3) = 5 - 3\]
After subtracting, we get 2. This technique allows us to simplify the process of adding negative numbers by transforming it into a subtraction problem, which many find more intuitive.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. These are the building blocks of most mathematical calculations. Each operation follows specific rules, especially when involving negative numbers. When consecutive arithmetic operations are required, as seen in the exercise \[5 + (-3) + (-5)\], after the first addition step, we proceed by adding the next number, which is another negative number (-5).

At this stage, we have:
\[2 + (-5)\],
which is the same as \[2 - 5\]. The essence of arithmetic operations is understanding how to combine numbers in sequence to arrive at the correct result.

Negative Number Arithmetic

Negative number arithmetic is a part of integer arithmetic that deals explicitly with the addition, subtraction, multiplication, and division of negative numbers. Adding negative numbers, as we've seen, can be thought of as subtraction. Conversely, subtracting a negative number becomes addition due to the double negative rule.

For instance, continuing from our previous example, after obtaining 2 from our first addition, we then address the addition of -5 to it, yielding:
\[2 + (-5) = 2 - 5\],
which results in -3. It's helpful to remember that a 'minus' sign next to another 'minus' sign turns the operation into addition, while a 'plus' sign next to a 'minus' sign indicates subtraction.