Problem 8 - Fundamentals of Mathematics

Question

Perform the indicated subtractions. $$ 8-(-10) $$

Step-by-Step Solution

Answer

The result is 18.

Step 1: Identify the Operation

The given expression is a subtraction problem, specifically \(8 - (-10)\). Here, we are supposed to subtract \(-10\) from 8.

Step 2: Understand the Double Negative

Subtracting a negative number can be counter-intuitive. In mathematics, subtracting a negative number is the same as adding its positive counterpart. So, \(-(-10)\) becomes \(+10\).

Step 3: Transform the Expression

Based on Step 2, transform the expression \(8 - (-10)\) into \(8 + 10\).

Step 4: Perform the Addition

Now, simply add the two positive numbers: \(8 + 10 = 18\).

Key Concepts

Subtraction of Integers

Subtraction of integers can sometimes be confusing, but understanding it deeply makes the process easy. When we talk about subtraction, we're essentially talking about removing or "taking away" a certain number from another. For example, subtracting 10 from 8, written as \( 8 - 10 \), means we reduce 8 by 10. This results in \( -2 \).

In the exercise, however, we're working with \( 8 - (-10) \). It's crucial to distinguish between directly subtracting a positive integer and dealing with subtracting a negative integer. When a negative integer is involved, subtraction doesn't quite work the same way as with positives.

Addition of Integers

Addition of integers is generally straightforward. You combine two numbers to get their sum. Let's think about what happens when you change subtraction of a negative integer to addition. This is where subtraction and addition meet in integer arithmetic.

In the exercise \( 8 - (-10) \), this operation transforms into \( 8 + 10 \). So you just add both numbers together. The reason subtraction turns into addition is due to the double negative concept. This helps simplify calculations greatly. As a quick example, when performing \( 8 + 10 \), the outcome is \( 18 \). The key is to notice that we’re adding when we see a double negative involved.

Double Negative Concept

The double negative concept is a fundamental part of integer arithmetic and helps clarify various mathematical operations. When you come across two negatives, as in subtracting a negative like \( 8 - (-10) \), these two negatives cancel each other out. This results in an addition problem.

To visualize this, consider \(-(-a) = a\). This means if you have \(-(-10)\), it equals \( +10 \).

• Double negatives result in addition.
• It's a way of thinking of subtraction in reverse.
• The opposing negative signs negate each other's effect.

Understanding the double negative concept clears up why and how we can transform subtraction problems into addition, thus simplifying the arithmetic process for any integer-based calculations.