Problem 35 - Fundamentals of Mathematics

Question

Rewrite each expression in simpler form. $$ 1-(-18) $$

Step-by-Step Solution

Answer

The expression simplifies to 19.

Step 1: Rewrite the Expression

Start with the expression given in the exercise, which is the subtraction of a negative number: \[ 1 - (-18) \]

Step 2: Apply the Rule for Subtracting Negative Numbers

Recall the rule: subtracting a negative number is equivalent to adding the corresponding positive number. So, apply this rule to the expression:\[ 1 - (-18) = 1 + 18 \]

Step 3: Perform the Addition

Now, simply add the two numbers together:\[ 1 + 18 = 19 \]

Step 4: Conclusion: Simplified Expression

Thus, the expression simplifies to the following:\[ 19 \]

Key Concepts

Subtracting Negative Numbers

In mathematics, subtracting negative numbers may seem a bit tricky at first. But with a bit of explanation, it becomes quite simple. When we subtract a negative number, it is similar to adding a positive number. This is because subtracting a negative is like reversing the subtraction operation.
For instance, if you have the expression \( x - (-y) \), it can be changed to \( x + y \). Hence, \( 1 - (-18) \) becomes \( 1 + 18 \).
This change simplifies calculations by converting a potentially confusing operation into something more straightforward: addition. Understanding this rule helps not only in simplifying expressions but also in solving equations and other mathematical problems.

• Think of subtraction as moving left on a number line and addition as moving right.
• Subtracting a negative is like moving double right — a subtraction and an additional positive move forward.

Addition

The next step after rewriting the expression, \( 1 + 18 \), involves simple addition. Addition is one of the basic operations in arithmetic, where we combine two numbers to get their sum. It's a foundational concept that you'll encounter across all levels of mathematics.
Adding numbers is straightforward. For instance, to find \( 1 + 18 \), you simply start with 1 and add 18 more.
Here’s a straightforward way to think about it:

• Start at the number 1 on a number line.
• Move 18 steps to the right.
• You’ll arrive at 19, which is the sum.

Grasping addition is crucial since it serves as the building block for more complex operations. It also plays a key role in everyday calculations and financial transactions.

Arithmetic Simplification

Arithmetic simplification involves taking an expression and reducing it to its simplest form. This is achieved through various operations, such as addition, subtraction, multiplication, and division, to make the expression easier to understand and work with.
In the original problem, we started with \( 1 - (-18) \). By applying the rule for subtracting negative numbers, we transformed it into an addition problem: \( 1 + 18 \).
Simplifying expressions helps in:

• Reducing errors by making calculations more straightforward.
• Enabling easier manipulation of equations.
• Providing clear and concise answers.

Think of simplification as cleaning up an equation to reveal its simplest and most elegant form. In our example, simplifying gave us a straightforward answer of 19.