Problem 10 - Prealgebra

Question

Subtract. $$2-(-5)$$

Step-by-Step Solution

Answer

The result is 7.

Step 1: Identify the Signs

The expression given is \( 2 - (-5) \). Notice that you are subtracting a negative number, \(-5\), from 2.

Step 2: Understand Subtraction Rule

When you subtract a negative number, it is equivalent to adding its positive counterpart. Therefore, \( 2 - (-5) \) becomes \( 2 + 5 \).

Step 3: Perform the Addition

Add the numbers: \( 2 + 5 = 7 \).

Key Concepts

Integers

Integers are whole numbers that can be either positive, negative, or zero. They do not include fractions or decimals. Understanding integers is crucial when dealing with operations like addition, subtraction, and more. Integers can be visualized on a number line, where zero is at the center, positive integers extend to the right, and negative integers extend to the left.

• Positive integers are numbers greater than zero, like 1, 2, 3, etc.
• Negative integers are less than zero, like -1, -2, -3, etc.
• Zero is a neutral integer, neither positive nor negative.

When dealing with integers in mathematical problems, the sign is crucial as it tells you in which direction to move on the number line. In our exercise, both 2 and -5 are integers, making our operation a task of understanding how these integers interact under subtraction.

Addition

Addition is one of the basic arithmetic operations where two or more numbers are combined to form a sum. The process can involve positive numbers, negative numbers, or both, like in this example. With integers, addition is straightforward:

• When adding two positive numbers, the result is positive.
• When adding two negative numbers, the result is negative.
• If adding a positive and a negative number, you calculate the difference and give the sign of the larger absolute value.

In the context of our example, the step of converting subtraction to addition, i.e., changing \( 2 - (-5) \) to \( 2 + 5 \), uses the concept of 'adding the opposite.' In essence, when you subtract a negative number, it's the same as adding the positive equivalent of that number. This is why we switched from subtracting \(-5\) to adding 5.

Negative Numbers

Negative numbers are essential for understanding a full spectrum of integer arithmetic. They represent quantities below zero. In mathematics, we're often faced with operations that involve negative numbers, and understanding their behaviors is key. Here are a few important points:

• Negative numbers have a minus (-) sign before them, indicating they are less than zero.
• When subtracting a negative number, you are essentially performing an addition of its opposite. This is why \(2 - (-5)\) results in \(2 + 5\).
• On a number line, moving to the left indicates subtraction, while moving to the right indicates addition.

In subtraction involving negative numbers, like our exercise \( 2 - (-5) \), the operation turns into an addition because subtracting a negative is equivalent to adding its positive counterpart. This is a unique characteristic of negative numbers, making them crucial for various mathematical solutions.