Problem 34 - Fundamentals of Mathematics

Question

For the following 6 problems, rewrite each number in simpler form. $$ 7-(-3) $$

Step-by-Step Solution

Answer

The result is 10.

Step 1: Identify the Signs

First, observe the operation: subtraction of a negative number. This is written as \(7 - (-3)\).

Step 2: Recognize Double Negative

Understand that subtracting a negative number is equivalent to adding its positive counterpart. Thus, \(7 - (-3) = 7 + 3\).

Step 3: Perform the Addition

Add the two numbers: \(7 + 3 = 10\). So, the expression \(7 - (-3)\) simplifies to 10.

Key Concepts

Addition of Integers

Adding integers is a fundamental part of math, involving bringing together numbers to find a sum. When you add two integers, you are essentially collecting their values to create a new total.
Consider two positive numbers. Adding them is straightforward — you simply sum their values.

• For example, adding 3 and 5 gives 8, written as: \(3 + 5 = 8\).
• If the numbers have different signs, like adding 7 and -3, you actually perform a subtraction using the larger number's sign. In this case, \(7 + (-3) = 4\), since 7 is larger, the result is positive.

Utilizing the properties of integers can aid problem-solving. For instance, if you encounter subtracting a negative, it transforms into an addition. So, encountering \(7 + 3\) results in a straightforward addition because both numbers are positive, resulting in a sum of 10.

Subtraction of Integers

Subtraction of integers involves removing the value of one integer from another. This operation can become complex when negative numbers are involved.
With positive integers, you remove one number from another. For instance:

• \(10 - 6 = 4\), simple subtraction of smaller from larger value gives 4.

With negative numbers, subtraction becomes less intuitive. In the case of subtracting a negative, it translates to the addition of its positive counterpart. Consider,

• \(7 - (-3)\) becomes \(7 + 3\) because the negative sign before -3 flips it to positive. This is a case of "subtracting a negative is like adding a positive."

This can be visualized as moving to the right on a number line, thereby making the final value greater instead of smaller.

Negative Numbers

Negative numbers are numbers less than zero and are represented with a minus (-) sign. They are important in various contexts like temperatures, altitudes below sea level, and bank account overdrafts.
Understanding negative numbers includes knowing their rules in operations:

• Addition: When adding two negative numbers, their absolute values are added, resulting in a more negative number, e.g., \(-2 + (-3) = -5\).
• Subtraction: Subtracting a negative number is like adding the opposite. \(-5 - (-2)\) is the same as \(-5 + 2\), which equals \(-3\).
• Multiplication and Division: Negative × Negative = Positive, while Negative × Positive = Negative.

Negative numbers provide an essential balance to positive numbers, offering a complete picture of the number line and allowing for full arithmetic operations.