Problem 9 - Fundamentals of Mathematics

Question

Find the sums. \(4+(-3)\)

Step-by-Step Solution

Answer

The sum is 1.

Step 1: Understand the Problem

The expression we need to solve is the sum of two numbers: 4 and -3.

Step 2: Identify Positive and Negative Numbers

In the expression, 4 is a positive number, and -3 is a negative number. We need to consider their signs when adding them together.

Step 3: Calculate the Sum of the Numbers

When adding a positive number and a negative number, you actually subtract their absolute values. So, the calculation becomes: 4 - 3.

Step 4: Perform the Subtraction

Subtract the smaller absolute value from the larger one: 4 - 3 = 1.

Step 5: Consider the Sign of the Result

Since the larger absolute value is positive (4), the result will be positive. Thus, 4 + (-3) = 1.

Key Concepts

Positive and Negative Numbers

Positive and negative numbers are fundamental in math, especially in arithmetic involving integers. Positive numbers are greater than zero and are located to the right of zero on the number line. Examples are 1, 2, 3, and so on. Negative numbers are less than zero and are found to the left of zero on the number line. Examples of negative numbers include -1, -2, -3, etc.
When working with these numbers, their signs play a crucial role in operations like addition and subtraction. For example:

• Adding two positive numbers results in a positive sum, such as 2 + 3 = 5.
• Adding two negative numbers results in a negative sum, like -2 + -3 = -5.
• Adding a positive number with a negative number involves finding the difference between their absolute values.

Understanding the difference between positive and negative numbers and their respective operations is key to mastering integer arithmetic.

Absolute Value

Absolute value is a concept that refers to the distance a number is from zero on the number line, regardless of direction. It is always expressed as a non-negative number.
For instance:

• The absolute value of 4 is 4, written as \(|4| = 4\).
• The absolute value of -3 is 3, written as \(|-3| = 3\).

Absolute value is used to simplify operations like addition and subtraction of integers, especially when dealing with numbers of opposite signs.
In the example of adding 4 and -3, we consider the absolute values: 4 and 3. Knowing these values helps us to determine the result of the calculation by subtracting the smaller absolute value from the larger one. This approach allows us to correctly compute sums and differences.

Integer Arithmetic

Integer arithmetic involves performing operations such as addition, subtraction, multiplication, and division with whole numbers, which can be either positive or negative.
To successfully manage arithmetic with integers, it helps to understand and apply their properties:

• Addition and Subtraction: When adding a positive and a negative integer, subtract the smaller absolute value from the larger one and take the sign of the larger absolute value.
• Multiplication: The product of two positive or two negative integers is positive, while the product of a positive and a negative integer is negative.
• Division: Similar to multiplication, dividing two integers with the same sign yields a positive result, whereas dividing integers with different signs results in a negative outcome.

In our specific exercise, adding 4 (a positive integer) to -3 (a negative integer) required us to subtract their absolute values to find the result, which was positive owing to the larger absolute value belonging to the positive integer. This demonstrates the intricacies of integer arithmetic and the importance of understanding how signs affect operations.