Problem 22 - Introductory and Intermediate Algebra for College Students 4th

Question

Find each sum without the use of a number line. $$-7+3$$

Step-by-Step Solution

Answer

The sum of -7 and 3 is -4.

Step 1: Identify the Numbers

Here we have two numbers -7 and +3, -7 is a negative number and 3 is a positive number.

Step 2: Apply Rule for Adding Positive and Negative Numbers

Since one number is negative and the other is positive, we subtract the small number from the large number. Here, 7 is larger than 3 and it's negative, so we subtract 3 from 7 which gives us 4. Keep the sign of the larger number, which is negative in this case. Thus, the result is -4.

Key Concepts

Algebraic Addition

Algebraic addition is the process of combining numbers (which can be both positive and negative) to get a sum. This concept is foundational in mathematics and is used extensively across numerous applications from simple daily calculations to complex mathematical models. When adding two numbers with different signs, such as \( -7 \) and \( +3 \) from the exercise, the process differs from simple addition. The key is to visualize each number's value and direction relative to zero on a number line. If they are opposite signs, the smaller absolute value (3) is subtracted from the larger absolute value (7), and the resulting number (4) takes the sign of the number with the larger absolute value, giving us \( -4 \). Comprehending algebraic addition lays the groundwork for understanding more advanced algebraic concepts.

Negative Numbers

Negative numbers are values less than zero, often represented with a minus sign (e.g., \( -7 \)). They play a crucial role in arithmetic operations, allowing for the representation of debt, temperature below zero, or any quantity that has decreased below a defined level. Negative numbers have a direction opposite to positive numbers on the number line. While positive numbers grow larger as they move away from zero to the right, negative numbers do so as they move to the left. In our exercise, \( -7 \) needs to be seen as 7 units to the left of zero. Understanding how to work with negative numbers is essential for solving many real-world problems and advanced mathematical equations.

Positive Numbers

In contrast to negative numbers, positive numbers are values greater than zero and are typically written without a plus sign, making \( 3 \) in the exercise a positive number. They indicate quantities that exist above a certain reference point or zero. When graphed on a number line, they are plotted to the right of zero. Positive numbers are what we commonly encounter in everyday life, such as distance, height, and money earned—quantities that can be physically accumulated or measured. Recognizing positive numbers and knowing how to combine them with negative numbers, as in algebraic addition, is a skill that underlies many areas of mathematics.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. These are the building blocks of basic mathematics and are integral to problem-solving across various disciplines. When dealing with operations involving both positive and negative numbers, special rules apply. For addition, as the sum in our example suggests, if the numbers have opposite signs, you subtract the smaller number from the larger, irrespective of the operation originally being addition. The sign of the answer depends on the sign of the larger number. Mastery of these operations is necessary for advancing in mathematics and for understanding concepts such as algebra, calculus, and beyond.