Problem 106 - Introductory Algebra for College Students

Question

A subtraction is expressed as addition of an opposite. Find this sum, indicated by a question mark. -8-13=-8+(-13)=?

Step-by-Step Solution

Answer

-21

Step 1: Understand the concept of subtraction as addition of an opposite

Subtraction of a number is equivalent to the addition of its opposite. In this context, the opposite of a positive number is the corresponding negative number and vice versa. Therefore, the expression -8-13 can be rewritten as -8 + (-13).

Step 2: Perform the addition

The expression after rewriting the subtraction as an addition operation is -8 + (-13). To perform this addition, just add the absolute values of the numbers without considering the sign and append the minus sign because both numbers are negative. This gives the result as -21. Concretely, |-8| + |-13| = 21. Therefore, -8 + (-13) equals -21.

Key Concepts

Negative Numbers

Negative numbers are crucial in understanding operations like subtraction as addition of opposites. When we talk about negative numbers, we're dealing with numbers that are less than zero. They are usually written with a minus (-) sign, like -8 or -13. Negative numbers allow us to express quantities below zero, such as debts or temperatures.
To grasp the subtraction as addition of opposites, remember that subtracting a number is the same as adding its negative counterpart. For example, instead of subtracting 13 from -8, you add -13 to -8, effectively performing -8 + (-13). This switch makes arithmetic operations easier to handle.
A helpful way to visualize negative numbers is on a number line. Imagine zero in the center, with positive numbers on the right and negative numbers stretching to the left. Counting leftwards from zero takes you deeper into negatives: -1, -2, -3, and so on. This visualization helps clarify how negative numbers work in operations.

Absolute Value

The concept of absolute value simplifies handling negative and positive numbers. In its simplest form, the absolute value is the distance of a number from zero on the number line, regardless of direction. Therefore, the absolute value \( |x| \), is always positive or zero. For instance, the absolute value of -8 is 8, and that of 13 is 13. Navigation of the number line is easier with absolute values.

In practical terms, when you subtract like in the expression \( -8 - 13 \), rewriting it as \( -8 + (-13) \) helps. You can think of it as combining the distances they represent from zero, ultimately leading you through the operation smoothly. Because both \( -8 \) and \( -13 \) are negative, you ignore signs, add their absolute values \( 8 + 13 = 21 \), and then incorporate the negative sign back due to their original negativity, resulting in -21. Absolute values thus act as the fluency key in calculations with negative numbers.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition and subtraction. They serve as the language of algebra, allowing representation of various practical situations. For example, the expression \( -8 - 13 \), or its equivalent \( -8 + (-13) \), presents the usefulness of algebraic shorthand in simplifying and solving mathematical problems.

In the realm of algebra, expressions allow for manipulation and understanding of the relationships between different quantities. Evaluating expressions, even with negative numbers, can be straightforward with proper rewritten forms and knowledge of operations. Algebraic expressions also provide flexibility. Instead of numerical constants, expressions might include variables like \( x \) or \( y \), which can represent unknown values needing calculation.
Understanding simple transformations in expressions, such as converting subtraction to addition of opposites, paves the way for tackling more complex algebraic equations. Recognizing these patterns helps immensely, eventually leading to mastery of even advanced topics in mathematics.