Problem 5 - Fundamentals of Mathematics

Question

Perform the indicated subtractions. $$ -8-12 $$

Step-by-Step Solution

Answer

The result of the subtraction is \(-20\).

Step 1: Understand the Problem

The exercise requires us to perform the subtraction of two numbers: \(-8\) and \(12\). This involves combining the two values with subtraction.

Step 2: Rewrite as Addition

Subtraction can be rewritten as the addition of a negative number. Here, \(-8 - 12\) can be rewritten as \(-8 + (-12)\).

Step 3: Combine the Numbers

Adding two negative numbers involves adding their absolute values and keeping the negative sign. Thus, \(-8 + (-12) = -(8 + 12) = -20\).

Step 4: Write the Result

After performing the addition of the absolute values, the result of the subtraction is a negative number: \(-20\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are represented with a minus sign (-) in front, like

• -1
• -5
• -12.

Negative numbers are used to represent values below a certain reference point, like temperatures below freezing or debts below zero balance.
Whenever you see a negative number, think of it as a position on a number line that is to the left of zero.
This concept helps in solving problems involving subtraction, since when subtracting a positive number from a negative number, it means you are moving further left on the number line.

- Think of subtraction as asking "how much deeper?" into the negatives you are going.
- For example, subtracting 12 from -8 is like moving 12 more steps to the left of -8.

Absolute Values

The absolute value of a number is essentially how far a number is from zero on the number line, without considering its direction.
For example,

• The absolute value of -8 is 8.
• The absolute value of 12 is 12.

We use absolute values when adding or subtracting negative numbers because they help us focus purely on the magnitude, or size, of the numbers involved.
To handle operations involving subtraction of negative numbers, first, we look at the magnitude of each number.
When subtracting, we rewrite the subtraction as adding a negative number. So,

• Instead of \(-8 - 12\) we think of it as \(-8 + (-12)\).
• Next, we combine the absolute values of 8 and 12, which is \(8 + 12 = 20\).

Keeping the negative sign shows the result is \(-20\).

Integer Arithmetic

Integer arithmetic involves math operations with whole numbers, whether they're positive, negative, or zero. Addition and subtraction are two basic operations.
When we subtract integers:

• Converting to addition makes handling the numbers smoother.
• Remember: Subtracting a positive number is like adding a negative number.

In our exercise, \(-8 - 12\), we rewrite it as \(-8 + (-12)\). Adding these integers involves finding the sum of their absolute values while keeping a negative sign.

• Sum of absolute values: \((8 + 12 = 20)\).
• Then, \(-8 + (-12)\) becomes \(-20\).

Thus, integer arithmetic ensures that we handle and simplify operations with integers correctly, using strategies like rewriting subtraction as addition.