Problem 55 - Fundamentals of Mathematics

Question

Perform each operation. $$ -1-6 $$

Step-by-Step Solution

Answer

The result is -7.

Step 1: Understand the Problem

The expression given is \(-1 - 6\). This means we need to subtract 6 from -1.

Step 2: Apply the Rule for Subtracting Integers

Subtracting a positive number is equivalent to adding its negative. Therefore, \(-1 - 6\) is the same as adding \(-6\). So, we rewrite the expression as \(-1 + (-6)\).

Step 3: Perform the Addition

Adding two negative numbers involves adding their absolute values and keeping the negative sign. The absolute values are 1 and 6, which sum up to 7. Therefore, \(-1 + (-6) = -7\).

Key Concepts

Negative Numbers

When we talk about negative numbers, we're referring to numbers that are less than zero. These numbers appear to the left of zero on a number line. In this context, negative numbers have a "-" sign in front of them. They are essential when dealing with concepts such as temperature below freezing, depth below sea level, or financial debt.

Understanding how to work with negative numbers involves seeing them as opposites of positive numbers. For example,

• -1 is one unit away from zero but in the opposite direction of +1.
• Negative numbers diminish the value when added to a positive number and increase the magnitude when subtracted from a larger negative number.

By learning how to navigate these numbers, you'll be better equipped to solve a variety of mathematical and real-world problems.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. This value is always a non-negative number. For example, both -5 and 5 have the same absolute value, which is 5.

Absolute value is crucial when adding or subtracting negative numbers because it shows how far a number is from zero:

• For the number -1, the absolute value is 1.
• For 6, the absolute value is 6.

When you add negative numbers, like in the expression o(-1 + (-6), understanding their absolute values helps you concentrate on the magnitude rather than the direction before determining the final sign of the answer.

Integer Addition

Integer addition can seem tricky, but it becomes simple once you understand the rules. The key is to add up the absolute values and then determine the sign of the sum based on the problem.

When adding two negative numbers, like in the example o(-1 + (-6), you:

• Calculate the sum of their absolute values: 1 + 6 = 7.
• Keep the negative sign in the final answer, resulting in -7.

The concept is similar when adding positive and negative numbers. First, find the difference between the absolute values, then use the sign of the number with the larger absolute value.

Practicing these steps with different combinations of integers helps in solidifying your understanding of integer addition.

Mathematics Problem-solving

Problem-solving in mathematics involves breaking down problems into manageable steps and following logical rules. It is the application of the mathematical concepts you learn, like negative numbers and integer addition, to derive solutions. To tackle o(-1 - 6), you'll need to:

• Recognize it requires subtracting 6 from -1, which translates to adding -6 to -1.
• Apply integer addition rules, focusing on absolute values.
• Achieve the final result by following a systematic approach: in this case, obtaining -7.

Consistent practice with varied problems reinforces these techniques, making it easier to tackle more complex equations in the future. As you practice problem-solving, you'll grow more comfortable with each of these concepts and improve your mathematical confidence.