Problem 18 - Fundamentals of Mathematics

Question

Find the value of each of the following. Use a calculator to check each result. $$ (-5)(-2) $$

Step-by-Step Solution

Answer

The product of \((-5)(-2)\) is 10.

Step 1: Identify the Given Numbers

The problem provides us with the numbers \(-5\) and \(-2\). Our task is to find the product of these two numbers.

Step 2: Understand the Multiplication Rule for Negative Numbers

When multiplying two negative numbers, the product is a positive number. This is because the rule states that the product of two numbers with the same sign (either both positive or both negative) results in a positive product.

Step 3: Multiply the Numbers

Compute \((-5) \times (-2)\). Using the rule that a negative times a negative is a positive, we get: \[ (-5) \times (-2) = 10 \]

Step 4: Verify Using a Calculator

Using a calculator, input \(-5 \times -2\) and confirm the result. You should obtain 10 as the result, which matches our manual calculation.

Key Concepts

Product of Integers

The product of integers is the result you get when you multiply two or more integers. Integers include all whole numbers and their negative counterparts, like -3, -1, 0, 1, 2, etc. When dealing with the product of integers, it’s important to understand how the signs of the numbers affect the outcome.

There are a few simple rules to remember:

• A positive number times a positive number is positive.
• A positive number times a negative number is negative.
• A negative number times a negative number is positive.

These rules help us determine whether the product will be positive or negative, simplifying calculations without a calculator.

For example, the multiplication of \[ (-5) \times (-2) \] follows the rule that two negative numbers multiply to produce a positive number. Therefore, the answer is 10.

Multiplication Rules

Multiplication rules are fundamental guidelines that help us quickly find products without error. In integer arithmetic, these rules are especially useful since signs significantly impact the results.

Here are the core multiplication rules:

• When you multiply two numbers with **the same sign**, the product is positive. This applies to both positive numbers and negative numbers.
• When you multiply two numbers with **different signs**, the product is negative.

Understanding these rules is essential as they simplify complex calculations and solve problems quickly.

In the case of \[ (-5) \times (-2) \], both numbers have the same sign (negative), hence according to the rules, the product should be positive 10.

Integer Arithmetic

Integer arithmetic describes operations such as addition, subtraction, multiplication, and division applied to integers. These operations follow a specific set of rules, especially when dealing with positive and negative integers.

One unique aspect of integer arithmetic is **handling negative numbers**. The rules are designed to ensure consistency and predictability when performing calculations.

For multiplication:

• **Same signs:** The result is always positive.
• **Different signs:** The result is always negative.

For example, the multiplication \[ (-5) \times (-2) \] is handled as follows: since both numbers are negative, their signs are the same and the result is positive, giving us a product of 10.

Mastering integer arithmetic is crucial for solving many mathematical problems accurately, especially when negatives are involved.