Problem 5 - Fundamentals of Mathematics

Question

Find the following products. $$ (-1)(4) $$

Step-by-Step Solution

Answer

The product is -4.

Step 1: Understand the Product of Two Numbers

Here, we are asked to find the product of two numbers: \((-1)\) and \(4\). This means we need to multiply these two numbers together.

Step 2: Apply the Rule of Signs for Multiplication

When multiplying two numbers, if one is negative and the other is positive, the product is always negative. Therefore, the product of \((-1)\) and \(4\) will be negative.

Step 3: Multiply the Absolute Values

Multiply the absolute values of the numbers: The absolute value of \(-1\) is \(1\), and the absolute value of \(4\) is \(4\). Now, multiply these two absolute values: \(1 \times 4 = 4\).

Step 4: Attach the Negative Sign

Since the product of a negative and a positive number is negative, we attach a negative sign to the result from Step 3: The product is \(-4\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero, represented with a minus sign. Understanding negative numbers is crucial in mathematics because they extend the number line to the left of zero. In daily life, they can represent things like loss, debt, or temperature below zero.

Key characteristics of negative numbers include:

• They are always less than zero.
• They have the same magnitude as their positive counterparts but in the opposite direction.
• When compared with positive numbers, negatives are always less.
• An interesting fact to remember is that adding two negative numbers will give a more negative number.

Knowing how negative numbers interact with other numbers, especially in operations like multiplication, is essential for solving many mathematical problems.

Product of Numbers

When we talk about the product of numbers, we are referring to the result of multiplying two or more numbers together. Multiplication is one of the four fundamental arithmetic operations, and it's a quick way to add the same number several times.

Here’s a simple way to think about products:

• The product of a number and one is the number itself. For example, \(1 \times 4 = 4\).
• If you multiply a number by zero, the product is always zero. This is because adding nothing (zero times) results in nothing.
• The product of two positive numbers is always positive.
• However, when you introduce negative numbers into multiplication, the result depends on the combination of positive and negative numbers, relating to their sign.

The result of multiplying can also show how repeated additions or groupings are related, making it a versatile and powerful mathematical operation.

Sign Rule for Multiplication

The sign rule for multiplication is a key part of multiplying numbers when negative signs are involved. It helps predict whether the product will be positive or negative. This rule can be summarized simply:

• Multiplying two positive numbers results in a positive product.
• Multiplying two negative numbers also results in a positive product because two negatives "cancel out" each other.
• However, multiplying a positive number with a negative number gives a negative product.
• If you multiply more than two numbers, consider the number of negative numbers involved. An odd count of negative numbers will result in a negative product, while an even count will be positive.

For our example of \((-1)(4)\), we apply this rule: one number is negative and the other is positive, so the product is negative. This logical rule reduces confusion and helps ensure accurate multiplication involving negative numbers.