Problem 50 - Fundamentals of Mathematics

Question

Determine each value. $$ |-(3 \cdot 4)| $$

Step-by-Step Solution

Answer

The value is 12.

Step 1: Calculate the Product

First, calculate the product inside the absolute value notation. Multiply 3 by 4 to get the result.\[ 3 \cdot 4 = 12 \]

Step 2: Apply the Negative Sign

Next, apply the negative sign to the product calculated in Step 1. The resulting value will be:\[ -(3 \cdot 4) = -12 \]

Step 3: Evaluate the Absolute Value

Now, find the absolute value of the result from Step 2. The absolute value of any number is its distance from zero on the number line, without considering its sign. Therefore, the absolute value of -12 is 12.\[ |-(3 \cdot 4)| = | -12 | = 12 \]

Key Concepts

Multiplication

Multiplication is one of the fundamental operations in mathematics that involves combining equal groups. It can be viewed as repeated addition. For instance, if we consider multiplying 3 by 4, it is equivalent to adding the number 3, four times:

• 3 + 3 + 3 + 3 = 12

The result, 12, is called the product of multiplying 3 and 4.
When you see a multiplication expression like 3 \( \cdot \) 4, the \( \cdot \) symbol means you are multiplying the two numbers.
Understanding multiplication is crucial because it is a building block for more advanced mathematical concepts.
It's important to remember that multiplication is commutative, meaning that changing the order of the numbers does not change the product, so 3 \( \cdot \) 4 = 4 \( \cdot \) 3.

Negative Numbers

Negative numbers are the numbers less than zero, represented by a minus sign (-). They are crucial in understanding mathematical concepts like subtraction, finance, and temperatures below zero.
In the exercise, we worked with the expression \(-12\). This means taking the number 12 and finding its opposite on the number line.
The multiplication of a positive number by a negative number will result in a negative product.

• For example, \(-(3 \cdot 4)\) results in \(-12\).
• The negative sign in front of the number influences the result to be less than zero.

Understanding negative numbers and how they operate within mathematical operations, like multiplication, is essential for solving equations and word problems effectively.

Number Line

A number line is a visual tool used to represent numbers and their relationships. It is a straight line with numbers placed at equal intervals along its length. Zero is usually placed in the center, with positive numbers to the right and negative numbers to the left.
In understanding absolute value, the number line becomes a valuable tool as it helps visualize distance.

• The absolute value of a number is its distance from zero on the number line.
• For example, the absolute value of \(-12\) is 12 because \(-12\) is 12 units away from zero.

The concept of the absolute value makes the number line especially helpful in grasping why numbers become positive when we "remove" their negative sign.