Problem 58 - Fundamentals of Mathematics

Question

Perform each operation. $$ 5 \cdot(-2) $$

Step-by-Step Solution

Answer

The result of the operation is -10.

Step 1: Understanding the Problem

We are asked to perform a multiplication operation between two numbers: 5 and -2.

Step 1: Multiply the Absolute Values

Ignore the signs first and multiply the absolute values of the numbers. The absolute value of 5 is 5, and the absolute value of -2 is 2. Multiply these: \[ 5 imes 2 = 10 \]

Step 2: Determine the Sign of the Result

According to the rules of multiplication for signed numbers, the product of a positive number and a negative number is negative. Thus, the product of 5 and -2 is negative.

Step 3: Write the Final Answer

Apply the negative sign to the product from step 1. The result is:\[ 5 imes (-2) = -10 \]

Key Concepts

Signed Numbers

Signed numbers are numbers that come with a positive (+) or negative (−) sign. They are used to indicate the direction or position of a value on the number line.
For instance, a positive number, like +3, signifies a position to the right of zero, while a negative number, like -3, indicates a position to the left of zero.
This concept is crucial when performing operations such as addition, subtraction, and multiplication, as the rules vary depending on the signs of the numbers involved.

• Positive numbers: Numbers greater than zero.
• Negative numbers: Numbers less than zero.
• Zero: Neutral element on the number line, neither positive nor negative.

Understanding signed numbers helps in solving problems involving direction, like example calculations in physics or tracking gains and losses in finance.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative.
It's a simple concept, but very important when you need to focus on the size or magnitude of a number, without considering its sign.
For any number, whether positive or negative, its absolute value is always positive.

• The absolute value of 5 is \( |5| = 5 \).
• The absolute value of -2 is \( |-2| = 2 \).

In multiplication operations like \( 5 \cdot (-2) \), we first multiply the absolute values \( 5 \cdot 2 = 10 \) and then determine the sign result as to whether it should be positive or negative based on the original signs of the numbers.

Negative Multiplication

Negative multiplication involves multiplying numbers where at least one of the numbers is negative. The rules for determining the sign of the result in multiplication operations are straightforward:
- If you multiply two numbers with the same sign, the result is positive.
- If you multiply two numbers with different signs, the result is negative.
This is why when we multiply 5 (positive) by -2 (negative), we end with \( -10 \).
The sign rules help align the operation contextually based on real-world signification, like understanding profit and loss scenarios or changes in direction in physics.
By following these rules, dealing with signed numbers becomes predictable and manageable, making it easier to approach and solve mathematical problems.