Problem 69 - Algebra A Combined Function

Question

Multiply -11 by 11 .

Step-by-Step Solution

Answer

-121

Step 1: Understand the Multiplication Problem

We need to multiply two numbers: -11 and 11. This is a straightforward arithmetic operation, but we must pay attention to the signs of the numbers involved.

Step 2: Consider the Signs

We have one negative number (-11) and one positive number (11). According to the rules of multiplication, a negative number multiplied by a positive number results in a negative product.

Step 3: Multiply the Absolute Values

Disregard the signs temporarily and multiply the absolute values of the numbers. The absolute value of -11 is 11, and the absolute value of 11 is 11. Therefore, we calculate: \[ 11 \times 11 = 121 \]

Step 4: Apply the Sign Rule

Since the product of a negative number and a positive number is negative, the result from Step 3 (121) will be negative. Therefore, the final product is -121.

Key Concepts

Negative Numbers

Negative numbers are values that are less than zero. They are found to the left of zero on a number line. Often, they are used to represent debts, temperatures below zero, or positions below a fixed point. For example,

• -11 is a negative number because it is 11 units below zero.
• Knowing how to work with negative numbers is essential in different mathematical operations.

To identify them, we look for a minus sign (-) in front of a number.

Sign Rules

The sign rules are key to understanding how to multiply and divide integers. When multiplying numbers, the sign of the result depends on the signs of the numbers you are multiplying.

• Two positive numbers produce a positive product.
• Two negative numbers also result in a positive product.
• A negative and a positive number yield a negative product.

For example, when we multiply 11 and -11, we are using the rule that the product of a positive and a negative number is a negative number: (-11) times 11 equals -121.
The concise rule is: back-to-same = positive and mixed = negative.

Absolute Value

Absolute value signifies the distance of a number from zero, regardless of direction. It's like ignoring any negative sign and just looking at how far away a number is from zero. The symbol used to represent absolute value is two vertical bars surrounding the number, like this:

• \( |-11| = 11 \)
• \( |11| = 11 \)

This means the absolute value of -11 is 11 because it is 11 units away from zero, without considering the direction.

Arithmetic Operations

Arithmetic operations are the basic operations of mathematics, which include addition, subtraction, multiplication, and division. In this context, we focus on multiplication, specifically the problem (-11) times 11. To multiply these:

• First, find the absolute values:
  • \( |-11| = 11 \)
  • \( |11| = 11 \)
• Next, multiply these absolute values: \[ 11 \times 11 = 121 \]
• Finally, apply the sign rule. Since one number is positive and the other negative, the result is: \(-121\)

This straightforward process shows how basic operations can be applied to numbers with different signs.