Problem 1 - Beginning and Intermediate Algebra

Question

Fill in each blank with one of the following. positive,negative,0 The product or the quotient of two numbers with the same sign is ______

Step-by-Step Solution

Answer

positive

Step 1: Identify the operation

Determine that the exercise is asking about the product (multiplication) or the quotient (division) of two numbers.

Step 2: Understand the sign rules

Recall the rule for multiplying or dividing two numbers. Numbers with the same sign (both positive or both negative) will result in a positive outcome.

Step 3: Fill in the blank

Based on the rule, the product or the quotient of two numbers with the same sign is positive.

Key Concepts

positive numbers

Positive numbers are numbers greater than zero. They are used to represent quantities that increase, gain, or move forward.
For instance, if you have 3 positive steps, you move forward by 3 units.
Positive numbers can be whole numbers like 1, 2, and 3, or they can be decimals like 1.5 or 3.25.
When you multiply or divide two positive numbers, the result is always a positive number.
Examples:

• Multiplication: 3 × 4 = 12
• Division: 12 ÷ 4 = 3

In both cases, the positive results make sense because you're combining or splitting positive quantities.

negative numbers

Negative numbers are numbers less than zero. They represent quantities that decrease, losses, or movement backward.
For example, if you have 3 negative steps, you move backward by 3 units.
Negative numbers are written with a minus sign (-) in front. Common examples include -1, -2, and -3.
When you multiply or divide two negative numbers, the result is positive. This might seem strange, but think of it as reversing a reversal.
Examples:

• Multiplication: (-3) × (-4) = 12
• Division: (-12) ÷ (-4) = 3

Even though both numbers are negative, the negative signs cancel each other out, resulting in a positive outcome.

sign rules

Sign rules help us determine the sign (positive or negative) of the result when we multiply or divide numbers. These rules vary based on the signs of the numbers involved.
Here are the basic sign rules for multiplication and division:

• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative
• Negative × Positive = Negative

The same rules apply to division:

• Positive ÷ Positive = Positive
• Negative ÷ Negative = Positive
• Positive ÷ Negative = Negative
• Negative ÷ Positive = Negative

Remembering these rules makes it easier to solve problems involving multiplication and division of numbers with different signs.
For example:
Multiplication:

• 3 × -4 = -12
• (-3) × 4 = -12

Both examples above show that multiplying a positive number with a negative number results in a negative product.