Problem 2 - 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 different signs is _____

Step-by-Step Solution

Answer

Negative

Step 1 - Understand the signs of numbers

Recall that numbers can be either positive, negative, or zero. Positive numbers are greater than zero. Negative numbers are less than zero.

Step 2 - Identify the nature of the operation

The problem involves either multiplying (product) or dividing (quotient) two numbers that have different signs.

Step 3 - Determine the result of different signs

When multiplying or dividing two numbers with different signs, a positive and a negative, the result is always negative. This is because a positive times a negative or a negative divided by a positive is negative by definition.

Step 4 - Fill in the blank

Since the result of the product or quotient of two numbers with different signs is negative, fill in the blank with 'negative'.

Key Concepts

positive numbers

Positive numbers are numbers that are greater than zero. They are found to the right of zero on a number line. Examples of positive numbers include 1, 2, 3, and so on. When you multiply or divide two positive numbers, the result is always positive. For example, \(3 \times 4 = 12\) and \(20 ÷ 5 = 4\). Positive numbers are important in everyday life as they help us count, measure, and compare. Knowing whether a number is positive plays a key role in many math operations, as it sets the stage for correct calculations.

negative numbers

Negative numbers are numbers less than zero. They are found to the left of zero on a number line, typically represented with a minus sign (-). Examples of negative numbers include -1, -2, -3, etc. When you multiply or divide two negative numbers, the result is positive. For instance, \( (-3) \times (-4) = 12 \) and \( (-20) ÷ (-5) = 4 \). Negative numbers often represent values below a certain reference point, such as temperatures below freezing or debts in a financial context. Understanding negative numbers is crucial for performing more complex integer operations accurately.

integer operations

Integer operations involve basic arithmetic with whole numbers, including positive, negative, and zero. The primary operations include addition, subtraction, multiplication, and division. It's essential to remember some key rules for these operations:

• Adding two positive integers always gives a positive result, e.g., \( 5 + 3 = 8 \).
• Adding two negative integers always gives a negative result, e.g., \( (-5) + (-3) = -8 \).
• Adding a positive integer and a negative integer involves finding the difference, e.g., \( 5 + (-3) = 2 \).
• Multiplying or dividing two integers with the same sign gives a positive result.
• Multiplying or dividing two integers with different signs gives a negative result, such as \( 5 \times (-3) = -15 \) and \( (-15) ÷ 3 = -5 \).

Understanding these rules helps simplify complex problems and ensures accuracy in calculations.