Question

Prep Exercise 1 When multiplying two numbers that have the same sign, the result is ____. Prep Exercise 2 When multiplying two numbers that have different signs, the result is ____. multiply. $$4 \cdot(-2)$$

Short answer

Answer: To determine the sign of the result when multiplying two numbers, follow these rules: 1. If the numbers have the same sign (both positive or both negative), the result will be positive. 2. If the numbers have different signs (one positive and one negative), the result will be negative. In the given example, \(4 \cdot(-2)\), the numbers have different signs (4 is positive and -2 is negative), so the result will be negative. Multiply the numbers as if they were both positive (4 * 2 = 8) and then apply the negative sign, resulting in a final answer of -8.

Step-by-step solution

Identifying result of the same signs multiplication

When multiplying two numbers that have the same sign (either both positive or both negative), the result will always be a positive number. Prep Exercise 2

Identifying result of the different signs multiplication

When multiplying two numbers that have different signs (one positive and one negative), the result will always be a negative number. Multiply

Multiplying 4 by -2

Following the rule from Prep Exercise 2, since these numbers have different signs (4 is positive and -2 is negative), the result will be a negative number. Multiply the numbers as if they were both positive: 4 * 2 = 8. Then apply the negative sign, resulting in a final answer of -8. Thus, \(4 \cdot(-2) = -8\).

Key concepts

Positive and Negative Numbers

Positive and negative numbers are fundamental in mathematics. They help us express value in different directions, such as gain or loss, profit or debt.

• Positive numbers: Typically greater than zero. They indicate a forward motion or addition. Examples include 1, 2, 3, and so on.
• Negative numbers: Less than zero. They suggest a backward motion or subtraction. These numbers are always preceded by a minus sign, like -1, -2, -3, etc.

When you think about positive and negative numbers, imagine them on a number line. Positive numbers extend to the right of zero, and negative numbers stretch to the left. This idea is key in understanding how their interactions will behave when performing operations, such as multiplication.
Both types of numbers are used in various real-world contexts, from temperatures below freezing to debt in finances. Knowing how to work with them is crucial.

Sign Rules in Multiplication

Understanding the sign rules in multiplication is essential for correctly multiplying numbers. These rules determine the sign (positive or negative) of the product based on the signs of the numbers you are multiplying.

• Same Signs: When you multiply two numbers with the same sign, the result will be positive. This is true for both positive numbers (like 3 and 4) and negative numbers (-3 and -4). For example, \[3 \times 4 = 12\] and \[-3 \times -4 = 12\].
• Different Signs: When you multiply numbers with different signs, one positive and one negative, the result will always be negative. For instance, \[3 \times -4 = -12\]. This rule applies no matter the order of the numbers: \[-4 \times 3 = -12\].

Remembering these rules helps in solving equations and understanding real-life situations involving gains and losses.

Integer Multiplication Rules

Integer multiplication rules build directly on the concepts of positive and negative numbers, plus the sign rules outlined above. Integers include all whole numbers, both positive and negative, including zero.

• Multiplying by Zero: Any integer multiplied by zero results in zero, regardless of the sign of the other number. For example, \[5 \times 0 = 0\] and \[-5 \times 0 = 0\].
• Multiplying Positives: The product of two positive integers is always positive. Example: \[6 \times 7 = 42\].
• Multiplying Negatives: When two negative integers are multiplied, the product is a positive integer. For instance, \[-7 \times -8 = 56\].
• Mixed Signs: If one integer is positive and the other is negative, the product is negative. Example: \[9 \times -2 = -18\].

These rules ensure consistent results and are fundamental in algebra and arithmetic, helping to solve complex problems by providing a clear path through calculations.