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. $$(-2) \cdot(-32)$$

Short answer

Answer: Positive

Step-by-step solution

Rules for multiplying numbers

When multiplying numbers, it is important to remember the rules for multiplication. If two numbers have the same sign, the result will be positive. If the numbers have different signs, the result will be negative.

Prep Exercise 1 Solution

As both numbers have the same sign, the result will be positive. Therefore, the answer to Prep Exercise 1 is "positive."

Prep Exercise 2 Solution

Since the two numbers have different signs, the result will be negative. Therefore, the answer to Prep Exercise 2 is "negative."

Apply these rules to the example

Now, we will multiply the given example $$(-2)\cdot(-32)$$ applying the rules we discussed. Since both numbers are negative, they have the same sign. According to the rules, the result should be positive. Now, proceed to multiply the numbers: $$(-2)\cdot(-32) = 2 \cdot 32 = 64$$. The result, 64, is a positive number, as expected based on the rules for multiplying numbers with the same sign.

Key concepts

Understanding Negative Numbers

Negative numbers are values less than zero. They are usually preceded by a minus (-) sign and found on the left side of zero on the number line. Negative numbers are used in many real-life situations. For example, they can represent temperatures below zero, depths below sea level, or losses in financial statements. When multiplying negative numbers, remember that they follow rules just like any other number. A negative number times another negative number results in a positive number. This is because the two negatives "cancel" each other out. For example:

• -2 times -3 equals 6.
• You can visualize this with numbers on a number line moving to the right when multiplying two negatives.

The Positivity of Positive Numbers

Positive numbers are greater than zero. They are situated to the right of zero on the number line and typically do not have a sign, though you can denote them with a plus (+) sign if needed. These numbers are what we use to count and measure objects, population, money, and much more. They are instinctively understood as regular numbers. When multiplying two positive numbers, the result is always positive. Positive numbers, owing to their nature, reinforce each other when multiplied. For example:

• 3 times 5 equals 15.
• The product is firmly located on the positive side of the number line.

Mastering Multiplication Sign Rules

Understanding multiplication sign rules is crucial when dealing with both negative and positive numbers. Let's break down the sign rules further for clarity: - Multiplying two positive numbers gives a positive result. No change in sign occurs. For instance,

• 4 times 7 equals 28.

- Multiplying two negative numbers results in a positive number. The negative signs cancel each other out. Consider

• -4 times -6 equals 24.

- A positive number multiplied by a negative number yields a negative result. The result's sign takes after the number that had a stronger impact due to its negative factor. As an example,

• -5 times 3 equals -15.
• Likewise, 2 times -9 results in -18.

These rules help in quickly determining the sign of the product without much calculation.