Problem 28 - Elementary Algebra

Question

Simplify. $$ (-3) 3 $$

Step-by-Step Solution

Answer

The simplified expression is -9.

Step 1: Identify the Numbers

The expression given is \((-3) \times 3\). The two numbers we are dealing with here are \(-3\) and \(3\).

Step 2: Multiply the Values

To simplify the expression, we multiply the numbers together: \((-3) \times 3 = -9\). This involves multiplying 3 by 3 and then considering the sign: negative times positive yields negative.

Step 3: Confirm the Sign

In multiplication, multiplying a negative number by a positive number results in a negative number. Therefore, \((-3)\times 3\) equals \(-9\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers that are less than zero, and they typically appear with a minus sign in front. For example,

• -1,
• -5,
• -10.

Negative numbers are commonly used to represent values on the opposite side of a zero point, such as debts in finances or temperatures below freezing. In mathematics, negative numbers are crucial as they help to express subtraction beyond the limits of zero. This means you can always subtract a larger number from a smaller one, and the result will be negative. Negative numbers are important when dealing with calculations like multiplication and division with non-positive numbers. Always remember, a negative number, like ewline (-3), can change an entire calculus when involved with other numbers. This is because, while they are lesser than zero, they have a particular influence on operations, especially multiplication, due to different sign rules.

Grasping Positive Numbers

Positive numbers, on the other hand, are numbers greater than zero. They are traditionally written without any symbol or sometimes with a plus sign prefix. Common examples include:

• 1,
• 7,
• 20.

These numbers are often used in contexts like counting units, expressing sizes, or any situation where quantities increase positively. Within mathematics, positive numbers are easier to visualize and are typically understood as moving forward in a sequential manner from zero. They serve as a benchmark for measurements or comparisons, defining what it means to "add up" or "have more." In calculations like addition, multiplication, and so forth, the positivity of these numbers ensures the outcome is straightforward when they are combined with other positive numbers. Remember, when no sign is stated before a number, it’s assumed to be positive, simplifying many everyday calculations.

Sign Rules in Multiplication

Multiplication involving different signs can seem a bit tricky, but some simple rules make it easy:

• A positive number multiplied by a positive number results in a positive number.
• A positive number multiplied by a negative number results in a negative number.
• Similarly, a negative number multiplied by a positive number also yields a negative number.
• A negative number multiplied by a negative number results in a positive number.

These rules are essential during operations to determine the sign of the product. Take, for example, multiplying ewline (-3) by ewline (+3): ewline The rule to follow is that a negative multiplied by a positive gives a negative. This is why ewline (-3) × 3 = -9. Memorizing these rules helps in quickly simplifying expressions without confusion about the contribution of each number’s sign. Sign rules maintain consistency in mathematical operations, making arithmetic predictable and reliable every time a calculation involving different signs is performed.