Problem 2 - Technical Mathematics with Calculus

Question

Dividing Signed Numbers Divide, keeping the proper sign on your answer. $$(-15) \div(3)$$

Step-by-Step Solution

Answer

-5

Step 1: Understand the Signs

When dividing two numbers, if one number is negative and the other is positive, the result will also be negative.

Step 2: Perform the Division

Disregarding the signs, divide 15 by 3 to get the absolute value of the answer.

Step 3: Apply the Sign Rule

Since the dividend (-15) is negative and the divisor (3) is positive, the result will be negative. Apply the negative sign to the result obtained in Step 2.

Key Concepts

Arithmetic Operations

Arithmetic operations are the bread and butter of mathematics, encompassing addition, subtraction, multiplication, and division. At their most basic, these operations allow us to calculate important values and solve problems both simple and complex.

Division, in this context, is the process of determining how many times one number is contained within another. Think of it as sharing or distributing a quantity evenly. When we refer to dividing signed numbers, we are including the arithmetic operation of division but with an additional layer of complexity due to the presence of negative numbers.

Negative Numbers

Negative numbers are a numerical expression that denotes a value less than zero, symbolized by a minus sign (-). They play a crucial role in a wide range of mathematical concepts, including arithmetic operations. In the context of division, understanding the behavior of negative numbers is essential.

When dividing with negative numbers, the sign of the result depends on the signs of the numbers involved, and it's important to remember that two negatives make a positive. However, if only one of the numbers is negative, the result will always be negative. These rules are what give structure to our number system and guide us towards the correct solutions when performing calculations.

Division Rules

The rules of division when it comes to signed numbers are straightforward but vital for accuracy in calculations. Here's the key point: when the signs of the two numbers involved are the same, the result is positive; when the signs are different, the result is negative.

In the exercise provided \( (-15) \div (3) \), we are dividing a negative number by a positive number. According to our rules, this will result in a negative number. By first ignoring the signs and simply dividing the absolute values, 15 divided by 3 equals 5. We then apply our division rule for signs—since our dividend is negative and our divisor is positive, our final answer will be negative 5 \( (-5) \). This exemplifies the importance of understanding both the arithmetic operation and the role of signs in division to arrive at the correct answer.