Problem 1 - Technical Mathematics with Calculus

Question

Dividing Signed Numbers Divide, keeping the proper sign on your answer. $$14 \div(-2)$$

Step-by-Step Solution

Answer

-7

Step 1: Understanding Division of Signed Numbers

When dividing numbers with different signs, the result is always negative. Here, you have a positive number (14) divided by a negative number (-2).

Step 2: Performing the Division

Ignore the signs for a moment and simply divide the absolute values of the numbers. This gives you 14 divided by 2, which is 7.

Step 3: Assigning the Correct Sign

Since the numbers being divided have different signs, the result will be negative. Thus, your final answer should have a negative sign.

Key Concepts

Division with Negative Numbers

When dividing negative numbers, it's important to pay attention to the signs of the numbers involved. The general rule for division is that if the dividend (the number being divided) and the divisor (the number by which you are dividing) have the same sign, the result is positive. Conversely, if the signs are different, the answer is negative.

Let's walk through an example: for the division problem \(14 \div (-2)\), you start by dividing the numbers as if they were both positive, obtaining \(7\) as the quotient of \(14 \div 2\). To determine the correct sign of the answer, you look at the original numbers. Since \(14\) is positive and \(2\) is negative, their signs differ. Therefore, according to the rule, the quotient must be negative, making the correct answer \( -7 \).

Remembering the Rule of Signs

A helpful way to remember this is to think of friends and enemies. If the dividend and divisor are 'friends' (both positive or both negative), they work together, and the result is 'friendly' or positive. If they are 'enemies' (one positive and one negative), they conflict, and the result is 'unfriendly' or negative. This analogy can make it easier to remember how to handle signs in division.

Absolute Value in Division

The absolute value of a number is its distance from zero on the number line, without considering its direction. In division, sometimes we temporarily ignore the signs of numbers and focus on their absolute values to simplify the calculation.

Take the problem \(14 \div (-2)\) as an example. To perform this division, you first consider the absolute values of both numbers. The absolute value of \(14\) is \(14\), and the absolute value of \( -2 \) is \(2\). You then divide these absolute values \(14 \div 2 = 7\). Only after finding the quotient of the absolute values do you re-apply the sign rule to find the final signed answer.

Using Absolute Values for Simplicity

Using absolute values can simplify calculations and reduce the potential for mistakes. It helps to focus on getting the magnitude of the answer first before finalizing the sign based on the original numbers.

Arithmetic Operations with Signed Numbers

Performing arithmetic operations with signed numbers—positive and negative—requires understanding of certain rules that help correctly determine the sign of the answer. In addition to division, these rules also apply to multiplication, addition, and subtraction.

For multiplication and division, the rule is straightforward: if the signs are the same, the answer is positive; if the signs are different, the result is negative. However, with addition and subtraction, you'll need to consider both the signs and the magnitudes of the numbers.

Adding and Subtracting Signed Numbers

With addition, if the signs are the same, you add the absolute values and keep the common sign. With subtraction, you find the difference in absolute values and assign the sign of the number with the larger absolute value.
Understanding these conventions allows for consistent computation across various mathematical problems and is fundamental to working with signed numbers.