Question

Prep Exercise 6 When dividing two num bers that have the same sign, the result is ___. Prep Exercise 7 When dividing two numbers that have different signs, the result is ___. divide. $$0 \div(-12)$$

Short answer

Question: Explain the difference in results when dividing two numbers with the same sign and two numbers with different signs, and find the result of the division expression: $$0 \div(-12)$$ Answer: When dividing two numbers that have the same sign, the result is positive. In contrast, when dividing two numbers that have different signs, the result is negative. In the given division expression, $$0 \div(-12)$$, the numbers have different signs. However, since 0 divided by any nonzero number always results in 0, the result of this division is $$0 \div(-12) = 0$$.

Step-by-step solution

Same Sign Division Rule

When dividing two numbers that have the same sign, the result is positive.

Different Sign Division Rule

When dividing two numbers that have different signs, the result is negative. Now, we will use these rules to solve the given division problem:

Problem

We are asked to find the result of the division expression: $$0 \div(-12)$$

Identify the Signs

The dividend (0) is a non-negative number, and the divisor (-12) is a negative number. Since they have different signs, this division falls under the Different Sign Division Rule.

Apply Division Rule

According to the Different Sign Division Rule, the result of dividing two numbers with different signs is negative. However, 0 divided by any nonzero number always results in 0. Therefore, the result of this division is: $$0 \div(-12) = 0$$

Key concepts

Same Sign Division

Understanding the rules of division is crucial for solving mathematical problems accurately. One such rule is the same sign division rule. This concept is fairly straightforward: when you divide two numbers that carry the same sign, be it both positive or both negative, the result is always a positive number.

For instance, if you have two positive numbers such as \( 6 \div 3 \), or two negative numbers such as \( (-6) \div (-3) \), the outcome of both divisions is \( +2 \). The signs cancel each other out, leaving a positive result. It's important to remember this rule, as it often simplifies the process of calculation and helps in understanding the relationships between numbers.

Different Sign Division

In contrast to the same sign division rule, when you divide two numbers with different signs—one positive and one negative—the rule states that the result will be a negative number. This can sometimes be counterintuitive, but remembering this rule can help avoid making mistakes in calculations.

For example, take \( 8 \div (-2) \) or \( (-8) \div 2 \). Although the numbers being divided are different in each case, since one is positive and the other is negative, the outcome for both will be \( -4 \). The differing signs mean that they do not cancel out, resulting in a negative result. This rule is pivotal in algebra and other higher mathematical disciplines where expressions can include a mix of positive and negative numbers.

Positive Result

When solving division problems, it's essential to predict the sign of the result, as it helps in understanding the nature of the answer. A positive result in division signifies that the two numbers you're working with either are both positive or both negative. In other words, they have the same sign.

Positive results are often indicative of consistent or directly proportional relationships, common in real-world scenarios such as calculating rates or ratios. A positive outcome in division is also reassuring during equation solving, confirming that the steps followed align with the fundamental rules of arithmetic.

Negative Result

A negative result occurs in division when the two numbers involved have different signs. The presence of a negative result can have various implications depending on the context—such as indicating a loss or debt in financial settings, or directional change in physics.

In mathematics, getting a negative result reminds you to be cautious with subsequent calculations, especially when dealing with equations and inequalities, as the interplay of positive and negative values can affect the solution set. Negative results are equally valid as their positive counterparts; they simply denote the opposite sign of whatever quantity is being calculated.