If You Divide Two Negatives Is It A Positive

Dividing Two Negatives: Why a Negative Divided by a Negative is Positive

The question of whether dividing two negative numbers results in a positive number is a fundamental concept in mathematics, often encountered early in a student's education. While the answer is a resounding "yes," understanding why this is true requires exploring the underlying principles of arithmetic and algebra. This article will delve into the rationale behind this rule, providing a comprehensive explanation suitable for learners of all levels, from those just grasping the basics to those seeking a deeper mathematical understanding. We'll explore the concept through intuitive examples, visual representations, and a formal algebraic approach.

Introduction: Understanding the Number Line and Operations

Before diving into the specifics of dividing negative numbers, it's crucial to establish a firm grasp of the number line and the operations of addition, subtraction, multiplication, and division. The number line provides a visual representation of numbers, extending infinitely in both positive and negative directions. Zero sits at the center, with positive numbers to the right and negative numbers to the left.

• Addition: Moving to the right on the number line represents addition. For example, 2 + 3 means moving three units to the right of 2, landing on 5.
• Subtraction: Moving to the left on the number line represents subtraction. 5 - 3 means moving three units to the left of 5, landing on 2.
• Multiplication: Multiplication can be viewed as repeated addition. 3 x 2 means adding 2 three times (2 + 2 + 2 = 6).
• Division: Division is the inverse of multiplication. 6 ÷ 2 asks "how many times does 2 fit into 6?". The answer is 3.

Understanding these basic operations is essential before tackling the complexities of working with negative numbers.

Intuitive Explanation: The Concept of "Opposites"

One way to understand why a negative divided by a negative is positive is to consider the concept of "opposites." Multiplication and division are inverse operations; they "undo" each other.

Let's consider the multiplication problem: (-3) x (-2) = ?

We know that 3 x 2 = 6. Now, let's think about the signs. A negative multiplied by a positive gives a negative. Therefore, (-3) x 2 = -6. Conversely, a positive multiplied by a negative also gives a negative: 3 x (-2) = -6.

Logically, if a negative multiplied by a negative results in a positive, then the inverse operation – division – should follow suit. This is because division "undoes" multiplication. So, if (-3) x (-2) = 6, then 6 ÷ (-2) = -3 and 6 ÷ (-3) = -2. This implies that if we divide a positive number (6) by a negative number (-2 or -3), we get a negative number.

To extend this logic, if we have a negative number divided by a negative number, we end up with a positive outcome. For example: -6 / -2 = 3. This aligns with the property of inverse operations.

Visual Representation: The Number Line and Division

We can visualize division using the number line. Let's take the example of -6 ÷ -2. This is asking, "How many times can we subtract -2 from -6 to reach 0?"

• Start at -6 on the number line.
• Subtract -2 (which is the same as adding 2). This moves us to -4.
• Subtract -2 again (add 2 again). This moves us to -2.
• Subtract -2 one more time (add 2 again). This moves us to 0.

We subtracted -2 three times to reach 0. Therefore, -6 ÷ -2 = 3. This visual representation reinforces the concept that dividing two negative numbers results in a positive number.

Algebraic Explanation: Distributive Property and Inverses

A more formal algebraic approach involves the distributive property and the concept of additive inverses. Recall that the additive inverse of a number is the number that, when added to the original number, results in zero. For example, the additive inverse of 5 is -5 (5 + (-5) = 0).

Consider the expression: (-a) / (-b), where 'a' and 'b' are positive numbers.

We can rewrite this as: (-1)(a) / (-1)(b)

Using the properties of fractions, we can separate the -1 terms:

(-1)/(-1) * (a/b)

Since (-1)/(-1) = 1, the expression simplifies to:

1 * (a/b) = a/b

This demonstrates that dividing two negative numbers results in the same value as dividing their positive counterparts. The negative signs cancel each other out, resulting in a positive outcome.

Understanding the Pattern: Extending to More Complex Scenarios

The rule of dividing two negatives resulting in a positive extends beyond simple whole numbers. It applies to all real numbers, including fractions, decimals, and irrational numbers.

• Fractions: (-2/3) ÷ (-1/2) = (2/3) x (2/1) = 4/3. The negative signs cancel.
• Decimals: (-2.5) ÷ (-0.5) = 5. The negative signs cancel.
• Variables: (-x) / (-y) = x/y, assuming x and y are non-zero.

Addressing Common Misconceptions

Several misconceptions can arise when working with negative numbers. It's important to clarify these to avoid errors.

• Mixing Operations: It's crucial to perform operations in the correct order, following the order of operations (PEMDAS/BODMAS). Incorrect order can lead to incorrect results.
• Ignoring Signs: Always pay close attention to the signs of the numbers. A seemingly small error in sign can dramatically alter the final answer.
• Confusing Additive and Multiplicative Inverses: Remember that the additive inverse of a number changes its sign (e.g., the additive inverse of 5 is -5), whereas the multiplicative inverse (reciprocal) is the number that, when multiplied by the original number, results in 1 (e.g., the multiplicative inverse of 5 is 1/5).

Frequently Asked Questions (FAQ)

Q1: What happens if I divide a positive number by a negative number?

A1: The result will be a negative number. For example, 6 ÷ (-2) = -3.

Q2: What happens if I divide a negative number by a positive number?

A2: The result will also be a negative number. For example, -6 ÷ 2 = -3.

Q3: Is this rule applicable to all number systems (integers, rational numbers, real numbers, etc.)?

A3: Yes, this rule applies consistently across all number systems.

Q4: Why is this rule important?

A4: Understanding this rule is fundamental for performing accurate calculations in algebra, calculus, and many other areas of mathematics and science. It forms the basis for working with more complex equations and solving problems in various fields.

Q5: How can I practice this concept?

A5: Practice solving various division problems involving negative numbers. Start with simple examples and gradually increase the complexity. Work through practice problems in textbooks or online resources to reinforce your understanding.

Conclusion: Mastering Negative Number Division

Understanding why a negative divided by a negative equals a positive is crucial for mastering fundamental arithmetic and algebra. By understanding the concepts of opposites, using visual representations, and employing formal algebraic approaches, we can confidently and accurately perform divisions involving negative numbers. This understanding serves as a cornerstone for more advanced mathematical concepts and problem-solving. Remember to pay attention to signs, follow the order of operations, and practice consistently to solidify your understanding of this important mathematical rule. With consistent effort and practice, you can confidently navigate the world of negative numbers and their operations.