If You Divide Two Negatives Is It A Positive

Dividing two negative numbers might seem counterintuitive at first, but the result is always a positive number. This mathematical principle is fundamental to arithmetic and algebra, and understanding it unlocks more complex mathematical concepts.

The Foundation of Negative Numbers

Before diving into division, understanding the basics of negative numbers is crucial. Negative numbers are any real number less than zero. They are used to represent deficits, opposites, or quantities below a reference point. Think of a thermometer: temperatures below zero are negative. In finance, an overdraft in a bank account is a negative balance.

Visualizing Negative Numbers on a Number Line

The number line is an invaluable tool for visualizing negative numbers. It extends infinitely in both directions, with zero at the center. Positive numbers lie to the right of zero, increasing in value as you move rightward. Negative numbers lie to the left of zero, decreasing in value as you move leftward. Each negative number is the mirror image of its positive counterpart concerning zero. For instance, -3 is located three units to the left of zero, while +3 is three units to the right.

Basic Operations with Negative Numbers

Understanding how negative numbers behave under basic arithmetic operations is key to grasping why dividing two negatives yields a positive.

Addition: Adding two negative numbers results in a negative number. For example, -2 + (-3) = -5. Think of it as accumulating debt; if you already owe $2 and then borrow another $3, you now owe $5.
Subtraction: Subtracting a negative number is the same as adding its positive counterpart. For example, 5 - (-2) = 5 + 2 = 7. This can be visualized as removing a debt; if someone takes away a $2 debt, your net worth increases by $2.
Multiplication: Multiplying a positive number by a negative number results in a negative number. For example, 3 x (-4) = -12. Multiplying two negative numbers results in a positive number. For example, -3 x -4 = 12. This concept is crucial and will be further explained later.

The Concept of Division

Division is essentially the inverse operation of multiplication. It determines how many times one number (the divisor) is contained within another number (the dividend). The result of this operation is called the quotient. For example, 12 ÷ 3 = 4, because 3 fits into 12 four times.

Division and Multiplication: Inverse Operations

The relationship between division and multiplication is fundamental. If a ÷ b = c, then b x c = a. This inverse relationship is crucial for understanding why certain rules apply in division, especially when dealing with negative numbers.

Understanding the Sign Rules in Division

The sign rules in division are directly derived from the sign rules in multiplication.

Positive ÷ Positive = Positive: This is straightforward. For example, 10 ÷ 2 = 5.
Negative ÷ Positive = Negative: For example, -10 ÷ 2 = -5.
Positive ÷ Negative = Negative: For example, 10 ÷ -2 = -5.
Negative ÷ Negative = Positive: This is the core of the topic. For example, -10 ÷ -2 = 5.

Why Does Dividing Two Negatives Result in a Positive?

The rule that dividing two negative numbers results in a positive number can be explained through several approaches, including mathematical proofs and real-world analogies.

Mathematical Proof

One way to understand this rule is through algebraic manipulation. Consider the equation:

-a / -b = x

Multiplying both sides by -b, we get:

-a = x * -b

To isolate x, we can multiply both sides by -1:

(-1) * -a = (-1) * x * -b

a = x * b

Now, dividing both sides by b:

a / b = x

Therefore, -a / -b = a / b, which means that dividing a negative number by another negative number yields a positive number.

Using the Inverse Relationship with Multiplication

Division can be thought of as the inverse of multiplication. We know that a negative times a negative yields a positive. Let's consider the problem: -12 ÷ -3 = ?

We're looking for a number that, when multiplied by -3, equals -12. In other words:

-3 * ? = -12

The only number that satisfies this equation is 4, because -3 * 4 = -12. Therefore, -12 ÷ -3 = 4.

Real-World Analogies

Sometimes, understanding abstract mathematical concepts is easier with real-world examples. While these analogies might not be perfect, they can provide an intuitive understanding.

Debt Reduction: Imagine you have a debt of $20 (-$20). If you divide that debt equally among 4 people, each person assumes a debt of $5 (-$5). This can be represented as -20 ÷ 4 = -5. Now, consider the opposite: removing debt. Suppose a debt of $20 (-$20) is being removed (divided negatively) from 4 people. Removing a debt is the same as giving them money, so each person effectively gains $5. This can be represented as -20 ÷ -4 = 5.
Direction and Velocity: Think of velocity as movement in a certain direction. Positive velocity is movement forward, and negative velocity is movement backward. Time can also be positive (future) or negative (past). If a car has a negative velocity (-30 mph), it's moving backward. If you look at its position 2 hours ago (-2 hours), the distance it covered is positive (60 miles), because it was moving backward in the past. This can be represented as distance = velocity x time, or -60 = -30 x 2, so -60 / -30 = 2.

Common Mistakes to Avoid

While the rule itself is straightforward, mistakes can occur if the underlying concepts are not fully understood.

Confusing Addition/Subtraction with Multiplication/Division

A common mistake is to apply the rules of addition and subtraction to multiplication and division, or vice versa. Remember that adding two negative numbers results in a negative number, but multiplying or dividing two negative numbers results in a positive number.

Incorrectly Applying the Order of Operations

The order of operations (PEMDAS/BODMAS) is crucial in complex expressions. Always perform operations in the correct order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Sign Errors

Careless sign errors are common. Always double-check the signs of each number before performing any operation. It's helpful to rewrite the problem if necessary to avoid confusion.

Examples and Practice Problems

To solidify the understanding of dividing negative numbers, let's go through several examples:

Example 1

Simplify: -24 ÷ -6

Both numbers are negative, so the result will be positive. 24 ÷ 6 = 4. Therefore, -24 ÷ -6 = 4.

Example 2

Simplify: (-15 + 3) ÷ -2

First, simplify the expression in the parentheses: -15 + 3 = -12. Now, divide: -12 ÷ -2 = 6.

Example 3

Simplify: -36 ÷ (4 x -3)

First, simplify the expression in the parentheses: 4 x -3 = -12. Now, divide: -36 ÷ -12 = 3.

Practice Problems

-45 ÷ -9 = ?
-100 ÷ -5 = ?
(-8 x 3) ÷ -4 = ?
(-20 + 5) ÷ -3 = ?
-72 ÷ (-6 - 2) = ?

Solutions to Practice Problems

Advanced Applications

The principle of dividing two negative numbers extends to more advanced mathematical fields, including algebra, calculus, and complex numbers.

Algebra

In algebra, this principle is used extensively in solving equations and simplifying expressions. For instance, when solving for x in the equation -2x = -10, you divide both sides by -2, resulting in x = 5.

Calculus

In calculus, dealing with derivatives and integrals often involves negative numbers. Understanding how to divide them correctly is crucial for finding accurate solutions. For example, when evaluating limits or finding the slope of a curve.

Complex Numbers

Complex numbers, which have both a real and an imaginary part, also follow these rules. Division of complex numbers involves multiplying the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator. This process also relies on the principle of dividing two negatives to get a positive.

The Historical Context

The development of negative numbers and their acceptance into mathematics was a gradual process. Ancient mathematicians initially struggled with the concept of numbers less than zero. It wasn't until the Indian mathematicians of the 7th century that negative numbers were formally recognized and used in arithmetic operations.

Early Resistance to Negative Numbers

Early Greek mathematicians, such as Euclid and Pythagoras, did not acknowledge negative numbers. They primarily focused on geometry and positive quantities. The idea of a number representing a deficit or absence was difficult to grasp.

The Breakthrough in India

Indian mathematicians, such as Brahmagupta, made significant contributions to the understanding and use of negative numbers. In his book "Brahmasphutasiddhanta," he outlined the rules for dealing with negative numbers, including the rule that a negative divided by a negative is a positive.

Gradual Acceptance in Europe

It took several centuries for negative numbers to be fully accepted in Europe. Mathematicians like Fibonacci and Cardano used negative numbers in their work, but they were often referred to as "fictitious" or "absurd" numbers. It wasn't until the 17th and 18th centuries that negative numbers became a standard part of mathematical notation and practice.

Conclusion

Dividing two negative numbers resulting in a positive number is a fundamental rule in mathematics, supported by both mathematical proofs and real-world analogies. Understanding this rule is crucial for mastering basic arithmetic and progressing to more advanced mathematical concepts. By avoiding common mistakes and practicing regularly, you can confidently apply this principle in various mathematical contexts. From solving algebraic equations to tackling calculus problems, this rule is a cornerstone of mathematical understanding.