How To Subtract Two Negative Numbers

Subtracting negative numbers might seem tricky at first, but understanding the underlying principles can make it surprisingly straightforward. The key lies in remembering that subtracting a negative number is the same as adding a positive number. This article will delve into the mechanics of subtracting negative numbers, providing clear explanations, examples, and tips to master this essential mathematical concept.

Understanding Negative Numbers

Before diving into subtraction, it's crucial to have a solid grasp of negative numbers. Negative numbers are numbers less than zero. They represent quantities that are the opposite of positive numbers. Think of a number line: zero sits in the middle, positive numbers extend to the right, and negative numbers extend to the left.

• Examples of Negative Numbers: -1, -5, -10, -3.14, -100

Negative numbers are used in various real-world scenarios:

• Temperature: Temperatures below zero degrees Celsius or Fahrenheit.
• Finance: Representing debt or overdraft in a bank account.
• Altitude: Measuring heights below sea level.
• Games: Tracking points lost or penalties incurred.

The Core Principle: Subtracting a Negative is Adding a Positive

The foundation of subtracting negative numbers rests on a simple yet powerful principle: subtracting a negative number is equivalent to adding its positive counterpart. Mathematically, this is expressed as:

a - (-b) = a + b

Where 'a' and 'b' are any numbers. This principle stems from the concept of inverse operations. Subtraction is the inverse operation of addition, and a negative number is the additive inverse of its positive counterpart.

Why does this work?

Imagine you're on a number line. Subtracting a positive number moves you to the left, decreasing the value. Subtracting a negative number, conversely, moves you to the right, increasing the value. This movement to the right is precisely what addition does.

Analogy:

Think of a debt (negative number). If someone takes away your debt (subtracting a negative), they are essentially giving you money (adding a positive).

Step-by-Step Guide to Subtracting Two Negative Numbers

Let's break down the process of subtracting two negative numbers into manageable steps:

1. Rewrite the Expression:

The first and most crucial step is to rewrite the subtraction of a negative number as addition of a positive number. Apply the principle: a - (-b) = a + b.

Example:

• -5 - (-3) becomes -5 + 3

2. Simplify the Addition:

Now you have a simple addition problem involving a negative and a positive number (or two positive numbers, depending on the original expression).

3. Determine the Sign of the Result:

To add numbers with different signs, consider their absolute values (the distance from zero).

• If the absolute value of the positive number is greater than the absolute value of the negative number, the result is positive.
• If the absolute value of the negative number is greater than the absolute value of the positive number, the result is negative.
• If the absolute values are equal, the result is zero.

4. Calculate the Magnitude of the Result:

Subtract the smaller absolute value from the larger absolute value. The result is the magnitude of your answer.

5. Combine the Sign and Magnitude:

Combine the sign you determined in step 3 with the magnitude you calculated in step 4. This is your final answer.

Example Walkthrough: -5 - (-3)

1. Rewrite: -5 - (-3) becomes -5 + 3
2. Sign: The absolute value of -5 is 5, and the absolute value of 3 is 3. Since 5 > 3, the result will be negative.
3. Magnitude: 5 - 3 = 2
4. Combine: The result is -2

Therefore, -5 - (-3) = -2

Examples and Practice Problems

Let's work through several examples to solidify your understanding:

Example 1: -8 - (-2)

1. Rewrite: -8 - (-2) becomes -8 + 2
2. Sign: |-8| = 8, |2| = 2. Since 8 > 2, the result is negative.
3. Magnitude: 8 - 2 = 6
4. Combine: The result is -6

Therefore, -8 - (-2) = -6

Example 2: -10 - (-15)

1. Rewrite: -10 - (-15) becomes -10 + 15
2. Sign: |-10| = 10, |15| = 15. Since 15 > 10, the result is positive.
3. Magnitude: 15 - 10 = 5
4. Combine: The result is 5

Therefore, -10 - (-15) = 5

Example 3: -4 - (-4)

1. Rewrite: -4 - (-4) becomes -4 + 4
2. Sign: |-4| = 4, |4| = 4. Since the absolute values are equal, the result is zero.
3. Magnitude: Not applicable (since the absolute values are equal)
4. Combine: The result is 0

Therefore, -4 - (-4) = 0

Practice Problems:

Try these on your own and check your answers:

1. -3 - (-7) = ?
2. -12 - (-5) = ?
3. -1 - (-1) = ?
4. -6 - (-10) = ?
5. -20 - (-8) = ?

Answers:

1. 4
2. -7
3. 0
4. 4
5. -12

Common Mistakes to Avoid

Subtracting negative numbers can be confusing, so it's important to be aware of common pitfalls:

• Forgetting to Rewrite: The most frequent mistake is failing to rewrite the expression by changing subtraction of a negative to addition of a positive. Always make this your first step.
• Incorrectly Determining the Sign: When adding numbers with different signs, double-check which number has the larger absolute value to determine the correct sign of the result.
• Confusion with Multiplication: Remember that this rule applies to subtraction. Multiplying two negative numbers results in a positive number. For example, (-2) * (-3) = 6, which is different from -2 - (-3).
• Rushing the Process: Take your time and follow the steps carefully. Rushing can lead to careless errors.

Advanced Applications

The principles of subtracting negative numbers extend to more complex algebraic expressions and equations.

Example 1: Simplifying Algebraic Expressions

Simplify the expression: 3x - (-2x) + 5

1. Rewrite: 3x - (-2x) becomes 3x + 2x
2. Combine like terms: 3x + 2x + 5 = 5x + 5

Therefore, the simplified expression is 5x + 5.

Example 2: Solving Equations

Solve for x: x - (-4) = -2

1. Rewrite: x - (-4) becomes x + 4
2. Equation: x + 4 = -2
3. Subtract 4 from both sides: x = -2 - 4
4. Simplify: x = -6

Therefore, the solution is x = -6.

The Number Line Visual

Visualizing the subtraction of negative numbers on a number line is a great way to understand the concept intuitively.

Imagine a number line extending infinitely in both directions, with zero at the center.

• Start: Locate the first number (the minuend) on the number line.
• Subtracting a Positive: Subtracting a positive number means moving to the left on the number line.
• Subtracting a Negative: Subtracting a negative number means moving to the right on the number line. The distance you move is equal to the absolute value of the number being subtracted.

Example: -2 - (-5)

1. Start: Locate -2 on the number line.
2. Subtract -5: Since you're subtracting a negative number, move 5 units to the right.
3. End: You end up at +3.

Therefore, -2 - (-5) = 3

The number line provides a tangible representation of how subtracting a negative number results in a movement in the positive direction, effectively adding to the original value.

Connecting to Real-World Scenarios

Understanding subtraction of negative numbers isn't just an abstract mathematical skill. It has practical applications in various real-world contexts.

1. Temperature Changes:

Suppose the temperature is -5°C, and it's expected to increase by 8°C. This can be represented as:

-5 - (-8) = -5 + 8 = 3°C

The temperature will be 3°C. Subtracting the negative change (a decrease) is the same as adding the equivalent positive change (an increase).

2. Financial Transactions:

Imagine your bank account has an overdraft of -$20. If you receive a deposit of $30, the change in your balance can be calculated as:

-20 - (-30) = -20 + 30 = $10

Your new balance will be $10. Subtracting the negative debt (overdraft) by the deposit is the same as adding the deposit to your initial debt.

3. Altitude and Depth:

A submarine is at a depth of -150 meters. It ascends by 75 meters. The new depth is:

-150 - (-75) = -150 + 75 = -75 meters

The submarine is now at a depth of -75 meters. Subtracting the negative change (ascending, moving closer to the surface) is the same as adding the equivalent positive change.

Frequently Asked Questions (FAQ)

• Why does subtracting a negative number equal adding a positive number?

  This stems from the concept of inverse operations. Subtraction is the inverse of addition, and a negative number is the additive inverse of its positive counterpart. Essentially, subtracting a negative is the same as removing a debt, which is the same as gaining something.

• Is subtracting a negative number always the same as adding?

  Yes, absolutely. This is a fundamental rule in mathematics.

• What if I have multiple subtractions of negative numbers in a row?

  Simply apply the rule repeatedly, working from left to right. For example: -2 - (-3) - (-1) = -2 + 3 + 1 = 2

• Does this rule apply to fractions and decimals as well?

  Yes, the rule applies to all real numbers, including fractions and decimals. For example: -0.5 - (-0.25) = -0.5 + 0.25 = -0.25

• How can I remember this rule?

  Think of real-world examples, such as temperature changes or financial transactions. Visualizing the number line can also be helpful. The phrase "minus a minus is a plus" is a common mnemonic device.

Conclusion

Subtracting negative numbers doesn't have to be a source of confusion. By understanding the core principle of rewriting subtraction of a negative as addition of a positive, you can confidently solve these types of problems. Remember to follow the steps carefully, avoid common mistakes, and practice regularly to solidify your understanding. With a solid grasp of this concept, you'll be well-equipped to tackle more advanced mathematical concepts with ease.