Negative Numbers Lessons - Definition, Rules, Examples

Lesson Overview

• What Are Negative Numbers?
• Rules for Negative Numbers
• Adding And Subtracting Negative Numbers
• Multiplication and Division of Negative Numbers
• Negative Integers With Exponents
• Negative Numbers Examples

In mathematics, negative numbers are values that fall below zero on the number line. They expand our ability to represent quantities and solve problems, allowing us to describe concepts like temperatures below freezing, depths beneath sea level, or financial debts.

Their applications range from describing the charge of an electron (-1) to modeling changes in stock prices.

What Are Negative Numbers?

On a number line, negative numbers are positioned to the left of zero. Examples include -3, -1.2, -5/2. These numbers allow us to express values below a reference point, such as: 

• Temperature: -8°F (eight degrees Fahrenheit below zero)
• Elevation: -50 meters (50 meters below sea level)

Finance: -$25 (a debt of 25 dollars)

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Negative Integers Quiz 1

Rules for Negative Numbers

Negative numbers follow specific rules when added, subtracted, multiplied, or divided. These rules ensure consistent and accurate calculations, whether dealing with temperatures, finances, or scientific data.

Adding And Subtracting Negative Numbers

For adding and subtracting negative numbers, there are some rules. These rules help work with numbers below zero using simple techniques.

Adding Negative Numbers

When adding negative numbers, think of moving left on the number line. Combine values and apply the sign based on the rules.

Rule 1: Adding a negative to a negative
When two negative numbers are added, combine their values and keep the result negative.
Example:
-2 + (-5)

• Combine the values: 2 + 5 = 7
• Apply the negative sign: -7

So, -2 + (-5) = -7.

Rule 2: Adding a positive to a negative
Find the difference between their absolute values and keep the sign of the larger absolute value.
Example:
-6 + 4

• Find the difference: 6 - 4 = 2
• Since 6 has the larger absolute value, keep the negative sign.

So, -6 + 4 = -2.

Subtracting Negative Numbers

Subtraction of negative numbers involves changing the operation to addition. Adjust the second number's sign and follow addition rules.

Case 1: Subtracting a Positive Number from a Negative Number

Treat subtracting a positive number like adding a negative number.

Example:

-4 + (-3)

Change to: -4 + (-3)

Combine the values: -4 + (-3) = -7

So, -4 + (-3) = -7

Case 2: Subtracting a Negative Number from a Negative Number

Treat subtracting a negative number like adding a positive number.

Example:

-6 - (-2)

• Change to: -6 + 2
• Combine the values: -6 + 2 = -4

So, -6 - (-2) = -4.

Tips to Remember

1. On the number line, moving left means subtracting or adding a negative.
2. Moving right means adding a positive.
3. Always switch subtraction to addition and adjust the second number's sign.

This method works for all addition and subtraction problems involving negative numbers.

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Integers: Positive and Negative Numbers

Multiplication and Division of Negative Numbers

These operations also follow specific rules that determine the sign of the result.

Multiplying Positive and Negative Numbers

• Different signs: Multiply a negative number by a positive number, and the result is negative.
  Example: -5 × 4 = -20
• Same signs: Multiply two negative numbers, and the result is positive.
  Example: -5 × -4 = 20

Dividing Positive and Negative Numbers

• Different signs: Divide a negative number by a positive number, and the result is negative.
  Example: -30 ÷ 5 = -6
• Same signs: Divide two negative numbers, and the result is positive.
  Example: -30 ÷ -5 = 6

Tips to Remember for Multiplying and Dividing Positive and Negative Numbers

• For Different Signs: The result is always negative when multiplying or dividing numbers with different signs.
• For Same Signs: The result is always positive when multiplying or dividing numbers with the same signs.

Quick Check: Use a mental rule: same signs give positive results, different signs give negative results.

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Multiplying and Dividing Negative Numbers.

Negative Integers With Exponents

When dealing with negative integers and exponents, two important rules help determine the final result:

• Even Exponent: If the exponent is an even number, the result will always be positive.
  Example: (-3)^4 = (-3) × (-3) × (-3) × (-3) = 81
• Odd Exponent: If the exponent is an odd number, the result will always be negative.
  Example: (-2)^5 = (-2) × (-2) × (-2) × (-2) × (-2) = -32

Negative Numbers Examples

Example 1: Adding Negative Numbers

-12 + (-8)

• Since we are adding two negative numbers, the result will be negative.
• Add the absolute values of the numbers: 12 + 8 = 20
• Place the negative sign in front of the sum: -20
• Therefore, -12 + (-8) = -20

Example 2: Subtracting Negative Numbers

-5 - (-9)

• Subtracting a negative number is the same as adding its positive counterpart.
• Rewrite the problem: -5 + 9
• Find the difference between the absolute values: 9 - 5 = 4
• Since 9 has a larger absolute value and is positive, the result is positive.
• Therefore, -5 - (-9) = 4

Example 3: Multiplying Negative Numbers

-7 × 4

• We are multiplying numbers with different signs.
• The product of a negative and a positive number is always negative.
• Multiply the absolute values: 7 × 4 = 28
• Place the negative sign in front of the product: -28
• Therefore, -7 × 4 = -28

Example 4: Dividing Negative Numbers

-36 ÷ (-9)

• We are dividing numbers with the same sign.
• The quotient of two negative numbers is always positive.
• Divide the absolute values: 36 ÷ 9 = 4
• Therefore, -36 ÷ (-9) = 4

Example 5: Combined Operations

-10 + (-3) × 2 - (-5)

• Follow the order of operations (PEMDAS/BODMAS):
  • Multiplication: -3 × 2 = -6
  • Rewrite the expression: -10 + (-6) - (-5)
  • Addition: -10 + (-6) = -16
  • Rewrite the expression: -16 - (-5)
  • Subtraction: -16 - (-5) = -16 + 5 = -11
• Therefore, -10 + (-3) × 2 - (-5) = -11

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