Negative Numbers

Those numbers or integers that have a minus value are called negative numbers. Although they were rejected by early mathematicians and scholars, today negative numbers play a significant role in many fields like statistics and psychology. Below is a picture of how negative numbers are placed on the number line.

While performing basic operations like addition, subtraction, multiplication, and division when the given numbers are negative, there are specific rules to be followed, and they are mentioned below

• When two negative numbers are added together, the result is always negative.

• When positive and negative numbers are added together, the sign of the number with the higher value will be written in the answer. For example, when -11 is added to 2, the answer is -9.

• The result is negative when we multiply a negative number by a positive number. E.g., -3  5 = -15.

• When two negative numbers are multiplied, the product is always positive. For example, -2  -4 = -8. 

• Dividing two negative numbers will result in a positive number.

• Dividing a negative number by a positive number results in a negative number.

 1. Addition of negative numbers
  
When two negative numbers are added, the result is always a negative number. 

Example: Solve (-2) + (-6)
(-2) + (-6) = -8.

2. Addition of a negative number and a positive number

When adding a negative number and a positive number, the result is the difference between the two numbers with the sign of the larger number.

Example: Solve (-8) + 3
(-8) + 3 = -5.

3. Multiplication of a negative number and a positive number 

When a negative number and a positive number are multiplied, the result is negative.

(-) × (+) = (-).

Example: Solve  -4 × 3
-4 × 3 = -12
 

4. Multiplication of two negative numbers

The result is positive when two negative numbers are multiplied.
(-)  (-) = (+).

Example: Solve (-3) × (-5)
(-3) × (-5) = 15

5. Division of two negative numbers.

Dividing two negative numbers results in a positive number.
(-)/(-) = (+)

Example: Solve -12/-4
-12/-4 = 3.

6. Division of a positive number and a negative number.

Dividing a positive number and a negative number will be negative.

(-)/(+) = (-).

Example: Solve (-21)/(+3) 

(-21)/(+3) = -7.

In mathematics, the square root of negative numbers is not possible, which is why we use imaginary numbers, which are represented by i = √-1.

The square root of a negative number, -a, can be expressed as:
√-a  =  i√a, where ‘a’ is a positive number. 

For example: 

• √-16 = 4i
   
• √-9 = 3i

There are certain rules for exponents with negative numbers. When an exponent is added to a negative number, say -1, the result depends on the sign (positive or negative) of the exponent.

• If the exponent n is even, then (-1)ⁿ = 1
   
• If n is odd, then (-1)ⁿ = -1

For example, 

• -3⁴ = -3   -3   -3   -3 = 81 (the exponent (4) is an even number, so according to the rule stated above, the result is positive)

• -5³ = -5  -5  -5 = -125 (the exponent is odd, so the result is negative).

Negative numbers have many applications in the real world. Some of them have been mentioned below:

• Finance and Stock Market: In the stock market, drops in stock prices are represented by negative numbers.

• Business: Any business will have profit as well as losses. While the profit is represented by positive numbers, loss is represented by negative numbers.

• Weather Forecast: While measuring temperature, negative numbers indicate cold conditions below freezing.

• Measurements: While measuring sea levels, the negative values indicate locations below the sea level.

• Medicine and Health: Negative numbers indicate a massive drop in body temperature, which is called hypothermia. Negative numbers are also used in various medical reports.