Negative Numbers

Negative Numbers are the numbers that are represented on the negative side of the number line. Negative Numbers are the numbers whose value is less than zero. They are placed on the left-hand side of the zero on the number line. We apply the (-) minus sign before negative numbers to represent them. For example, -5 represents a number that is five units on the left side of zero in the number line.

In his article, we will learn about, negative numbers, operations on negative numbers, their properties, examples, and others in detail.

Negative Numbers Definition

Negative Numbers as the name suggest are the numbers that contain a negative (-) sign as the prefix. They are on the left side of zero on the number line. Negative numbers can be integers, rational numbers, decimals, etc. Various examples of negative numbers are -3, -1.12, -4/5, etc. The figure added below shows the negative numbers on the number line.

Negative numbers are -3, -4, -5, … and so on.

Rules of Negative Numbers

We can easily perform all the basic operations such as addition, subtraction, multiplication, and division on negative numbers and the basic properties of negative numbers are,

• Sum of two negative numbers is always a negative number. Example, (-24) + (-12) = -(24 + 12) = -36
• Sum of a positive number and a negative number is the difference of both numbers with the sign of the bigger number. Example, (-11) + 5 = -11 + 5 = -(11-5) = -6
• Product of two negative numbers is a positive number. Example, (-3) × (-6) = 18
• Product of a positive number and a negative number is a negative number. Example, (-3) × 4 = -12
• Division of two negative numbers is a positive number. Example, (-6)/(-3) = 2
• Divsion of a positive number and a negative number is a negative number. Example, (-32)/4 = -8

How to Add and Subtract Negative Numbers?

Negative numbers can be easily added or subtracted performing addition and subtraction on negative numbers we must follow the following rules.

Addition of Negative Numbers

Addition of negative numbers is studied under two following headings:

• Case 1: Addition of two Negative Numbers
• Case 2: Addition of a Negative Number and a Positive Number

Now let’s learn the same in detail.

Case 1: Addition of two Negative Numbers

In the case of adding two negative numbers, we simply add the two numbers and then we put a negative (minus) sign in front of the result. This can be explained by following the example

Example: Simplify (-11) + (-7)

Solution:

= (-11) + (-7)

= -(11 + 7)

= -(18)

Case 2: Addition of a Negative Number and a Positive Number

In the case of the addition of a Negative Number and a Positive Number we simply find the difference of the two numbers and then put the sign of the bigger number. This can be explained by following the example,

Example: Simplify (-11) + 17

Solution:

= (-11) + 17

= (17 – 11)

= 6

Subtraction of Negative Numbers

Two negative numbers are subtracted by following the rule that says, “Change the operation of subtraction to addition and also change the sign of the second number.” This rule is explained by the example as,

Example: Simplify (-12) – (-14)

Solution:

(-12) – (-14)

= (-12) + 14

= (14 – 12)

= 2

Multiplication and Division of Negative Numbers

Multiplication and Division of Negative Numbers are done in the same manner as in the case of a normal taking into consideration the minus or the negative sign. The multiplication and division of two negative numbers always result in a positive number while the multiplication and division of a negative number with a positive number give a negative number as a result. Let’s look into the detail

Multiplication of Negative Numbers

Multiplication of negative numbers is achieved by following the rules as discussed below, there are two ways of multiplying the negative numbers,

• Case 1: Multiplying a Negative Number with a Positive Number
• Case 2: Multiplying Negative Number with Negative Number

Let’s learn the same in detail.

Multiplying a Negative Number with a Positive Number

Negative numbers are multiplied by positive numbers then the result is obtained by finding the product of two numbers and then applying the negative sign, this is explained as,

(-a) × (b) = -(ab)

Thus, the product of a negative number and the positive number is a positive number. We can explain the same with same by the example,

Example: Multiply the same, (-3) × (6)

Solution:

= (-3) × (6)

= -18

Multiplying a Negative Number with a Negative Number

Negative numbers are multiplied by negative numbers then the result is obtained by finding the product of two numbers and then applying the positive sign, this is explained as,

(-a) × (-b) = (ab)

Thus, the product of a negative number and the negative number is a positive number. We can explain the same with same by the example,

Example: Multiply the same, (-3) × (-6)

Solution:

= (-3) × (-6)

= 18

Division of Negative Numbers

Division of negative numbers is achieved by following the rules as discussed below, there are two ways of division of the negative numbers,

• Case 1: Divison Negative Number with a Positive Number
• Case 2: Divison Negative Number with Negative Number

Let’s learn the same in detail.

Division by Negative Number with Positive Number

Negative numbers are divided by positive numbers then the result is obtained by dividing two numbers and then applying the negative sign, this is explained as,

(-a) / (b) = -(a/b)

Thus, the division of a negative number and the positive number is a positive number. We can explain the same with same by the example,

Example: Divide (-6) by (3)

Solution:

= (-6) / (3)

= -2

Dividing a Negative Number with a Negative Number

When Negative numbers are divided by negative numbers then the result is obtained by finding the quotient of two numbers and then applying the positive sign, this is explained as,

(-a) / (-b) = a/b

Thus, the division of a negative number and the negative number is a positive number. We can explain the same with same by the example,

Example: Divide (-6) by (-3)

Solution:

= (-6) / (-3)

= 2

Negative Numbers with Exponents

The exponents of the negative number are calculated by following the rules,

• (-1)ⁿ = 1 if n is even
• (-1)ⁿ = (-1) if n is odd

These rules are explained as,

If n is even then the value of the exponent is positive and the exponent is calculated normally,

For example, -4⁴ = (-4) × (-4) × (-4) × (-4) = 256

If n is odd then the value of the exponent is negative and the exponent is calculated normally,

For example, -4³ = (-4) × (-4) × (-4) = -64

Learn More, Laws of Exponents

Square Root of Negative Numbers

From the above heading, we know that if a negative number is raised to even power then the result is positive. Hence, the square of a negative number is positive. We can find the square root of a number only if its square is defined. Hence, the square root of a negative number is not a real number rather it is an imaginary number. Let’s see an example

Example: Find the square root of -4

Solution:

√(-4) = √(-1 ⨯ 4) = √-1 ⨯ √4 = 2i

Here i is called iota and i = √-1 . Here i is imaginary number.

Read More,

• Whole Numbers
• Real Numbers
• Prime Numbers

Examples on Negative Numbers

Example: Simplify (-21) – (-18)

Solution:

= (-21) – (-18)

= (-21) + 18

= – (21 – 18)

= -3

Example: Simplify (-34) + (-19)

Solution:

= (-34) + (-19)

= (-34) – 19

= – (34 + 19)

= -53

Example: Simplify (-12) / (2)

Solution:

= (-12) / (2)

= – 12/2

= – 6

Example: Simplify (-12) × (-6)

Solution:

= (-12) × (-6)

= 12 × 6

= 72

FAQs on Negative Numbers

1. What are Negative Numbers in Mathematics?

Negative numbers are the numbers in mathematics that are always left side of zero on the number line. These numbers have a minus sign (-) in front of these numbers to represent that they are negative numbers.

2. What are Some Examples of Negative numbers?

Some examples of negative numbers are -2, -3.78, -11/9, etc.

3. What are the Properties of Negative Numbers?

There are various properties associated with negative numbers and some of the important properties of negative numbers are,

• Sum of two negative numbers is always a negative number.
• Multiple of two negative numbers is always a positive number.
• Difference between two negative numbers is either a positive number or a negative number.
• Division of two negative numbers is always a positive number, etc.

4. What is the Sum of Two Negative Numbers?

The sum of two negative numbers is always a negative number, suppose we have two negative numbers (-a), and (-b) then their sum is found as,

(-a) + (-b) = -(a+b)

5. What is the Product of Two Negative Numbers?

The product of two negative numbers is always a positive number, suppose we have two negative numbers (-a), and (-b) then their product is found as,

(-a) + (-b) = -(a+b)

6. What are Negative Numbers used for?

Negative numbers are used to represent various sorts of things in daily life, suppose when we deposit money in our bank account then the amount is credited to our account (positive number), whereas, when we withdraw money from our bank account then the amount is debited from our account (negative number), etc.

7. What is the Square of a Negative Number?

The square of a negative number is a positive number.

8. What is the Square Root of Negative Numbers?

The square root of negative numbers is an imaginary number

9. What is the Mod of Negative Number?

The Mod of Negative Numbers is the positive value of the Number. For Example, mod of -2 i.e. |-2| = 2.

10. What is the Log of Negative Numbers?

The log of negative numbers is not defined.

11. What is Factorial of Negative Numbers?

The Factorial of Negative Number is a complex number

12. Can Prime Numbers be Negative?

No, the prime numbers can’t be negative. Prime Numbers are always positive.