Competitive Exams

IIT JEE, NEET, ESE, GATE, AE/JE, Olympiad

Only IAS

UPSC, State PSC

School Preparation

Foundation (Class 6-12), Commerce, Arts, CuriousJr (1st - 8th), Science, International Boards

Govt Exam

Judiciary, SSC, Defence, Teaching, JAIIB & CAIIB, BIHAR EXAMS WALLAH, UP Exams, Railway, Nursing Exams, Banking, WB Exams

UG & PG Entrance Exams

MBA, IPMAT, IIT JAM, LAW, CUET UG, UGC NET, GMAT, Design & Architecture, Pharma, CSIR NET, CUET PG, NEET PG

FINANCE

CA, CS, Finance Courses, ACCA

Others

English Wallah, Online Degrees

English Proficiency Test

IELTS, TOEFL

Agriculture

Additive Inverse: Definition, Properties, and Examples

Additive inverse is the number that balances another to zero. Understand the additive inverse of real and complex numbers with solved examples here.

Chandni 21 Jan, 2025

Additive Inverse: Every number has an opposite that perfectly cancels it out. This concept, called the additive inverse, is fundamental to understanding mathematical relationships and solving equations.

Whether dealing with real numbers, complex numbers, or algebraic expressions, the additive inverse helps balance and simplify calculations. In this blog, we will explore the applications of the additive inverse in real and complex numbers with solved examples.

[video width="1920" height="1080" mp4="https://www.pw.live/exams/wp-content/uploads/2024/12/Curious-Jr-Ad-3-1-1.mp4"][/video]

Additive Inverse Definition

The additive inverse of a number is the value that, when added to the original number, gives a sum of zero. Simply put, it’s the opposite of the number. If the number is positive, its additive inverse is negative, and if the number is negative, its additive inverse is positive. This works because a number and its opposite cancel each other out. For example, the additive inverse of 6 is -6, because 6 + (-6) equals 0. Similarly, the additive inverse of -9 is 9, because -9 + 9 also equals 0. Even zero has an additive inverse, which is itself, because 0 + 0 equals 0. Imagine you have 5 candies and you give all 5 away. Now, you have nothing left. This is like adding 5 and -5 to get 0. Similarly, if you have a debt of $10 (represented as -10) and repay $10, your balance becomes zero. This concept of canceling out a number with its opposite is what defines the additive inverse. [video width="1920" height="1080" mp4="https://www.pw.live/exams/wp-content/uploads/2024/12/curious-jr.mp4"][/video]

Properties of Additive Inverse

The properties of the additive inverse, derived from negating the original number, are outlined below.

Double Negation: −(−a) = a
Square Property: (−a) ² = a ²
Distributive Over Addition: −(a+b) = (−a)+(−b)
Distributive Over Subtraction: −(a−b) = (−a)+b
Subtraction of Negative: a−(−b) = a+b
Multiplication with a Negative Factor: (−a)⋅b = a⋅(−b) =−(a⋅b)
Product of Two Negatives: (−a)⋅(−b) = a⋅b

Additive Inverse of Real Numbers

The additive inverse of a real number is the value that, when added to the original number, results in a sum of zero. Essentially, the additive inverse is the opposite of the given number. The concept applies to all real numbers, which include natural numbers, whole numbers, integers, fractions, decimals, rational numbers, and irrational numbers. Let’s understand this in detail with clear examples for each type of number.

Additive Inverse of Natural Numbers

Natural numbers are counting numbers starting from 1, and they are always positive. The additive inverse of a natural number is its negative counterpart. For example, the additive inverse of 5 is −5, because 5 + (−5)=0.

Additive Inverse of Whole Numbers

Whole numbers include all natural numbers and zero. The additive inverse of a positive whole number is its negative. For instance, the additive inverse of 7 is −7. However, zero is an exception because its additive inverse is itself: 0 + 0 = 0.

Additive Inverse of Integers

Integers include positive numbers, negative numbers, and zero. For positive integers, the additive inverse is the negative version. For example, the additive inverse of 3 is −3. For negative integers, the additive inverse is the positive version, as multiplying by −1 reverses the sign. For example, the additive inverse of −6 is 6, because −6 + 6 = 0. For zero, its additive inverse remains zero.

Additive Inverse of Fractions

Fractions are positive unless stated otherwise. The additive inverse of a fraction is the negative version. If a fraction is a/b, its additive inverse is −a/b For instance, the additive inverse of 2/3 is −2/3 because 2/3 + (−2/3) = 0.

Additive Inverse of Rational Numbers

Rational numbers can be either positive or negative. The additive inverse is found by multiplying the number by −1. For a positive rational number p/q, its additive inverse is −p/q. For a negative rational number −p/q, its additive inverse is p/q. For example, the additive inverse of 5/4 is −5/4, and the additive inverse of −3/7 is 3/7.

Additive Inverse of Decimals

A decimal consists of a whole part and a fractional part. The additive inverse involves changing the sign of the entire number. For example, the additive inverse of 2.35 is −2.35, and the additive inverse of −0.2 is 0.2.

Additive Inverse of Irrational Numbers

Irrational numbers include non-terminating, non-repeating decimals and roots of non-perfect squares or cubes. The additive inverse is obtained by multiplying the number by −1. For example, the additive inverse of √2 is -√2, because √2

. Similarly, the additive inverse of

2 - \sqrt 3

, the additive inverse is

, because

2 - \sqrt 3 + (

Additive Inverse of Complex Numbers

A complex number is written in the form a + ib , where a and b are real numbers, and i represents the imaginary unit (√-1). The additive inverse of a complex number is another complex number that, when added to the original, results in zero (0 + i0). In other words, for a complex number a + ib , its additive inverse is c + id , such that:

How It Works:

To find the additive inverse of a + ib , we simply take the negative of both the real and imaginary parts. This means the additive inverse is -a - ib . For example:

The additive inverse of -3 + 2i is 3 - 2i .
The additive inverse of a + 3i is -a - 3i .

Additive Inverse of Algebraic Expressions

The additive inverse concept applies to algebraic expressions , which are combinations of variables and constants. To find the additive inverse of an algebraic expression, multiply every term in the expression by -1. This flips the sign of each term (positive becomes negative, and negative becomes positive). When the original expression and its additive inverse are added, the result is always zero.

How It Works:

For any algebraic expression P(x), the additive inverse is −P(x) If P(x) = ax ⁿ + bx ⁿ⁻¹ + ⋯+c, its additive inverse is: −P(x) =−ax ⁿ − bx ⁿ⁻¹ −⋯−c

Difference Between Additive Inverse and Multiplicative Inverse

The following table outlines the differences between additive and multiplicative inverses, incorporating insights from the provided references and practical examples.

Difference Between Additive Inverse and Multiplicative Inverse
Aspect	Additive Inverse	Multiplicative Inverse
Definition	The additive inverse of a number a is the number −a.	The multiplicative inverse of a non-zero number a is 1/a.
Notation	Represented as −a	Represented as 1/a or $a^{-1}$
Objective	Balances the number to zero when added.	Balances the number to one when multiplied.
Applicability	Applicable to all real numbers, including zero.	Applicable only to non-zero numbers (zero has no multiplicative inverse)
Outcome	Adding a number to its additive inverse gives zero: a+(−a)=0	Multiplying a number by its multiplicative inverse gives one: a× (1/a)= 1.
Similarities	Both involve operations that "cancel out" the original number in their respective contexts.	Both are used to simplify equations and mathematical expressions.
Applications	Additive inverses are used in mathematics, physics, and economics to solve equations, simplify expressions, and balance values.	Multiplicative inverses are used in algebra to isolate variables (e.g., and in calculus to simplify functions like

Additive Inverse Solved Examples

Example 1 : Find the additive inverse of 2/3x−4/5. Solution: The additive inverse of each term in the expression is obtained by multiplying by −1: Additive inverse = − ( 2/3 x − 4/5 ) = − 2/3 x + 4/5 Verification: (2/3x−4/5) + (−2/3x+4/5) = 2/3x − 2/3x−4/5 + 4/5 = 0

Example 2: Find the additive inverse of 2+3i−5

Solution: The additive inverse of each part is obtained by flipping the sign: Additive inverse= −1⋅(2+ √3i−√5) = −2− √3i + √5 Verification: (2+√3i−√5) + (−2−√3i+√5) = 2−2+√3i − √3i − √5 +√5 = 0

Example 5: Find the additive inverse of 4x ² y−7xy ² +5z−8

Solution: Multiply each term in the expression by −1. Additive inverse = −(4xy ² − 7xy ² + + 5z − 8) = 4x ² y + 7xy ² - 5z + 8 Verification: (4x ² y − 7xy ² + 5z − 8) + (-4x ² y + 7xy ² - 5z + 8)

Related Articles
Area of Rectangle	Isosceles Triangle
Composite Numbers	Differentiation
Perimeter of Rectangle	Surface Area of Cylinder

Additive Inverse FAQs

How does the additive inverse differ from the multiplicative inverse?

The additive inverse cancels a number to make the sum zero, while the multiplicative inverse cancels a number to make the product one.

Can the additive inverse simplify logarithmic equations?

Yes, in log⁡(a)+log⁡(−a)=, the additive inverse negates the log arguments

How does the additive inverse work in modular arithmetic?

Imodular arithmetic, the adn ditive inverse of a is the number that, when added to a, gives a remainder of zero.

What is the additive inverse in three-dimensional space?

In 3D space, the additive inverse of a point (x,y,z)is (−x,−y,−z)

🔥 Trending Blogs

Essay on Pollution: Understand Causes, Types, and Impact

What are Extracurricular Activities, Types and Benefits

NCERT Solutions for Class 5 Maths Chapter 9 Boxes and Sketches

Mental Maths Questions With Solutions, Download PDF

Natural Phenomena for Class 8 Junior Explorers

Talk to a counsellorHave doubts? Our support team will be happy to assist you!

7019243492

07019243492

Check out these Related Articles

CURIOUS JUNIOR

Addition Word Problems for Class 3, Download PDF

Read Full Story

1 Aug 2025, 01:14 pm

CURIOUS JUNIOR

Even Numbers: Definition, Example, Properties, List 1 to 1000

Read Full Story

1 Aug 2025, 01:13 pm

CURIOUS JUNIOR

Separation of Substances: Class 6 Science

Read Full Story

1 Aug 2025, 01:09 pm

CURIOUS JUNIOR

Trapezium: Properties, Types, Formulas, and Applications

Read Full Story

1 Aug 2025, 01:12 pm

CURIOUS JUNIOR

Adjectives - Definition, Types, Forms, Usage, Examples

Read Full Story

1 Aug 2025, 01:08 pm

CURIOUS JUNIOR

List of Homophones Words A to Z

Read Full Story

1 Aug 2025, 01:07 pm

CURIOUS JUNIOR

How to Write 15000 in Words in English, Check Here

Read Full Story

1 Aug 2025, 01:06 pm

CURIOUS JUNIOR

Class 8 Sound: Adventure into the World of Sound

Read Full Story

1 Aug 2025, 01:05 pm

CURIOUS JUNIOR

What are Length, Width, and Height: Key Differences with Examples

Read Full Story

1 Aug 2025, 01:04 pm

Join 15 Million students on the app today!

Live & recorded classes available at ease

Dashboard for progress tracking

Millions of practice questions at your fingertips

Free Learning Resources

PW Books

Notes (Class 10-12)

Class 10 Math's Notes

Class 10 Chemistry Notes

Class 10 Physics Notes

lass 10 Biology Notes

Aptitude & Reasoning

Class 10 Geography

Physics Class 11 Notes

Class 11 Chemistry Notes

Maths Notes Class 11

Zoology Class 11

Class 11 Botany Notes

Physics Class 12 notes

Chemistry Class 12

Maths Notes Class 12

Zoology class 12

Class 12th Botany Notes

PW Study Materials

Arjuna JEE Study Material

JEE 5 PYQs

JEE Mind Maps

Yakeen Study Material

NCERT Punch

Arjuna Neet Study Material

NEET PYQs

CBSE Sample Paper

CBSE Question Bank

Udaan For Class 10th Study Material

Notes (Class 6-9)

Class-6 Theory & Notes

Math's Notes for class 7

Science Notes for class 7

Class 8 Math Notes

Class 8 Chemistry Notes

Class 8 Physics Notes

Class 8 Biology Notes

Class 8 SST Notes

Class 9 Math's Notes

Class 9 Physics Notes

Class 9 Chemistry Notes

Class 9 Biology Notes

Ncert Solutions

NCERT Solutions For Class 6

NCERT Solutions For class 7

NCERT Solutions for class 8

NCERT Solutions for class 9

NCERT Solutions for class 10

NCERT Solutions for class 11

NCERT Solutions for Class 12

NCERT Solutions For Class 6

Govt Exams

Entrance Exams

Our Other Websites

Class 6th to 12th Online Courses

Class 12 Online Course

Class 11 Online Course

Class 10 Online Course

Class 9 Online Course

Class 8 Online Course

Govt Job Exams Courses

SSC Online Coaching

Bank Exam Online Coaching

TET Online Coaching

JAIIB & CAIIB Online Coaching

Bihar Exam Online Coaching

UPSC Coaching

UPSC Online Coaching

UPSC Offline & Hybrid Coaching

State PSC Online Coaching

UPPSC Online Coaching

BPSC Online Coaching

Defence Exam Coaching

NDA Online Coaching

CDS Online Coaching

AFCAT Online Coaching

Agniveer Online Coaching

Gate Exam Coaching

Civil Engineering Courses

Mechanical Engineering Courses

Formulas

Other Exams

Company Secretary Exam

Know about Physics Wallah

Physics Wallah is an Indian edtech platform that provides accessible & comprehensive learning experiences to students from Class 6th to postgraduate level. We also provide extensive NCERT solutions, sample paper, NEET, JEE Mains, BITSAT previous year papers & more such resources to students. Physics Wallah also caters to over 3.5 million registered students and over 78 lakh+ Youtube subscribers with 4.8 rating on its app.

We Stand Out because

We provide students with intensive courses with India’s qualified & experienced faculties & mentors. PW strives to make the learning experience comprehensive and accessible for students of all sections of society. We believe in empowering every single student who couldn't dream of a good career in engineering and medical field earlier.

Our Key Focus Areas

Physics Wallah's main focus is to make the learning experience as economical as possible for all students. With our affordable courses like Lakshya, Udaan and Arjuna and many others, we have been able to provide a platform for lakhs of aspirants. From providing Chemistry, Maths, Physics formula to giving e-books of eminent authors like RD Sharma, RS Aggarwal and Lakhmir Singh, PW focuses on every single student's need for preparation.

What Makes Us Different

Physics Wallah strives to develop a comprehensive pedagogical structure for students, where they get a state-of-the-art learning experience with study material and resources. Apart from catering students preparing for JEE Mains and NEET, PW also provides study material for each state board like Uttar Pradesh, Bihar, and others

[video width="1920" height="1080" mp4="https://www.pw.live/exams/wp-content/uploads/2024/12/Curious-Jr-Ad-3-1-1.mp4"][/video]

Additive Inverse Definition

Properties of Additive Inverse

The properties of the additive inverse, derived from negating the original number, are outlined below.

Double Negation: −(−a) = a
Square Property: (−a) ² = a ²
Distributive Over Addition: −(a+b) = (−a)+(−b)
Distributive Over Subtraction: −(a−b) = (−a)+b
Subtraction of Negative: a−(−b) = a+b
Multiplication with a Negative Factor: (−a)⋅b = a⋅(−b) =−(a⋅b)
Product of Two Negatives: (−a)⋅(−b) = a⋅b

Additive Inverse of Real Numbers

Additive Inverse of Natural Numbers

Additive Inverse of Whole Numbers

Additive Inverse of Integers

Additive Inverse of Fractions

Additive Inverse of Rational Numbers

Additive Inverse of Decimals

Additive Inverse of Irrational Numbers

. Similarly, the additive inverse of

2 - \sqrt 3

, the additive inverse is

, because

2 - \sqrt 3 + (

Additive Inverse of Complex Numbers

How It Works:

To find the additive inverse of a + ib , we simply take the negative of both the real and imaginary parts. This means the additive inverse is -a - ib . For example:

The additive inverse of -3 + 2i is 3 - 2i .
The additive inverse of a + 3i is -a - 3i .

Additive Inverse of Algebraic Expressions

How It Works:

For any algebraic expression P(x), the additive inverse is −P(x) If P(x) = ax ⁿ + bx ⁿ⁻¹ + ⋯+c, its additive inverse is: −P(x) =−ax ⁿ − bx ⁿ⁻¹ −⋯−c

Difference Between Additive Inverse and Multiplicative Inverse

The following table outlines the differences between additive and multiplicative inverses, incorporating insights from the provided references and practical examples.

Difference Between Additive Inverse and Multiplicative Inverse
Aspect	Additive Inverse	Multiplicative Inverse
Definition	The additive inverse of a number a is the number −a.	The multiplicative inverse of a non-zero number a is 1/a.
Notation	Represented as −a	Represented as 1/a or $a^{-1}$
Objective	Balances the number to zero when added.	Balances the number to one when multiplied.
Applicability	Applicable to all real numbers, including zero.	Applicable only to non-zero numbers (zero has no multiplicative inverse)
Outcome	Adding a number to its additive inverse gives zero: a+(−a)=0	Multiplying a number by its multiplicative inverse gives one: a× (1/a)= 1.
Similarities	Both involve operations that "cancel out" the original number in their respective contexts.	Both are used to simplify equations and mathematical expressions.
Applications	Additive inverses are used in mathematics, physics, and economics to solve equations, simplify expressions, and balance values.	Multiplicative inverses are used in algebra to isolate variables (e.g., and in calculus to simplify functions like

Additive Inverse Solved Examples

Example 2: Find the additive inverse of 2+3i−5

Example 5: Find the additive inverse of 4x ² y−7xy ² +5z−8

Related Articles
Area of Rectangle	Isosceles Triangle
Composite Numbers	Differentiation
Perimeter of Rectangle	Surface Area of Cylinder