The additive inverse of a number is the value that adds to it to give zero. The additive inverse of 9 is −9, because 9 + (−9) = 0, so it is the same thing as its opposite. To find it, change the sign: the additive inverse of −14 is 14, and 0 is its own.
A number and its additive inverse are mirror images across 0, so moving by one and then the other brings you back to 0. 0 is the only number that is its own additive inverse.
Formal definition and notation
For every real numbera, its additive inverse is the unique number−a with a + (−a) = 0. This is the inverse property of addition. On the number line, a and −a are the same distance from 0 on opposite sides. Subtraction is defined with it: a − b = a + (−b).
Symbol
Read it as
Meaning
−a
the opposite of a (or negative a)
the additive inverse of a; if a = −3, then −a = 3
a + (−a) = 0
a plus the opposite of a equals zero
the inverse property of addition
−(−a) = a
the opposite of the opposite of a is a
taking the additive inverse twice gives back the number
Key facts
Every number has exactly one additive inverse. To find it, change the sign.
0 is its own additive inverse, because 0 + 0 = 0. It is the only such number.
The additive inverse of a negative number is positive: the additive inverse of −7.5 is 7.5.
Subtracting a number means adding its additive inverse: 6 − (−9) = 6 + 9 = 15.
For a sum, change every sign: the additive inverse of x − 4 is −x + 4.
Inverse property of addition
Inverse property of addition
a + (−a) = 0
a
any number: positive, negative, a fraction or a decimal
−a
its additive inverse: the same number with the opposite sign
0
the additive identity; adding 0 changes nothing, and inverses always combine to it
The same idea turns subtraction into addition: 10 − 4 = 10 + (−4) = 6.
Additive Inverse examples, step by step
Example 1: Change the sign
Warm-up
Find the additive inverse of 23 and of −5.6.
The additive inverse of 23 is −23, since 23 + (−23) = 0.
The additive inverse of −5.6 is 5.6, since −5.6 + 5.6 = 0.
Each pair sits the same distance from 0 on opposite sides of the number line.
Answer: The additive inverses are −23 and 5.6.
Example 2: Subtract by adding the inverse
Core
Find 6 − (−9).
Subtracting a number means adding its additive inverse: a − b = a + (−b).
The additive inverse of −9 is 9, so 6 − (−9) = 6 + 9.
Add: 6 + 9 = 15.
Answer:6 − (−9) = 15.
Example 3: Use the inverse to solve an equation
Test-style
Solve x + 12 = 5.
To get x alone, cancel the 12. Its additive inverse is −12.
Add −12 to both sides: x + 12 + (−12) = 5 + (−12).
The left side becomes x + 0 = x. The right side is 5 − 12 = −7.
Check: −7 + 12 = 5. It works.
Answer:x = −7.
Common mistakes
✗ MistakeTreating it like absolute value: the additive inverse of 8 is 8.
✓ InsteadThe additive inverse of 8 is −8, because 8 + (−8) = 0.
Do not confuse additive inverse with absolute value. The additive inverse of −8 is 8, but |−8| is also 8 for a different reason: one flips the sign, the other measures distance from 0.
✗ MistakeUsing the reciprocal: the additive inverse of 5 is 15.
✓ InsteadThe additive inverse of 5 is −5: 5 + (−5) = 0.
15 is the multiplicative inverse, because 5 × 15 = 1. Additive inverses add to 0; multiplicative inverses multiply to 1.
✗ MistakeThinking the inverse must be negative: the additive inverse of −3 is −3.
✓ InsteadThe additive inverse of −3 is 3, because −3 + 3 = 0.
The additive inverse has the opposite sign, so a negative number has a positive inverse. −3 + (−3) = −6, not 0.
✗ MistakeChanging only the first sign: the additive inverse of x − 4 is −x − 4.
The multiplicative inverse multiplies to 1: for 4 it is 14. The additive inverse adds to 0: for 4 it is −4.
Quick check: 3 questions
Question 1 of 3
What is the additive inverse of −14?
Show answer and explanation
Answer: D.−14 + 14 = 0, so the additive inverse is 14. Choice A keeps the same number, choice B is the reciprocal (the multiplicative inverse), and 0 is the sum, not the inverse.
Question 2 of 3
Which sum equals 0?
Show answer and explanation
Answer: B.−7 and 7 are additive inverses, so their sum is 0. Choice A adds a reciprocal, and choices C and D add two numbers with the same sign, which moves away from 0.
Question 3 of 3
Which expression has the same value as 3 − 8?
Show answer and explanation
Answer: C. Subtracting 8 is the same as adding its additive inverse, −8: 3 − 8 = 3 + (−8) = −5. Choices A and B both equal 5, and choice D adds 8 instead of its inverse.
Additive Inverse: frequently asked questions
Is the additive inverse the same as the opposite?
Yes. For real numbers, the additive inverse of a number is its opposite: the number the same distance from 0 on the other side of the number line. The additive inverse of 12 is −12, and the opposite of 12 is −12. ‘Opposite’ describes where the number sits; ‘additive inverse’ describes what it does: it adds with the number to make 0.
What is the additive inverse of 0?
The additive inverse of 0 is 0 itself, because 0 + 0 = 0. Zero is the only number that is its own additive inverse. Every other number has an inverse with the opposite sign, so the two are different numbers: 2 and −2, or −0.4 and 0.4.
What is the difference between an additive inverse and a multiplicative inverse?
An additive inverse adds to a number to give 0; a multiplicative inverse, or reciprocal, multiplies with it to give 1. For 5, the additive inverse is −5 and the multiplicative inverse is 15. For −23, they are 23 and −32. Zero has an additive inverse (0) but no multiplicative inverse.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 7.NS.A.1.b, CCSS 7.NS.A.1.c, CCSS 6.NS.C.6.a
Cite this entry
Effortless Math Team. “Additive Inverse.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/additive-inverse/
Flashcard
TermAdditive Inversewhat adds to make 0
DefinitionThe number that adds to a given number to make 0; change the sign. The additive inverse of −14 is 14, because −14 + 14 = 0.
Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2Number LineA number line is a straight line with numbers placed at equal distances along it, so every point stands for a number. Zero sits in the middle, positive numbers go to the right and negative numbers to the left. Numbers increase from left to right, so a number line shows order, distance and adding as moving.Number System · Grade 2, Grade 3, Grade 6SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2SubtractionSubtraction is the operation that finds how much is left when you take one amount away from another, or how far apart two numbers are. It is written with a minus sign: 9 − 4 = 5. The answer is called the difference. Subtraction undoes addition, so you can check 9 − 4 = 5 by adding 5 + 4 = 9.Operations · Grade 1, Grade 2, Grade 3Negative NumberA negative number is a number less than zero, written with a minus sign, such as −3, −0.5 or −23. On a number line, negative numbers sit to the left of 0. They describe amounts below a starting point: temperatures below zero, depths below sea level, or money owed. Zero itself is neither negative nor positive.Number System · Grade 6, Grade 7Absolute ValueThe absolute value of a number is its distance from zero on the number line, so it is never negative. It is written with two vertical bars: |−6| = 6 and |6| = 6. A positive number and 0 keep their value; a negative number loses its minus sign.Number System · Grade 6, Grade 7, Grade 8ReciprocalThe reciprocal of a number is 1 divided by that number, so the two multiply to 1. For a fraction, swap the numerator and denominator: the reciprocal of 34 is 43, and the reciprocal of 5 is 15. Zero has no reciprocal. Dividing by a fraction is the same as multiplying by its reciprocal.Number System · Grade 5, Grade 6, Grade 7DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3IntegerAn integer is a whole number, its opposite, or zero: the counting numbers, their negatives, and zero. The integers are …, −3, −2, −1, 0, 1, 2, 3, … and go on forever in both directions. Integers have no fractional or decimal part, so 4.5 and 12 are not integers.Number System · Grade 6, Grade 7, Grade 8ZeroZero (0) is the number that represents none or no quantity. It is neither positive nor negative and sits in the middle of the number line, between the negative and positive numbers. Adding zero to a number does not change it, as in 9 + 0 = 9, so zero is called the additive identity.Number System · Grade 3, Grade 4, Grade 5EquationAn equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.Algebra · Grade 6, Grade 7, Grade 8Inverse OperationsInverse operations are operations that undo each other. Addition and subtraction are inverses: 9 + 6 − 6 = 9. Multiplication and division are inverses: 7 × 4 ÷ 4 = 7. You use them to check answers and to solve equations, undoing each step in reverse order until the variable is alone.Operations · Grade 1, Grade 3, Grade 6SolutionA solution is a value that makes an equation or inequality true. To test a value, substitute it for the variable and check that the statement holds: x = 4 solves x + 2 = 6 because 4 + 2 = 6. A solution can also be an ordered pair, a point on a graph that satisfies the relationship.Algebra · Grade 6, Grade 7, Grade 8OppositeThe opposite of a number is the number the same distance from 0 on the other side of the number line. The opposite of 7 is −7, and the opposite of −7 is 7. Opposites add to 0, and 0 is its own opposite. The opposite of a negative number is positive.Number System · Grade 6, Grade 7, Grade 8Rational NumberA rational number is any number that can be written as a fraction of two integers, ab, where b is not 0. Fractions, integers, terminating decimals, and repeating decimals are rational numbers: 34, −5 = −51, 1.6 and 0.333… all qualify. Numbers such as √2 and π are irrational.Number System · Grade 7, Grade 8, Algebra 1