How to Find the Additive Inverse of a Number (With Examples)
The concept of additive inverse plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Grasping the additive inverse helps students easily solve equations, identify patterns, and avoid common mistakes between "inverse" types in maths.
What Is Additive Inverse?
An additive inverse is defined as a number which, when added to the original number, results in zero. In simple terms, the additive inverse of a number is its “opposite sign” value. You’ll find this concept applied in areas such as algebra, rational numbers, and the rules for adding integers.
Key Formula for Additive Inverse
Here’s the standard formula: \( \text{Additive Inverse of } x = -x \ )
For any number x, its additive inverse is the same number with an opposite sign. If x is negative, its inverse is positive, and vice versa. If x is zero, its additive inverse is zero.
Additive Inverse Property
Additive inverse property states: For every real, rational, or complex number a, there exists an additive inverse (–a) such that a + (–a) = 0. Zero is the unique additive identity because any number plus its additive inverse equals zero.
| Number | Additive Inverse | Sum |
|---|---|---|
| 5 | -5 | 0 |
| -3 | 3 | 0 |
| 0 | 0 | 0 |
| 2/3 | -2/3 | 0 |
| -7/9 | 7/9 | 0 |
How to Find the Additive Inverse
Follow these easy steps for any type of number:
- For a positive number: Change its sign to negative. Example: 4 → -4
- For a negative number: Change its sign to positive. Example: -8 → 8
- For a fraction or rational number: Change the sign of the numerator. Example: -3/5 → 3/5
- For zero: The additive inverse of 0 is 0.
General rule: Simply multiply the number by -1.
Worked Examples (Integers, Fractions, Rational Numbers, Zero)
| Example | Additive Inverse | Check |
|---|---|---|
| 2/3 | -2/3 | 2/3 + (–2/3) = 0 |
| -3/5 | 3/5 | –3/5 + 3/5 = 0 |
| -7 | 7 | –7 + 7 = 0 |
| 0 | 0 | 0 + 0 = 0 |
| 15/7 | –15/7 | 15/7 + (–15/7) = 0 |
Stepwise Illustration
- Find the additive inverse of –3/5:
1. The number is –3/52. Change the sign (–3 becomes 3): Additive inverse = 3/53. Check: –3/5 + 3/5 = 0 - Find the additive inverse of 0:
Additive inverse of 0 is 0, because 0 + 0 = 0
Additive Inverse for Different Types of Numbers
| Type of Number | Example | Additive Inverse |
|---|---|---|
| Integer | 9 | -9 |
| Negative Integer | –8 | 8 |
| Fraction | 5/11 | –5/11 |
| Rational Number | –12/15 | 12/15 |
| Zero | 0 | 0 |
Difference Between Additive and Multiplicative Inverse
| Property | Additive Inverse | Multiplicative Inverse |
|---|---|---|
| Operation | Added to number to get 0 | Multiplied to number to get 1 |
| Formula | –x | 1/x |
| Example | 5 + (–5) = 0 | 5 × (1/5) = 1 |
| Also called | Opposite Number | Reciprocal |
Frequent Errors and Misunderstandings
- Confusing additive inverse with reciprocal (multiplicative inverse).
- Forgetting to change only the sign, not the value.
- Assuming additive inverse of a fraction is its reciprocal (wrong!).
- Thinking zero has no additive inverse—actually 0 is its own additive inverse.
Speed Trick or Quick Tip
When you see “find the additive inverse”, just flip the sign on the number! For fractions and decimals, only the sign changes — never the denominator or decimal value.
For a complex number a + bi, the additive inverse is –a – bi.
Try These Yourself
- What is the additive inverse of 17?
- Find the additive inverse of –11/9.
- Write the additive inverse of 0.6.
- What is the sum of 3/8 and its additive inverse?
- Does zero have an additive inverse?
Relation to Other Concepts
The idea of additive inverse connects closely with additive identity and multiplicative inverse. Mastering this helps with understanding rules for solving equations, especially in integers, rational numbers, and fractions.
Classroom Tip
A quick way to remember the additive inverse: use the number line. The additive inverse is always the mirror image of the number across zero. Vedantu’s teachers often use this visual strategy to simplify the concept for all learners in live online maths sessions.
We explored additive inverse—from definition, formula, examples, common mistakes, and its relation to different number types. Continue your practice with Vedantu to solve additive inverse problems quickly and confidently!
Related Links for Deeper Learning:
Multiplicative Inverse |
Additive and Multiplicative Identity |
Rational Numbers |
Integers |
Fractions |
Addition and Subtraction of Fractions |
Properties of Integers |
What is an Integer? |
Addition of Integers Rules
FAQs on Additive Inverse in Mathematics: Definition, Rules & Examples
1. What is the additive inverse of a number?
The additive inverse of a number is the number that, when added to the original number, results in zero (the additive identity). It's also known as the opposite of the number. For example, the additive inverse of 5 is -5, because 5 + (-5) = 0.
2. How do you find the additive inverse of a fraction?
To find the additive inverse of a fraction, simply change the sign of the fraction. The additive inverse of a/b is -a/b. For example, the additive inverse of 2/3 is -2/3, and the additive inverse of -3/5 is 3/5.
3. What is the additive inverse of 0?
The additive inverse of 0 is 0 itself, since 0 + 0 = 0.
4. What is the additive inverse of -3/5?
The additive inverse of -3/5 is 3/5 because -3/5 + 3/5 = 0.
5. Is additive inverse the same as reciprocal?
No, additive inverse and multiplicative inverse (reciprocal) are different. The additive inverse results in zero when added to the original number. The multiplicative inverse (reciprocal) results in one when multiplied by the original number. The reciprocal of a number a/b is b/a.
6. What is the additive inverse property?
The additive inverse property states that for every real number 'a', there exists an additive inverse '-a' such that a + (-a) = 0. This property is fundamental to arithmetic and algebra.
7. What is the additive inverse of a complex number (a + bi)?
The additive inverse of a complex number a + bi is -a - bi. This is because (a + bi) + (-a - bi) = 0.
8. How is the additive inverse used in solving equations?
The additive inverse is crucial for solving equations. To isolate a variable, you add its additive inverse to both sides of the equation. For example, to solve x + 5 = 10, you add -5 (the additive inverse of 5) to both sides: x + 5 + (-5) = 10 + (-5), simplifying to x = 5.
9. What's the difference between the additive inverse and the additive identity?
The additive inverse is the number you add to get zero. The additive identity is the number you add to any number to get that same number back (which is 0).
10. Can you provide examples of additive inverses with integers?
Here are some examples using integers: • The additive inverse of 7 is -7 (7 + (-7) = 0) • The additive inverse of -12 is 12 (-12 + 12 = 0) • The additive inverse of 0 is 0 (0 + 0 = 0)
11. How is additive inverse related to subtraction?
Subtraction can be viewed as adding the additive inverse. For example, 5 - 3 is the same as 5 + (-3).
12. What are some common mistakes students make with additive inverses?
Common mistakes include confusing additive inverses with multiplicative inverses (reciprocals) and incorrectly handling signs, especially when dealing with negative numbers.
