Additive inverse of a number is what you add to the original number to get a sum of zero. On the other hand, multiplicative inverse of a number is what you multiply the original number by to get a product of one.
Let's learn about the Additive Inverse and Multiplicative Inverse with the help of solved examples.
Additive Inverse Definition
The additive inverse of a number is a value that, when added to the original number, results in a sum of zero
For a number 'a', it is denoted as '-a'. This represents the value that, when added to 'a,' results in a sum of zero.
For example, the additive inverse of 5 is -5.
Additive Inverse of 0
For any number 'a,' the additive inverse of 0 is still 0. This is because when 0 is added to 0, the result is, unsurprisingly, 0.
While other numbers have both positive and negative additive inverses, 0 is unique in having an additive inverse that is the same number itself.
Symbolically, 0 + 0 = 0.
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Multiplicative Inverse Definition
The multiplicative inverse, or reciprocal, of a number is a value that, when multiplied with the original number, results in a product of one. It is also called the reciprocal of a number.
The multiplicative inverse, denoted as '1/a' or 'a-1' is the reciprocal of a non-zero number 'a,' such that their product equals 1.
For instance, the multiplicative inverse of 3 is 1/3, as 3 × (1/3) equals 1.
Multiplicative Inverse of 0
The multiplicative inverse, or reciprocal, of 0 is undefined. In other words, there is no real number 'a' such that 0 multiplied by 'a' equals 1.
Symbolically, 0×a=1 has no real solution for 'a'.
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Difference between Additive Inverse and Multiplicative Inverse
Let's discuss the differences between additive and multiplicative inverses.
Additive Inverse vs. Multiplicative Inverse | ||
|---|---|---|
Aspect | Additive Inverse | Multiplicative Inverse |
Definition and Notation | The additive inverse of a number 'a' is represented as '-a.' | The multiplicative inverse of a non-zero number 'a' is denoted as '1/a' or 'a-1.' |
Similarities | Both inverses involve operations with a given number to result in a specific outcome: zero for additive inverses and one for multiplicative inverses. | Both concepts play fundamental roles in algebraic manipulations and equation-solving. |
Properties | Adding a number to its additive inverse yields zero: a+(−a)=0. | Multiplying a number by its multiplicative inverse results in one: a x (1/a) =1. |
Applications | Used in solving equations, balancing expressions, and understanding symmetries. | Applied in solving equations involving division, scaling, and proportionality. |
Additive Inverse and Multiplicative Inverse Examples
Let's solve some example questions on Additive Inverse and Multiplicative Inverse.
1. Find the additive inverse of -12.
The additive inverse of -12 is 12, as -12 + 12 equals 0.
2. Determine the multiplicative inverse of 1/3.
The multiplicative inverse of 1/3 is 3, as (1/3) × 3 equals 1.
3. Solve for 'x' in the equation 2x + 5 = 0 using additive inverses.
Subtract 5 from both sides to get 2x = -5. Then, divide by 2 to find x = -5/2.
4. Apply the multiplicative inverse to solve 4y = 8.
Divide both sides by 4 to find y = 2.
5. Use additive inverses to balance the equation 2a - 7 = 5.
Add 7 to both sides to get 2a = 12. Then, divide by 2 to find a = 6.