What is an Additive Inverse?
In Mathematics, the additive inverse of a number is defined as a value which on adding with the original number obtains the result 0. The additive inverse of the original number ‘p’ is represented by unary minus i.e. - p.
For Example:
The additive inverse of 29 is -29 because 29 - 29 = 0
The additive inverse of -29 is 29 because -29 + 29 = 0
The additive inverse of - 0.4 is 0.4, as -0.4 + 0.4 = 0
The additive inverse is also known as the opposite number, sign change, negation. For a real number, it changes its sign. The additive inverse of a positive number is always negative whereas the additive inverse of a negative number is always positive. The ‘0’ is the only digit with its additive inverse being itself. Hence, the additive inverse of the number 0 is 0.
What is the General Formula of Additive Inverse of a Number?
The general formula of the additive inverse of a number is derived as:
Additive Inverse Formula = -1 * Number
Additive Inverse Number of Complex Number
A complex number is defined as a sum of a real number ‘a’ and imaginary number ‘bi’. Hence, the complex number of Z can be represented as a sum of real number ‘a’ and imaginary number ‘bi’.
Accordingly,
Z = a + bi
In order to find the additive inverse of Z, we need to find a number z’ which when added to z, obtains the result 0.
Let us say that z’ is the complex number c + di
Accordingly,
z’ = c + di
To find the inverse of z, let's add z and z’ to get the result 0.
z + z’ = 0
Substituting the value of z = a + bi, and z’ = c + d in the above equation
We get, a + bi + c + di = 0
The above equation can be written as (a + c) ( bi + di) = 0
We will now remove the imaginary number ‘i’ from the bracket and consider it as a common number
(a + c) + (b + d)i = 0
As we know, 0 is a complex number and can be written as 0 + 0i
Accordingly,
(a + c) + (b + d)i = 0 + 0i
Lastly, because of the definition of what makes two complex number equal, we know that for the equation given above, the real part of the left side will be equal to the real part of the right-hand side, and the imaginary number of the left-hand side will be equal to the imaginary number of the right-hand side.
Accordingly,
(a + c) + (b + d)i = 0 + 0i
a + c = 0 and,
b + d = 0
Hence, the value of c and d are -a and -b respectively.
Lets now substitute the value of c and d in z’ and see what happens
z’ = c + di
z’ = -a +( -bi)
z’ = -a - bi
Hence, the additive inverse of complex number z = a + bi is z’ = -a - bi
Additive Inverse of Rational Number
Additive Inverse of a rational number is the same original number with the opposite sign. For example, the additive inverse of 2/3 is - 2/3.
We can also determine the additive inverse of a rational number by multiplying the original number with -1.
For Example,
The additive inverse of 15/7 is
15/7 * -1 = -15/7
Solved Examples
1. Add the rational number 5/3 and its additive inverse.
Solution:
Step 1:
The additive inverse of 5/3 is -5/3.
Hence, we have to find 5/3 + (- 5/3)
Step 2:
As we can see in the above addition, both the rational numbers have opposite signs, we have to determine the absolute difference of them without the actual sign.
Hence,
|5/3 - 5/3| = |0| = 0
So,
5/3 + ( - 5/3) = 0
2. What is the additive inverse of complex number 9 - 4i?
Solution:
Step 1:
To find the additive inverse of complex number 9 - 4i, we will first write the given complex number in the standard form i.e.
Z = 9 - 4i
Step 2:
Now, we will multiply the number by -1 as shown below:
-1 * (9 - 4i) = -9 + 4i
Hence, the additive inverse of complex number 9 - 4i is -9 + 4i.
We could have also calculated the inverse of complex number 9 - 4i by inverting the signs of real and imaginary terms.
Q1. Mention Some Points on the Difference Between Additive Inverse and Multiplicative Inverse?
Ans. The difference between additive inverse and the multiplicative inverse is listed below in a tabular form.
Additive Inverse | Multiplicative Inverse |
The additive inverse of a number is defined as a value which on adding with the original number obtains the result 0 | The multiplicative inverse of a number is defined as a value which multiplying with the original number obtains the result 1 |
It obtains the result 0 | It obtains the result 1 |
Sign of the original number is first changed and then added to the original number to obtain the result 0 | Reciprocal of the original number is multiples with the original number to obtain the result 1 |
Example, 76 + ( -76) = 0 | 76 * (1/76) = 1 |
Q2.What are the Properties of Additive Inverse?
Ans. Some of the properties of additive inverse are listed below:
Additive inverse of a number is strongly linked with the subtraction, which can be observed as the addition of opposite:
y - z = y + (-z)
Conversely, the additive inverse can be examined as a subtraction from 0.
-y = 0 - y
In addition to above, some of the other properties of additive inverse are listed below:
- (-z) = z
- (y + z) = (-y) + (-z)
- (y - z) = z - y
y - (-z) = y + z
(-y) * z = y * (-z) = - ( y * z)
( -y) * ( -z) = y * z
(-y)² = z