Additive Identity and Inverse of Complex Numbers

Additive Inverse is one of the most fundament topics in the stream of Algebra which is a branch of mathematics that deals with variables and numbers. Additive Inverse is also an important topic under the other branches such as Number Theory and Abstract Algebra. This article deals with this concept of Additive Inverse in good detail and also the method of finding Additive Inverse for each type of number such as Integers, Rational Numbers, Real Numbers, Complex Numbers, etc. as well as other mathematical structures such as Matrices.

Definition of Additive Inverse

For any real number a, x is called its additive inverse iff a + x = Additive Identity i.e., 0.

Examples of Additive Inverse

Some examples of additive inverse are:

• -3 is the additive inverse of 3 and vice versa.
• -2/5 is the additive inverse of 2/5 and vice versa.
• 3√2 is the additive inverse of -3√2 and vice versa.
• a + ib is the additive inverse of -a -ib and vice versa. 
• \begin{bmatrix} a&b\\c & d \end{bmatrix}       is the additive inverse of \begin{bmatrix} -a & -b\\-c & -d \end{bmatrix}       and vice versa.

Properties of Additive Inverse

There are several properties of  Additive Inverse, some of which are as follows:

Every Number has an Additive Inverse:

For any number a, there exists a number b such that a+b=0. This additive inverse is unique, meaning that there is only one number b that satisfies this equation.

Additive Inverse is Symmetric:

If a is the additive inverse of b, then b is the additive inverse of a. This can be written as (-a) = b and (-b) = a.

Additive Inverse and Subtraction:

Subtraction of a number a from another number b is equivalent to adding the additive inverse of a to b. For example, b - a = b +(-a).

Additive Inverse and Zero:

The additive inverse of zero is zero itself, since 0 + 0 = 0. This is sometimes referred to as the "trivial" additive inverse.

Additive Inverse and Addition:

The sum of a number and its additive inverse is always zero. In other words, a + (-a) = 0. This property is sometimes called the "cancellation law."

Uniqueness of Additive Inverse

Every real number has a unique additive inverse. This means that for any real number a, there is only one real number b such that a+b=0.

Proof:

Suppose there are two numbers b and c that are both additive inverses of a. 

i.e., a + b =0 and a + c = 0. Then, we can add b and c to get:

b + c = b + c + (a+b) + (a+c)
= (b+a) + (b+c) + c
= 0 + 0 + c
= c

So we have b+c = c, which implies that b=0, contradicting the assumption that b is the additive inverse of a. 

Therefore, there can only be one additive inverse for a given number.

How to Find Additive Inverse?

To find the Additive Inverse of any number, we can use the following steps:

Step 1: Start with the given number.

Step 2: Change the sign of the number by placing a minus sign (-) in front of it.

Step 3: The opposite or negative of the original number is the additive inverse of the given number.

Finding Additive Inverse of a Rational Number

We can easily find the inverse of the rational number by just taking the negative value of the given rational number. Suppose we are given a national number as p/q them its additive inverse is -p/q. Some examples of the same are,

Example: Find the additive inverse of

• 9
• -11

Solution:

• Additive Inverse of 9 = -(9) = -9
• Additive Inverse of -11 = -(-11) = 11

Finding Additive Inverse of an Integer

As we know, the additive inverse of a number is the number which when added to the original number, yields a sum of 0. Thus, this holds for Integers as well. 

To find the additive inverse of an Integer, we simply change the sign of the integer i.e., if the integer is positive, its inverse is a negative number of the same and if the inverse is negative then its inverse is the positive number of the same. For example, the additive inverse of 7 is -7, and the additive inverse of -2 is 2.

Finding Additive Inverse of a Rational Number

A number of the p/q where p and q both are integers and q can't be equal to 0, is called Rational Number and similar to Integers, additive inverse for Rational Numbers can be found by changing the polarity (negative or positive sign) of the number. For example, the additive inverse of -3/4 is 3/4, and the additive inverse of 7/5 is -7/5.

Finding Additive Inverse of a Complex Number

An ordered pair of numbers (a, b) represented in the a + ib is called a complex number where a and b are the real number. For a complex number a + ib, the additive inverse is defined as a complex number c + id such that

(a + ib) + (c + id) = 0 + i0)

For example, 3 - 2i is the additive inverse of -3 + 2i, a + 3i is the additive inverse of -a - 3i, etc.

Finding Additive Inverse of a Polynomial

Similar to complex numbers, it is also possible to find the additive inverse of a polynomial. In algebra, a polynomial is an expression consisting of variables and coefficients and can be written in the form of a sum of terms that are powers of those variables. An additive inverse for a polynomial can be found just by changing each coefficient in the Polynomial.

For example, the additive inverse of the polynomial 3x³ + 2x² - 5x + 7 is the polynomial -3x³ - 2x² + 5x - 7, since their sum is equal to 0. Similarly, the additive inverse of the polynomial -2x² + x - 3 is the polynomial 2x² - x + 3.

Finding Additive Inverse of a Matrix

In linear algebra, a matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Just like numbers and polynomials, matrices also have additive inverses. The additive inverse of a matrix A is another matrix B such that their sum is equal to the additive identity matrix i.e., null matrix, denoted by the symbol 0. The additive identity matrix is a square matrix with all its elements equal to zero, except for the diagonal elements, which are equal to one.

For example, if A = \begin{bmatrix}1&2\\ 3&4 \end{bmatrix}      , then its additive inverse B is  \begin{bmatrix}-1&-2\\ -3&-4 \end{bmatrix}      . 

Since A = \begin{bmatrix}1&2\\ 3&4 \end{bmatrix}

 \because A + B = \begin{bmatrix}1&2\\ 3&4  \end{bmatrix} + \begin{bmatrix}-1&-2\\ -3&-4 \end{bmatrix} = \begin{bmatrix} 0&0\\ 0&0\end{bmatrix}

Learn more about, Inverse of a Matrix

Additive Inverse Vs Multiplicative Inverse

There are some key differences between Additive Inverse and Multiplicative that are discussed in the table below,

Solved Problems on Additive Inverse

Problem 1: What is the additive inverse of -12?

Solution:

Let the additive inverse of -12 be x. Then, we know that for a number a and its additive inverse b,

a + b = additive identity i.e., 0

So, we have:

-12 + x = 0

⇒ x = 12

Thus, 12 is the additive inverse of -12.

Problem 2: Determine the additive inverse of the fraction 3/5.

Solution:

Let the additive inverse of 3/5 be x. Then, we know that for a number a and its additive inverse b,

a + b = additive identity i.e., 0

So, we have:

3/5 + x = 0

⇒ x = -3/5

Thus, 3/5 is the additive inverse of -3/5.

Problem 3: If a is the additive inverse of b, what is the additive inverse of a?

Solution:

If a is the additive inverse of b, then we know that:

a + b = 0

Let's assume x is the additive inverse of a, then by the definition of additive inverse

a + x = 0

But as a + b = 0 and a + x = 0

⇒ x = b

Thus, b is the additive inverse of a.

Problem 4: What is the additive inverse of the complex number 4 + 5i?

Solution:

Let the complex number 4 + 5i have an additive inverse x + yi. 

For the additive inverse of 4 + 5i, we know,

(4 + 5i) + (x + yi) = 0 + 0i

⇒ (4 + x) + (5 + y)i = 0 + 0i

Real part: 4 + x = 0

⇒ x = -4

Imaginary part: 5 + y = 0

⇒ y = -5

Therefore, the additive inverse of 4 + 5i is -4 - 5i. 

Problem 5: Determine the additive inverse of the polynomial 2x² - 3x + 1.

Solution:

As the additive inverse of a polynomial is simply the polynomial with all its coefficients negated. 

Therefore, the additive inverse of 2x² - 3x + 1 is -2x² + 3x - 1.

To verify the solution, we can use the property of additive inver a + a^-1 = 0,

Thus, 2x² - 3x + 1+ (-2x² - (-3)x - (-1))  = 0

Thus, additive inverse of 2x² - 3x + 1 is -2x² + 3x - 1.

Problem 6: Find the additive inverse of the matrix \bold{A = \begin{bmatrix} 3& -4\\2 & -1 \end{bmatrix}}

Solution:

Let the matrix A have an additive inverse matrix X. 

For the additive inverse of matrix A, we know,

A + X = 0

where 0 is the null matrix i.e., matrix with all elements equal to zero.

⇒ X = -A

Thus, additive inverse of matrix A is \bold{-A = \begin{bmatrix} -3& 4\\-2 & 1 \end{bmatrix}} .

Unsolved Practice Problems

1. Find the additive inverse of -7.
2. Determine the additive inverse of the fraction 5/8​.
3. If 3 + x = 0 what is the value of x?
4. Find the additive inverse of the complex number 2 - 6i.
5. What is the additive inverse of the polynomial 4x³ - 3x+2?
6. Determine the additive inverse of the matrix [5 −3].
7. If a= - 10 and b= - 15, what is the additive inverse of a + b?
8. Find the additive inverse of the rational number −7/9.
9. What is the additive inverse of the polynomial −x² + 4x - 7?
10. For the matrix [0 1], find its additive inverse.

Related Articles:

• What is the Multiplicative Identity and Multiplicative Inverse of the complex number?
• Additive Identity and Inverse of Complex Numbers
• Additive Identity vs Multiplicative Identity