What is the additive inverse of the complex number 9 4i?

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What is the additive inverse of the complex number $2-4 i ?$

In this tutorial you will learn how to find the additive inverse of a complex number.

But before we dig into the additive inverse of complex numbers, let’s quickly recap what exactly an additive inverse is. We will start with the real numbers.

(Note, the additive inverse of a complex number is not the same as its multiplicative inverse. Check out our multiplicative inverse tutorial for everything you need to know about multiplicative inverses of complex numbers.)

The additive inverse of a real number x, is a number that when added to x, gives an answer of 0.

If the additive inverse is denoted as x’, then for a given x , if there exists x′ such that x + x′ = 0 , then x′ is called the additive inverse of x.

In other words, adding the additive inverse of a number to itself is sort of like “undoing” the number, so that you end up back at 0. For this reason, the additive inverse is also sometimes called the opposite number. When we get to the additive inverse of complex numbers in a bit, you will see why.

So how do you calculate the additive inverse of a real number?

Well, intuitively, we know that the additive inverse of 3 is –3, because when we add (-3) to 3, the answer is 0.

3 + (-3) = 0

In a similar way, the inverse of –14.5 is 14.5 and the inverse of is .

To formalize this, let me ask you – what did we do to the number’s above to find their additive inverse?

That’s right, we just switched the sign of the number. In other words, the additive inverse of a any real number is found by multiplying the real number by -1.

In general, the additive inverse of a number n is –n = -1 x n.

Next, let’s see if this holds true for complex numbers as well.

The Additive Inverse Of A Complex Number

A complex number z can be expressed as the sum of a real number a and an imaginary number bi.

z = a + bi

From the definition of additive inverses, in order to find the additive inverse of z, we need to find a number z’ which when added to z, will give us an answer of 0.

In other words, we need a z’ such that z + z’ = 0.

Let’s say that z’ is the complex number c + di.

z’ = c + di

Let’s add z and z’ together and set them equal to 0 to find the inverse of z.

We can remove the imaginary unit i from the second bracket as a common factor.

Remember that the number 0 is actually a complex number which can be written as 0 + 0i.

An lastly, due to the definition of what makes two complex numbers equal, we know that for the above equation, the real part of the left hand side equals the real part of the right hand side, and the imaginary part of the left hand side equals the imaginary part of the right hand side.

Remember we defined z’ as c+di. So let’s plug the above values for c and d into z’ and see what happens.

So the additive inverse of any complex number z = a + bi is z’ = -a – bi

Now do you recall how at the beginning of the tutorial we said that to find the additive inverse of any real number you need to multiply that number with -1?

Well, this holds for complex numbers too!

Let’s check it out!

If z = a + bi, we can easily calculate z.(-1)

So multiplying a complex number by -1 results in the additive inverse of the complex number – just like it does with real numbers. i.e. The additive inverse of a complex number z is -z.

To state it formally: The additive inverse of the complex number  where is .

So now, using this, it becomes pretty straightforward to find the additive inverse of any complex number.

How To Find The Additive Inverse Of Complex Numbers

To calculate the additive inverse of a complex number z:

1. Write the complex number in standard form z = a + bi
2. Multiply the number by -1
3. The additive inverse of z is -z = -a – bi

Or, an even easier way – once the complex number z is in the standard a+bi form, just change the sign of the real term and the imaginary term.

Let’s look at an example.

What is the additive inverse of the complex number 9 – 4i?

First, let z = 9 – 4i
Then z’, the inverse of z, is (-1)(z)

= -1(9 – 4i)
= -9 + 4i

So the additive inverse of 9 – 4i is –9 + 4i. We could have also got to this answer by simply inverting the sign of the real and imaginary terms.

Visualizing the Additive Inverse Of A Complex Number

Remember at the beginning of the tutorial we said that the additive inverse of a number is sometimes called the opposite value of the number? Let’s see why this is especially true when it comes to the additive inverse of a complex number.

Sticking with the example of z = 9 – 4i, let’s plot z on the complex plane.

Now, let’s put the inverse of z (z’= -9 + 4i) on the same plot.

Can you see how z’ is on the opposite side of the complex plane from z? The additive inverse of a complex number has the effect of rotating the complex number 180 degrees around the origin (as you can see in the image below).

This is why the additive inverse is sometimes called the opposite number.

Properties Of The Additive Inverse Of Complex Numbers

Many of the properties of the additive inverse of real numbers hold true for complex numbers as well.

For a complex number z, and it’s inverse z’:

1. z + z’ = 0 : This follows from the definition of the additive inverse of a complex number.
2. The inverse of z’ is z : Taking the inverse of an inverse gives you the original number back
3. z’ is unique : To see this, let x′ and x″ each be additive inverses of x; then x′ = x′ + 0 = x′ + (x + x″) = (x′ + x) + x″ = 0 + x″ = x″

The Additive Inverse Of Complex Numbers – Examples

What is the additive inverse of -2+3i?

Solution:

If z = -2+3i, then it’s additive inverse z’ = (-1)(z)

z’ = (-1)(-2 + 3i)
z’ = 2 – 3i

(Alternatively, we could have just used the quick cut and inverted the signs of the real and imaginary parts of z to get to the additive inverse 2 – 3i)

What is the additive inverse of 4-7i?

Solution:

If z = 4-7i, then it’s additive inverse z’ = (-1)(z)

z’ = (-1)(4 – 7i)
z’ = -4 + 7i

(Alternatively, we could have just used the quick cut and inverted the signs of the real and imaginary parts of z to get to the additive inverse -4+ 7i)

What is the additive inverse of -8 + 3i?

Solution:

If z = -8 + 3i, then it’s additive inverse z’ = (-1)(z)

z’ = (-1)(-8 + 3i)
z’ = 8 – 3i

(Alternatively, we could have just used the quick cut and inverted the signs of the real and imaginary parts of z to get to the additive inverse 8 – 3i)