# Complex Numbers: z^0=1?

• Hepic
In summary, complex numbers consist of a real part and an imaginary part, written in the form a + bi. To raise a complex number to the power of 0, the imaginary part is set to 0 and the real part is set to 1, resulting in a value of 1. This is because any number raised to the power of 0 is equal to 1. Additionally, any complex number raised to the power of 0 yields a real number. Complex numbers can be represented geometrically on a complex plane, where the real and imaginary parts are plotted on the x-axis and y-axis respectively. The distance from the origin to the point representing the complex number is known as the modulus, and the angle formed by the

#### Hepic

Lets say z!=0, and zeC(is complex).
So for example is z=2+3i.
z^0=1 => (2+3i)^0=1. I am correct? I know that all numbers in zero make us one,but it works with complex numbers too?

Yes.
It works in the case of complex numbers as well.

In general, though, the topic of exponentiation and logarithms of complex numbers is a nasty subject, where we have to be very careful when dealing with them.

Alright. Thank you !

## 1. What are complex numbers?

Complex numbers are numbers that consist of a real part and an imaginary part. They are written in the form a + bi, where a is the real part and bi is the imaginary part with i being the square root of -1.

## 2. How do you raise a complex number to the power of 0?

To raise a complex number to the power of 0, you simply have to set the imaginary part to 0 and the real part to 1. This results in the value of 1.

## 3. Why does z^0 equal 1 for complex numbers?

This is because any number raised to the power of 0 is equal to 1. In the case of complex numbers, the imaginary part is set to 0, resulting in the real part being the only value left, which is 1.

## 4. Are there any other special properties of raising complex numbers to the power of 0?

Yes, another special property is that any complex number raised to the power of 0 yields a real number. This is because the imaginary part is set to 0, leaving only the real part.

## 5. How can complex numbers be represented geometrically?

Complex numbers can be represented geometrically on a complex plane, where the real part is plotted on the x-axis and the imaginary part is plotted on the y-axis. The distance from the origin to the point representing the complex number is known as the modulus, and the angle formed by the point and the positive x-axis is known as the argument or phase angle.

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