What is z Inverse in Complex Number?

The inverse of a complex number z, also known as its multiplicative inverse, is simply 1/z.

Understanding the Complex Number Inverse

Let's delve deeper into what this means. If z is a complex number, its inverse, denoted as z^-1 or 1/z, is the complex number that, when multiplied by z, results in 1 (the multiplicative identity).

Calculation

If z = a + bi, where a and b are real numbers and i is the imaginary unit (√-1), then the inverse of z is calculated as follows:

1/z = 1/(a + bi)

To express this in the standard form of a complex number (x + yi), we need to rationalize the denominator by multiplying both the numerator and denominator by the complex conjugate of z, which is a - bi:

1/z = (1/(a + bi)) ((a - bi)/(a - bi*))

1/z = (a - bi) / (a² - (bi)²)

Since i² = -1, we have:

1/z = (a - bi) / (a² + b²)

Therefore, the inverse of z = a + bi is:

1/z = a/(a² + b²) - (b/(a² + b²))*i

Example

Let's say z = 2 + 3i. Then:

1/z = 1/(2 + 3i)

Multiply numerator and denominator by the conjugate (2 - 3i):

1/z = (2 - 3i) / ((2 + 3i)(2 - 3i))

1/z = (2 - 3i) / (4 + 9)

1/z = (2 - 3i) / 13

1/z = 2/13 - (3/13)*i

So, the inverse of 2 + 3i is 2/13 - (3/13)*i.

Summary

In summary, the inverse of a complex number z allows us to perform division in the complex number system, and it is calculated by dividing the complex conjugate of z by the square of its magnitude. It's important for various mathematical operations involving complex numbers.