If mod2z=1 for a non zero complex number then which is necessarily true?

When we have a non-zero complex number \( z \) such that \( \text{mod} \, 2z = 1 \), we can derive some interesting properties about \( z \). The notation \( \text{mod} \, 2z \) typically refers to the modulus (or absolute value) of the complex number \( 2z \). So, we can express this mathematically as follows:

Understanding the Modulus of a Complex Number

The modulus of a complex number \( z = a + bi \) (where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit) is given by:

\( |z| = \sqrt{a^2 + b^2} \)

In our case, we need to find the modulus of \( 2z \):

\( |2z| = |2(a + bi)| = |2a + 2bi| = \sqrt{(2a)^2 + (2b)^2} = \sqrt{4a^2 + 4b^2} = 2\sqrt{a^2 + b^2} = 2|z| \)

Setting Up the Equation

Given that \( |2z| = 1 \), we can substitute this into our earlier expression:

\( 2|z| = 1 \)

From this equation, we can solve for \( |z| \):

\( |z| = \frac{1}{2} \)

Conclusion About the Complex Number

Thus, we have established that if \( \text{mod} \, 2z = 1 \) for a non-zero complex number \( z \), it necessarily follows that:

• The modulus of \( z \) is \( \frac{1}{2} \).

This means that the distance of the complex number \( z \) from the origin in the complex plane is \( \frac{1}{2} \). This property is crucial in various applications, including signal processing and complex analysis, where understanding the magnitude of complex numbers plays a significant role.