91Ó°ÊÓ

In complex analysis, visualizing the position of complex numbers on the complex plane helps us intuitively understand their properties. The given condition \(|1-z|<1\) refers to a geometric representation. This inequality represents a disc with center at the point \((1, 0)\) on the complex plane and a radius of 1 unit. Any complex number \(z\) that satisfies this condition will fall within this disc.
This means that \(z\) always remains closer to the point \((1, 0)\) than to the origin \((0, 0)\), and does not extend beyond this disc's boundary.
Visualizing it, since the center of our disc is at \(x = 1\), the entirety of the disc lies to the left of the vertical line \(x = 1\). Therefore, any point \(z\) that lies within this disc has its real component \(x\) less than 1.

• Center of the disc: \((1, 0)\)
• Radius: 1
• All points \(z\) satisfy \(x < 1\)

The argument of a complex number, often denoted as \( \operatorname{Arg}(z) \), is the angle between the line connecting the origin and the point \(z\), and the positive real axis. This angle is measured in radians and can be positive or negative, depending on whether \(z\) is above or below the real axis.
In this context, any point \(z\) within the disc (centered at \((1, 0)\) with radius 1) has an argument less than \(\pi/2\) radians and greater than \(-\pi/2\) radians. Why? Because the disc lies entirely to the left of \(x = 1\), ensuring that \(z\)’s argument, when viewed from the origin, creates an angle confined between these two values.
Consequently, this geometric constraint guarantees that \(|\operatorname{Arg}(z)| < \pi/2\), which maintains that the angle does not reach the rightmost \(x = 1\) line.

• The argument represents the angle in the complex plane.
• The angle lies between \(-\pi/2\) and \(\pi/2\).

Inequalities like \(|1-z|<1\) in complex analysis translate into spatial constraints within the complex plane, such as limiting the region where \(z\) can exist. This type of inequality is evaluated through modulus.
The modulus \(|1-z|<1\) indicates that the distance between the complex number \(z\) and point \((1, 0)\) is always less than 1 unit. Geometrically, it confines \(z\) to a circle's interior, preventing it from straying beyond this perimeter.
Understanding inequalities this way becomes crucial because it segments the complex plane into regions with different properties and constraints. Such disciplined comprehension offers both qualitative and quantitative insights into complex numbers' behavior and their allowed regions.

• Modulus measures the distance between two complex points.
• Inequality restrictions determine usable boundaries for \(z\).