Problem 96 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$|z|=5$$

Step-by-Step Solution

Answer

The graph of complex numbers satisfying |z|=5 is a circle with radius 5 centered at the origin (0,0) in the complex plane.

Step 1: Understanding Complex Number Graph

In the complex plane, a complex number is illustrated as a point. The horizontal axis represents the real part and the vertical axis represents the imaginary part of the complex number. The origin (0,0) corresponds to the complex number 0. The modulus (or absolute value) of a complex number, denoted |z|, is a measure of how distant it is from the origin.

Step 2: Understanding the equation |z|=5

The equation |z|=5 means the distance from z to the origin is 5. This describes a circle with radius 5 in the complex plane with the origin (0,0) being the center.

Step 3: Sketching the graph

On graph paper or a digital graphing tool, plot the center of the circle at (0,0). Starting from the center, measure out 5 units in all directions and create a closed circle. Any point on this circle represents a complex number z with |z|=5.

Key Concepts

Complex Plane

Imagine a typical X-Y coordinate system, but in the complex plane, the horizontal axis (X-axis) is known as the real axis, and the vertical axis (Y-axis) is called the imaginary axis. Each point in this plane corresponds to a complex number, with the horizontal component representing the real part and the vertical component the imaginary part.

For example, the complex number 3 + 4i would be graphed by moving 3 units along the real axis and 4 units up the imaginary axis, much like plotting points on a traditional Cartesian grid. This graphical representation is fundamentally important as it allows us to visualize mathematical concepts of complex numbers, such as addition, subtraction, and finding the modulus.

Modulus of Complex Numbers

The modulus of a complex number, written as \(|z|\), is a numerical value representing the distance of the point from the origin on the complex plane. It's similar to finding the length of the hypotenuse in a right-angle triangle using the Pythagorean theorem.

Mathematically, if we have a complex number \( z = a + bi \), where \( a \) and \( b \) are real numbers, the modulus is calculated with \( |z| = \sqrt{a^2 + b^2} \). The modulus is always a non-negative real number and provides a way to compare the 'sizes' of complex numbers without considering their direction from the origin.

Complex Number and Polar Coordinates

Polar coordinates offer another way to represent complex numbers, emphasizing their angle and distance from the origin. A complex number in polar form is denoted by \( r(\cos \theta + i\sin \theta) \), where \( r \) is the modulus (indicating how far from the origin the number lies) and \( \theta \) is the argument (the angle measured counterclockwise from the positive real axis).

Conversion between rectangular coordinates \( (a,b) \) and polar coordinates \( (r,\theta) \) involves trigonometry: \( a = r\cos \theta \) and \( b = r\sin \theta \). This polar form is particularly helpful in multiplication or division of complex numbers as it simplifies the arithmetic to dealing with moduli and adding angles.

Interpreting Absolute Value of Complex Numbers

When we discuss the absolute value or modulus of a complex number, we're talking about its 'size' without concern for its direction. This is analogous to how we might consider the length of a vector in physics or the magnitude of an earthquake in seismology.

Interpreting the equation \(|z| = 5\) is straightforward: any complex number whose distance from the origin is 5 fits this condition. This set of numbers forms a circle on the complex plane with a radius of 5. It's a valuable concept in complex analysis, as it provides a geometric interpretation of the solutions to equations involving complex numbers.