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Chapter 1: Problem 36

What is the only complex number with modulus 0 ?

Short Answer

Expert verified
The complex number zero (0) is the only complex number with modulus 0.

Step by step solution

01

Understanding Modulus of a Complex Number

The modulus of a complex number \( z = a + bi \) is given by \( |z| = \sqrt{a^2 + b^2} \). It represents the distance of the complex number from the origin in the complex plane.
02

Setting the Modulus to Zero

For the modulus to be zero, the expression \( \sqrt{a^2 + b^2} \) must be equal to 0. This implies that the equation \( a^2 + b^2 = 0 \) must hold true.
03

Solving the Equation

The equation \( a^2 + b^2 = 0 \) implies that both \( a^2 = 0 \) and \( b^2 = 0 \). The only real numbers for which this is true are \( a = 0 \) and \( b = 0 \).
04

Identifying the Complex Number

Therefore, the complex number with \( a = 0 \) and \( b = 0 \) is \( z = 0 + 0i \), which simplifies to just \( z = 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Plane
The complex plane is a two-dimensional plane used to graphically represent complex numbers. Complex numbers have a part that is real, and a part that is imaginary. If we have a complex number of the form \( z = a + bi \), then \( a \) is the real part and \( b \) is the imaginary part.
In the complex plane:
  • The horizontal axis (x-axis) represents the real part (\(a\)).
  • The vertical axis (y-axis) represents the imaginary part (\(b\)).
Each complex number corresponds to a unique point on this plane.
This representation helps to visualize operations with complex numbers. Addition, subtraction, and even multiplication can be interpreted geometrically. For example, adding two complex numbers is similar to vector addition on this plane.
Modulus of a Complex Number
The modulus of a complex number can be thought of as its absolute value or distance from the origin in the complex plane. For a complex number \( z = a + bi \), the modulus is defined as:
\[|z| = \sqrt{a^2 + b^2}\]This modulus gives a measure of how "long" the vector representing the complex number is in the complex plane.
To break this down further:
  • \(a\) is the real component, representing horizontal distance.
  • \(b\) is the imaginary component, representing vertical distance.
To find the modulus, you are essentially using Pythagoras' theorem to find the hypotenuse of a right triangle where \(a\) and \(b\) are the sides.
This concept is the same as finding the magnitude of a vector in 2D space, making it a crucial tool in complex number analysis.
Zero Modulus
Zero modulus occurs when the modulus or absolute value of a complex number equals zero. In terms of the complex plane, this means that the complex number is located exactly at the origin.
For the modulus \(|z|\) to be zero, the equation \(\sqrt{a^2 + b^2} = 0\) must hold true. By solving this, we find that:
  • Both \(a\) and \(b\) must be zero.
Thus, the complex number is \(z = 0 + 0i\), which is simply \(z = 0\).
This neatly aligns with our geometric intuition. On the complex plane, the only point that has absolutely no distance from the origin is the origin itself. Zero modulus is a unique property attributed only to the complex number zero.

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Most popular questions from this chapter

Use a CAS as an aid in factoring the given quadratic polynomial. $$ i z^{2}-(2+3 i) z+1+5 i $$

Use a \(\mathrm{CAS}^{\S}\) to first find \(z^{n}=w\) for the given complex number and the indicated value of \(n\). Then, using the output and the same value of \(n\), determine whether \(w^{1 / n}=\left(z^{n}\right)^{1 / n}=z .\) If not, explain why not. $$ 1+3 i ; n=8 $$

Suppose that \(z=r(\cos \theta+i \sin \theta)\). Describe geometrically the effect of multiplying \(z\) by a complex number of the form \(z_{1}=\cos \alpha+i \sin \alpha\), when \(\alpha>0\) and when \(\alpha<0\).

For the complex numbers \(z_{1}=-1\) and \(z_{2}=5 i\), verify that: (a) \(\operatorname{Arg}\left(z_{1} z_{2}\right) \neq \operatorname{Arg}\left(z_{1}\right)+\operatorname{Arg}\left(z_{2}\right)\) (b) \(\operatorname{Arg}\left(z_{1} / z_{2}\right) \neq \operatorname{Arg}\left(z_{1}\right)-\operatorname{Arg}\left(z_{2}\right)\).

Use a CAS to solve the given polynomial equation. In Mathematical the command Solve will find all roots of polynomial equations up to degree four by means of a formula. $$ z^{4}+4 i z^{2}+10 i=0 $$
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