Problem 3
Question
Every complex number \(a+b \mathbf{i}\) may be represented as a point in the complex plane. The unit circle in the complex plane consists of all the complex numbers whose distance from the origin is 1 ; thus, clearly, the unit circle consists of all the complex numbers which can be written in the form $$ \cos x+\mathbf{i} \sin x $$ for some real number \(x\). Prove that \(f(x)=\) cis \(x\) is a homomorphism from \(\mathbb{R}\) onto \(T\).
Step-by-Step Solution
Verified Answer
\(f(x) = \text{cis } x\) is a homomorphism from \(\mathbb{R}\) onto \(T\), preserving addition and covering the unit circle.
1Step 1: Understanding the Unit Circle
The unit circle in the complex plane is defined as the set of all complex numbers with a magnitude (or modulus) of 1. A complex number represented as \(z = a + b \mathbf{i}\) is on the unit circle if and only if \(|z| = \sqrt{a^2 + b^2} = 1\).
2Step 2: Expressing Complex Numbers on the Unit Circle
Complex numbers on the unit circle can be expressed in terms of trigonometric functions. For a complex number \(z\) on the unit circle, we can write \(z = \cos x + \mathbf{i} \sin x\), where \(x\) is the angle (in radians) the line corresponding to \(z\), measured from the positive real axis. This is often denoted as \(\text{cis} x = \cos x + \mathbf{i} \sin x\).
3Step 3: Defining the Function \(f(x) = \text{cis } x\)
The function \(f(x) = \text{cis } x\) maps a real number \(x\) to a complex number on the unit circle. Since \(\cos^2 x + \sin^2 x = 1\), the complex number \(\cos x + \mathbf{i} \sin x\) indeed has a magnitude of 1, thus mapping into the unit circle \(T\).
4Step 4: Proving Homomorphism Property
To prove that \(f\) is a homomorphism, we need to show that it preserves the operation of addition: \(f(x + y) = f(x) \cdot f(y)\). We calculate \(f(x + y) = \cos(x + y) + \mathbf{i} \sin(x + y)\) and use angle sum identities: \(\cos(x+y) = \cos x \cos y - \sin x \sin y\) and \(\sin(x+y) = \sin x \cos y + \cos x \sin y\). This simplifies to \((\cos x \cos y - \sin x \sin y) + \mathbf{i}(\sin x \cos y + \cos x \sin y)\).
5Step 5: Verifying Multiplicative Property
Now, we compute \(f(x) \cdot f(y) = (\cos x + \mathbf{i} \sin x)(\cos y + \mathbf{i} \sin y) = (\cos x \cos y - \sin x \sin y) + \mathbf{i}(\sin x \cos y + \cos x \sin y)\).Observe that this expression matches \(f(x + y)\), confirming that \(f(x+y) = f(x) \cdot f(y)\). Therefore, \(f\) preserves addition, proving it is a homomorphism.
6Step 6: Onto Property
Function \(f\) is onto if for every \(z\) in the image \(T\), there exists an \(x\) in \(\mathbb{R}\) such that \(f(x) = z\). Since the unit circle is periodic with period \(2\pi\), every real number \(x\) corresponds to an angle, and thus a complex number on the unit circle, ensuring \(f\) is onto.
Key Concepts
Complex NumbersUnit CircleUnit CircleUnit CircleHomomorphismTrigonometric Functions
Complex Numbers
Complex numbers, often written as \(a + b \mathbf{i}\), extend the concept of one-dimensional number lines into the two-dimensional complex plane. Here, \(a\) is the real part and \(b\) is the imaginary part, with \(\mathbf{i}\) being the imaginary unit where \(\mathbf{i}^2 = -1\). Complex numbers can be viewed as vectors or points on this plane, making it easier to visualize operations like addition and multiplication.
The complex plane is structured like a grid where the horizontal axis represents the real part, and the vertical axis represents the imaginary part. Complex numbers are uniquely positioned on this plane, and their properties, including magnitude, or distance from the origin, can be calculated. The magnitude \(|z|\) of a complex number \(z = a + b \mathbf{i}\) is given by \(\sqrt{a^2 + b^2}\).
The complex plane is structured like a grid where the horizontal axis represents the real part, and the vertical axis represents the imaginary part. Complex numbers are uniquely positioned on this plane, and their properties, including magnitude, or distance from the origin, can be calculated. The magnitude \(|z|\) of a complex number \(z = a + b \mathbf{i}\) is given by \(\sqrt{a^2 + b^2}\).
- Magnitude: represents the distance from the origin.
- Equal magnitudes for a point means it lies on a circle centered at the origin.
- Operations: Addition and multiplication have geometric interpretations involving rotations and scalings.
Unit Circle
The unit circle is a fundamental concept in mathematics, especially in trigonometry and complex analysis. It consists of all complex numbers that have a magnitude of 1, meaning each point on the unit circle is exactly one unit away from the origin.
In the complex plane, a point \(z = a + b \mathbf{i}\) is on the unit circle if \(|z| = 1\). This gives us the equation \(\sqrt{a^2 + b^2} = 1\), which simplifies to the requirement that \(a^2 + b^2 = 1\).
To represent these points, we use trigonometric functions:
In the complex plane, a point \(z = a + b \mathbf{i}\) is on the unit circle if \(|z| = 1\). This gives us the equation \(\sqrt{a^2 + b^2} = 1\), which simplifies to the requirement that \(a^2 + b^2 = 1\).
To represent these points, we use trigonometric functions:
- Cosine (\
Unit Circle
The unit circle is a fundamental concept in mathematics, especially in trigonometry and complex analysis. It consists of all complex numbers that have a magnitude of 1, meaning each point on the unit circle is exactly one unit away from the origin.
In the complex plane, a point \(z = a + b \mathbf{i}\) is on the unit circle if \(|z| = 1\). This gives us the equation \(\sqrt{a^2 + b^2} = 1\), which simplifies to the requirement that \(a^2 + b^2 = 1\).
To represent these points, we use trigonometric functions:
In the complex plane, a point \(z = a + b \mathbf{i}\) is on the unit circle if \(|z| = 1\). This gives us the equation \(\sqrt{a^2 + b^2} = 1\), which simplifies to the requirement that \(a^2 + b^2 = 1\).
To represent these points, we use trigonometric functions:
- Cosine (\
Unit Circle
The unit circle is a fundamental object that appears frequently in mathematics, especially in trigonometry and complex analysis. It consists of all complex numbers \(z = a + b \mathbf{i}\) with a magnitude (or modulus) of 1. The equation \(a^2 + b^2 = 1\) describes the unit circle, signifying that each point \(z\) is one unit from the origin. This characteristic confines all these points to a circle centered at the origin.
Every point on the unit circle can be expressed using trigonometric functions:
Every point on the unit circle can be expressed using trigonometric functions:
- \(z = \cos x + \mathbf{i} \sin x\), where \(x\) is an angle measured in radians.
- \(x\) denotes the angle the radius vector makes with the positive real axis.
Homomorphism
A homomorphism is a function between two algebraic structures that preserves the operations of those structures. In the context of complex numbers, this relates to how functions map real numbers onto complex numbers on the unit circle while maintaining their structural properties.
For the function \(f(x) = \text{cis } x = \cos x + \mathbf{i} \sin x\), it serves as a homomorphism from the real numbers \(\mathbb{R}\) to the unit circle \(T\) in the complex plane. We aim to show that \(f\) preserves the operation of addition. Thus, for any real numbers \(x\) and \(y\):
For the function \(f(x) = \text{cis } x = \cos x + \mathbf{i} \sin x\), it serves as a homomorphism from the real numbers \(\mathbb{R}\) to the unit circle \(T\) in the complex plane. We aim to show that \(f\) preserves the operation of addition. Thus, for any real numbers \(x\) and \(y\):
- \(f(x+y) = \cos(x+y) + \mathbf{i}\sin(x+y)\)
- Employ angle addition formulas: \(\cos(x+y) = \cos x \cos y - \sin x \sin y\) and \(\sin(x+y) = \sin x \cos y + \cos x \sin y\).
- The calculations confirm: \(f(x+y) = f(x) \cdot f(y)\).
Trigonometric Functions
Trigonometric functions, such as cosine and sine, are at the heart of understanding complex numbers, especially those on the unit circle. These functions relate angles in the circle to coordinates in the complex plane, facilitating the expression of complex numbers in terms of trigonometry.
The sine and cosine functions map each angle \(x\) (measured in radians) to points on the unit circle as \(\cos x\) and \(\sin x\), respectively. Thus:
The sine and cosine functions map each angle \(x\) (measured in radians) to points on the unit circle as \(\cos x\) and \(\sin x\), respectively. Thus:
- \(\cos x\) represents the horizontal coordinate, or real part, of the complex number on the unit circle.
- \(\sin x\) denotes the vertical coordinate, or imaginary part.
Other exercises in this chapter
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