Problem 72 - Algebra and Trigonometry

Question

Show that \(e^{i \pi}+1=0\)

Step-by-Step Solution

Answer

\( e^{i \pi} = -1 \), so \( e^{i \pi} + 1 = 0 \).

Step 1: Understand Euler's Formula

Euler's formula states that for any real number \theta, \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \).

Step 2: Substitute \theta with \pi

Plug in \theta = \pi into Euler's formula: \( e^{i\pi} = \cos(\pi) + i\sin(\pi) \).

Step 3: Evaluate Trigonometric Functions

Recall that \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \). So, \( e^{i\pi} = -1 + i \cdot 0 = -1 \).

Step 4: Add 1 to both sides

To show \( e^{i\pi} + 1 = 0 \), add 1 to the result from the previous step: \( -1 + 1 = 0 \).

Key Concepts

complex numbers

Complex numbers are a type of mathematical number that include both a real part and an imaginary part. The imaginary unit is denoted by i, where \text{\( i^2 = -1 \)}. This allows us to write complex numbers in the form of \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.
Complex numbers are essential in many fields of science and engineering since they enable the solution of equations that do not have any real solutions. For example, the equation \( x^2 + 1 = 0 \) does not have any real solution since there is no real number whose square is -1. However, using the imaginary unit \( i \), we can write \( x = i \) and \( x = -i \) as the solutions.
The addition, subtraction, multiplication, and division of complex numbers follow the usual arithmetic rules, with the additional property that \( i^2 = -1 \). Complex numbers can be represented in a plane known as the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part of the number.

trigonometric functions

Trigonometric functions are fundamental in the study of angles and periodic phenomena. They include sine (sin), cosine (cos), and tangent (tan). These functions are primarily used to relate the angles of a triangle to the lengths of its sides.
For Euler's formula, we particularly focus on cosine and sine functions. Euler's formula is a remarkable equation in complex number theory, written as \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \). This formula binds together complex exponentials and trigonometric functions.
When \theta = \pi, we identify important trigonometric values: \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \). This gives us Euler's formula in a special case: \( e^{i\pi} = -1 + i \cdot 0 = -1 \). This step is crucial for understanding and proving the equation \( e^{i\pi} + 1 = 0 \).

mathematical proof

Mathematical proofs are logical arguments demonstrating the truth of a given statement based on axioms and previously established results. Proofs can be approached in several ways, including direct proof, proof by contradiction, and proof by induction.
To prove that \( e^{i\pi} + 1 = 0 \), we use a direct approach involving Euler's formula:

• We start by stating Euler's formula for \( \theta = \pi \): \( e^{i\pi} = \cos(\pi) + i\sin(\pi) \).
• Next, we evaluate the trigonometric functions: \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \).
• This simplifies to: \( e^{i\pi} = -1 + i \cdot 0 = -1 \).
• Finally, by adding 1 to both sides, we get: \( e^{i\pi} + 1 = -1 + 1 = 0 \).

This completes the proof and demonstrates the elegance of Euler's formula. This is often known as Euler's identity, one of the most beautiful equations in mathematics.