Argument of a complex number

Definition

The argument of a complex number is the angle formed between the positive real axis and the line representing the complex number in the complex plane, typically measured in radians. It is an essential concept when working with complex exponents and logarithms, as it helps determine the direction of the number and plays a crucial role in expressing complex numbers in polar form.

5 Must Know Facts For Your Next Test

1. The argument is usually denoted by $$\arg(z)$$ for a complex number $$z$$.
2. The value of the argument can vary by multiples of $$2\pi$$ due to periodicity, meaning that multiple angles can represent the same complex number.
3. To find the argument, one can use the arctangent function: $$\arg(z) = \tan^{-1}(\frac{b}{a})$$ where $$a$$ is the real part and $$b$$ is the imaginary part.
4. The range of the principal argument is typically set between $$-\pi$$ and $$\pi$$, allowing for a unique representation within this interval.
5. Understanding the argument is crucial when applying Euler's formula, which links complex exponentials to trigonometric functions.

Review Questions

• How is the argument of a complex number calculated and what role does it play in its representation?
  • The argument of a complex number is calculated using the arctangent function, specifically $$\arg(z) = \tan^{-1}(\frac{b}{a})$$, where $$a$$ is the real part and $$b$$ is the imaginary part. This angle represents how far from the positive real axis the complex number lies in the complex plane. It plays a significant role in expressing the number in polar form, which highlights both its magnitude and direction.
• Discuss why understanding the periodic nature of the argument is important when working with complex exponents and logarithms.
  • The periodic nature of the argument means that for any complex number, there are infinitely many angles that can represent it due to the addition of multiples of $$2\pi$$. This is crucial when dealing with complex exponents and logarithms because it impacts how we express powers and roots. When taking logarithms of complex numbers, we must account for these multiple values to accurately express results, particularly when using formulas like Euler's formula.
• Evaluate how changing the argument affects the computation of powers of a complex number using De Moivre's Theorem.
  • Changing the argument directly impacts how we compute powers of a complex number through De Moivre's Theorem, which states that for any integer $$n$$, if a complex number is expressed in polar form as $$r( ext{cos} \theta + i\text{sin} \theta)$$, then its nth power can be computed as $$r^n( ext{cos}(n\theta) + i\text{sin}(n\theta))$$. Altering the argument leads to different angles for each power, resulting in distinct points on the unit circle in the complex plane. Thus, recognizing how changes in argument affect results is essential for accurate computations.

Related terms

Complex Plane: A two-dimensional plane used to represent complex numbers, where the x-axis corresponds to the real part and the y-axis corresponds to the imaginary part.

Polar Form: A way of expressing complex numbers using their magnitude (modulus) and argument, typically written as $$r( ext{cos} \theta + i\text{sin} \theta)$$ or as $$re^{i\theta}$$.

Modulus: The modulus of a complex number is its distance from the origin in the complex plane, calculated as the square root of the sum of the squares of its real and imaginary parts.