Problem 49 - Precalculus Mathematics for Calculus

Question

Write the complex number in polar form with argument \(\theta\) between 0 and \(2 \pi\). $$2+i$$

Step-by-Step Solution

Answer

The polar form of the complex number is \( \sqrt{5}(\cos 0.464 + i\sin 0.464) \).

Step 1: Identify the Components

A complex number can be written as \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. For the given complex number \( 2+i \), we have \( a = 2 \) and \( b = 1 \).

Step 2: Calculate the Modulus

The modulus \( r \) of a complex number \( a + bi \) is calculated using the formula \( r = \sqrt{a^2 + b^2} \). Substituting the values, we have:\[ r = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \]

Step 3: Determine the Argument

The argument \( \theta \) is calculated using the formula \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \). Plugging in our values:\[ \theta = \tan^{-1}\left(\frac{1}{2}\right) \]Using a calculator, this gives \( \theta \approx 0.464 \) radians. Since we are in the first quadrant, no adjustment is needed.

Step 4: Combine to Polar Form

The polar form of a complex number is \( z = r(\cos \theta + i\sin \theta) \). Substitute \( r \) and \( \theta \) into this form:\[ z = \sqrt{5}(\cos 0.464 + i\sin 0.464) \]

Key Concepts

Complex Numbers

When you first encounter complex numbers, they might seem a bit mysterious. A complex number combines both a real number and an imaginary number, expressed as \( z = a + bi \). Here, \( a \) is the real part, and \( bi \) is the imaginary part, with \( i \) representing the imaginary unit. The imaginary unit \( i \) is defined such that \( i^2 = -1 \). So, complex numbers let you extend real numbers with this new dimension of calculation. In practical terms, complex numbers allow us to solve equations that have no solutions in the real number system, such as roots of negative numbers.

Modulus

The modulus of a complex number is similar to the idea of "magnitude" or "length." It gives you an idea of how far a complex number is from the origin in the complex plane. To find the modulus of a complex number \( z = a + bi \), you use the formula:\[r = \sqrt{a^2 + b^2}\]

• It's like extending the Pythagorean theorem to complex numbers.
• For our example, the complex number \( 2 + i \) has a modulus of \( \sqrt{5} \).

This modulus helps us when converting a complex number from rectangular form to polar form, assisting in descriptions based on circles or rotations rather than linear dimensions.

Argument of Complex Number

Besides knowing how "long" a complex number is using the modulus, we also want to know its "direction." This is where the argument comes in. The argument of a complex number \( z = a + bi \) is the angle \( \theta \), measured in radians, between the positive real axis and the line connecting the origin with the point \( (a, b) \) in the complex plane.

The formula to find \( \theta \) is:

\[ \theta = \tan^{-1}\left(\frac{b}{a}\right)\]

• The result is typically adjusted based on which quadrant of the complex plane the number lies in.
• In our example, since \( 2+i \) is in the first quadrant, no adjustments are needed, resulting in \( \theta \approx 0.464 \) radians.

Understanding the argument is vital for representing complex numbers in polar form, as it describes the rotation angle from the positive x-axis.

Polar Coordinates

Polar coordinates offer a way to express complex numbers that highlight their distance from the origin and their angle, rather than their horizontal and vertical components. Polar form is particularly useful in fields like engineering and physics, where rotating and scaling actions are common.

To write a complex number in polar form, use:

\[ z = r(\cos \theta + i\sin \theta)\]Here, \( r \) is the modulus, and \( \theta \) is the argument.

• This form gives a succinct way to describe both the magnitude and direction of the number.
• In our example, \( 2+i \) converts to polar form as \( \sqrt{5}(\cos 0.464 + i\sin 0.464) \).

By understanding polar coordinates, you can more easily visualize and manipulate complex numbers in scenarios that involve rotations and phases.