Problem 36 - Algebra and Trigonometry Real Mathematics, Real People

Question

Represent the complex number graphically, and find the trigonometric form of the number. $$5-3 i$$

Step-by-Step Solution

Answer

The trigonometric form of the complex number \(5-3i\) is \( \sqrt{34} e^{- i arctan(3/5)} \). The number can be represented graphically in the fourth quadrant of the complex plane, with the coordinates (5, -3).

Step 1: Graph Representation

To represent the complex number \(5-3i\) graphically, we plot it on the complex plane. The real part, 5, is the x-coordinate, and the imaginary part, -3, is the y-coordinate.

Step 2: Calculating the Magnitude/Modulus

The magnitude (r) of a complex number \((a+bi)\) is represented as \(r=\sqrt{a^2+b^2}\). In this specific case, \(r=\sqrt{5^2+(-3)^2} = \sqrt{25+9} = \sqrt{34}\). Thus, the modulus of \(5-3i\) is \(\sqrt{34}\).

Step 3: Calculating the Argument/Angle

The argument (\(\theta\)) of a complex number can be determined by using the following formula: \(\theta=arctan(b/a)\). For the complex number \(5-3i\), applying this formula leads to: \(\theta= - arctan(3/5)\) . We use a negative sign because the complex number lies in the fourth quadrant where the angle is negative.

Step 4: Trigonometric Form

Once we have the modulus and argument, we can write the trigonometric form of the complex number as \(re^{i\theta}\), which here becomes \(\sqrt{34} e^{- i arctan(3/5)}\).

Key Concepts

Graphing Complex Numbers

Graphing complex numbers on the complex plane provides a visual representation of these numbers, which can be quite insightful when studying their properties. To graph a complex number like \(5-3i\), imagine a standard Cartesian coordinate system. The real part of the complex number, in this case 5, serves as the x-coordinate, and the imaginary part, -3, being the y-coordinate.

Picture placing a dot at the coordinates (5, -3). This point corresponds to our complex number on the complex plane, often referred to as the Argand diagram. The horizontal axis is known as the real axis, while the vertical axis is called the imaginary axis, where the 'i' in \(5-3i\) represents \(\text{\(\sqrt{-1}\)}\), an imaginary unit.

Magnitude of a Complex Number

Understanding the magnitude or modulus of a complex number is key to grasping its 'size'. The magnitude of a complex number \(a+bi\) is calculated using the formula \(r=\sqrt{a^2+b^2}\). It tells us how far the number is from the origin on the complex plane.

For the number \(5-3i\), we calculate the magnitude as follows: \(r=\sqrt{5^2+(-3)^2} = \sqrt{25+9} = \sqrt{34}\). This numerical value, \(\sqrt{34}\), represents the distance of our point (5, -3) from the origin (0, 0). It's akin to measuring the length of the hypotenuse in a right-angled triangle formed by the real and imaginary parts of the complex number.

Argument of a Complex Number

The argument of a complex number is its angle with the positive real axis on the complex plane, measured counterclockwise and usually expressed in radians. This angle provides direction from the origin to our number.

To compute the argument for \(5-3i\), we use the formula \(\theta=arctan(\frac{b}{a})\). Since \(5-3i\) is in the fourth quadrant (where both x and y coordinates, or real and imaginary parts, are positive and negative, respectively), the angle is negative, leading us to \(\theta= - arctan(\frac{3}{5})\). This negative angle reflects the fact that we would rotate in a clockwise direction from the positive real axis to reach our complex number.

Euler's Formula

Embracing the beauty of mathematics, Euler's formula establishes a profound connection between complex numbers, exponentiation, and trigonometric functions. The formula states that for any real number \(\theta\), \(e^{i\theta} = cos(\theta) + isin(\theta)\), where 'e' is the base of the natural logarithm, and 'i' is the imaginary unit.

This formula can transform complex numbers into their exponential form, which is particularly useful in applications such as signal processing and quantum mechanics. For our number \(5-3i\), whose magnitude and argument are \(\sqrt{34}\) and \(-arctan(\frac{3}{5})\) respectively, Euler's formula conveys it in its elegant exponential form: \(\sqrt{34} e^{-i arctan(\frac{3}{5})}\). This format highlights the rhythmic dance between algebra and geometry encapsulated in complex numbers.