Question

First simplify each of the following numbers to the \(x+i y\) form or to the \(r e^{i \theta}\) form. Then plot the number in the complex plane. $$\left(\frac{1+i}{1-i}\right)^{2}$$

Short answer

-1 is the simplified form. It is plotted at -1 on the real axis.

Step-by-step solution

- Simplify the expression

First, simplify the fraction inside the parentheses. To do this, multiply the numerator and the denominator by the conjugate of the denominator: \[\frac{1+i}{1-i} \cdot \frac{1+i}{1+i} = \frac{(1+i)^2}{(1-i)(1+i)}\]

- Simplify the denominator

The denominator simplifies as follows: \[(1-i)(1+i) = 1 - i^2 = 1 - (-1) = 1 + 1 = 2\]

- Simplify the numerator

The numerator simplifies as follows: \[(1+i)^2 = 1^2 + 2i + i^2 = 1 + 2i - 1 = 2i\]

- Combine results from steps 2 and 3

Putting it all together, we get: \[\left(\frac{2i}{2}\right)^2 = (i)^2 = -1\]

- Plot the result

To plot \(-1\) on the complex plane, note that it is a complex number in the form \(x + iy\) where \(x = -1\) and \(y = 0\). This is located on the real axis at \(-1\).

Key concepts

Complex Plane

The complex plane is a way to visualize complex numbers. It is similar to the Cartesian plane used for real numbers, but it has two axes:

• The horizontal axis, called the real axis, represents the real part of a complex number.
• The vertical axis, called the imaginary axis, represents the imaginary part of a complex number.

Any complex number can be represented as a point or a vector on this plane. For instance, the number \(1 + i\) would be plotted at the coordinates \( (1, 1) \). In the original exercise, we simplified the expression to \(-1 + 0i\), which is plotted at the point \(-1\) on the real axis.

Complex Number Arithmetic

Complex number arithmetic operates similarly to arithmetic with regular numbers, but it also involves the imaginary unit \i\, where \(i^2 = -1\). Here are some basic operations:

• Addition: \( (a + bi) + (c + di) = (a + c) + (b + d)i \)
• Subtraction: \( (a + bi) - (c + di) = (a - c) + (b - d)i \)
• Multiplication: Use the distributive property: \( (a + bi)(c + di) = ac + adi + bci + bdi^2 \)
• Division: This involves multiplying the numerator and the denominator by the conjugate of the denominator. A conjugate of a complex number \(a + bi\) is \(a - bi\).

In our original problem, we multiplied by the conjugate \((1 + i)\) to simplify the fraction.

Polar Form

The polar form of a complex number expresses it in terms of its magnitude and angle with respect to the positive real axis.
The general formula is: \(z = r e^{i \theta}\). Here, \(r\) is the magnitude (or modulus) and \(\theta\) is the argument (or angle).

• The magnitude \(r\) is calculated as \sqrt{x^2 + y^2}\ where \(x\) and \(y\) are the real and imaginary parts, respectively.
• The angle \theta\ is calculated as \tan^{-1} \frac{y}{x}\.

Converting between rectangular and polar forms helps with multiplication and division of complex numbers.
In our problem, although we didn't convert to polar form, knowing it would simplify computation of powers and roots of complex numbers.