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. $$(i+\sqrt{3})^{2}$$

Short answer

(i + \sqrt{3})^2 = 2 + 2i\sqrt{3}. Plot at (2, 2\sqrt{3}).

Step-by-step solution

Expand the given expression

Expand the expression \( (i + \sqrt{3})^{2} \) using the binomial theorem. \[ (i + \sqrt{3})^{2} = i^{2} + 2 \cdot i \cdot \sqrt{3} + (\sqrt{3})^{2} \]

Simplify individual components

Substitute and simplify each term: \ i^{2} = -1 \, \ 2 \cdot i \cdot \sqrt{3} = 2i\sqrt{3} \, and \ (\sqrt{3})^{2} = 3 \ \[ (i + \sqrt{3})^{2} = -1 + 2i\sqrt{3} + 3 \]

Combine and simplify

Combine the real and imaginary parts into a single complex number: \[ -1 + 3 = 2 \, thus \ (i + \sqrt{3})^{2} = 2 + 2i\sqrt{3} \]

Plot on the complex plane

To plot \( 2 + 2i \sqrt{3} \): measure 2 units along the real axis (x-axis) and \ 2 \sqrt{3} \ units (approximately 3.46 units) along the imaginary axis (y-axis). Then, place a point at this location.

Key concepts

binomial theorem

The binomial theorem is a powerful tool in algebra that allows us to expand expressions raised to a power. For any two numbers, \(a\) and \(b\), and a positive integer \(n\), the binomial theorem provides a way to expand \((a+b)^n\). In our example, we expanded \((i + \sqrt{3})^2\), which involves using the formula for the square of a binomial: \((a+b)^2 = a^2 + 2ab + b^2\). By substituting \(a = i\) and \(b = \sqrt{3}\), we get:

• \(i^2\)
• \(2 \cdot i \cdot \sqrt{3}\)
• \((\sqrt{3})^2\)

After expanding and simplifying, our expression becomes clear and simpler to work with.

complex plane

The complex plane is a graphical representation of complex numbers. Each complex number \(a + bi\) can be seen as a point in a two-dimensional plane where the x-axis represents the real part \(a\) and the y-axis represents the imaginary part \(bi\). For the number \(2 + 2i\sqrt{3}\), you measure 2 units along the real axis and approximately 3.46 units along the imaginary axis. This coordinate system makes it easier to visualize operations like addition and multiplication of complex numbers and can also help in understanding the geometry of complex numbers.

imaginary unit

The imaginary unit \(i\) is a fundamental concept in complex numbers, defined by the property that \(i^2 = -1\). It enables extending the real number system to handle equations that cannot be solved with only real numbers. In our problem, we use \(i\) to express complex numbers and to simplify terms. For instance, when we computed \(i^2\), we substituted \(-1\) as per the definition, which helped simplify the given expression. The imaginary unit makes the abstraction of imaginary and complex numbers possible, opening up a wide range of solutions and applications in various fields of science and engineering.