Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-20: Represent the complex number geometrically. $$ (1+2 i)^{2} $$

Step-by-Step Solution

Answer

The complex number \((1 + 2i)^2\) is \(-3 + 4i\) geometrically represented on the Argand plane.

Step 1: Write the Complex Number

The complex number given is \((1 + 2i)\). This number is in the form \(a + bi\), where \(a = 1\) and \(b = 2\).

Step 2: Compute the Square of the Complex Number

The problem requires squaring the complex number \((1 + 2i)\). Use the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\). Substituting \(a = 1\) and \(b = 2i\), we get:\[(1 + 2i)^2 = 1^2 + 2 \cdot 1 \cdot 2i + (2i)^2\]Calculate each part separately:- \(1^2 = 1\)- \(2 \cdot 1 \cdot 2i = 4i\)- \((2i)^2 = 4i^2 = 4(-1) = -4\)Putting them together, \((1 + 2i)^2 = 1 + 4i - 4 = -3 + 4i\).

Step 3: Plot the Resulting Complex Number on the Argand Plane

The resulting complex number is \(-3 + 4i\). Plot this on the complex plane, also known as the Argand plane. In this plane, complex numbers are represented where the horizontal axis is the real part and the vertical axis is the imaginary part.- Plot the point at \((-3, 4)\) where \(-3\) is on the real axis and \(4\) is on the imaginary axis.

Step 4: Verify the Plot

Ensure that the point \((-3, 4)\) is correctly placed on the Argand plane representing the complex number \(-3 + 4i\). This geometric representation is essentially a vector from the origin to the point \((-3, 4)\).

Key Concepts

Geometric Representation of Complex Numbers

Understanding the geometric representation of complex numbers is essential for visualizing their properties on a plane. A complex number, such as \((a + bi)\), has two components:

• "a" is the real part
• "b" is the imaginary part, where "i" denotes the imaginary unit with the property \(i^2 = -1\)

To represent a complex number geometrically, we think of it as a point or position vector in two-dimensional space. Here, the real part measures the horizontal distance, while the imaginary part measures the vertical distance from the origin.
For example, the complex number \( (1 + 2i) \) can be positioned at the point \((1, 2)\) on this plane.
This visualization aids in performing operations like addition, subtraction, and multiplication by allowing us to see the movements and transformations graphically.

Understanding the Argand Plane

The Argand plane, named after Jean-Robert Argand, is a special visual representation for complex numbers. It resembles the Cartesian plane, but here it specifically serves for complex analysis.

• The horizontal axis represents the real part of complex numbers
• The vertical axis represents the imaginary part

Every point on this two-dimensional plane corresponds to a unique complex number.
To place a complex number like \(-3 + 4i\) on the Argand plane, move 3 units left along the real axis (for \(-3\)) and 4 units up along the imaginary axis (for \(4i\)).
Visualizing on the Argand plane highlights the beauty of complex numbers and provides insight into how operations such as addition and multiplication graphically transform these points.

Exploring the Binomial Expansion

The binomial expansion is a fundamental concept used in many branches of mathematics, including the manipulation of complex numbers. When faced with an expression of the form \((a + bi)^n\), we can use the binomial theorem to expand it.

• The binomial theorem provides a way to expand expressions raised to a power, using the formula \((a + b)^2 = a^2 + 2ab + b^2\)

In our example \((1 + 2i)^2\), we first plug in \(a = 1\) and \(b = 2i\) to get:

• \(1^2 = 1\)
• \(2 \cdot 1 \cdot 2i = 4i\)
• \((2i)^2 = 4i^2 = 4(-1) = -4\)

Adding these results:\(1 + 4i - 4 = -3 + 4i\)This shows how binomial expansion simplifies complex algebraic expressions, making them easier to interpret and geometrically represent. By breaking it down, each component becomes more manageable, providing clarity in the overall calculation process.