Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The complex number is represented as the point (0, 2) on the complex plane.

Step 1: Simplify the Expression

To simplify the expression \((1+i)^2\), we use the formula for squaring a binomial: \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = 1\) and \(b = i\). Substitute these values into the formula:\[ (1+i)^2 = 1^2 + 2(1)(i) + i^2 \] Simplifying further:\[ 1 + 2i + (i^2) \].

Step 2: Substitute and Simplify

Recall that \(i^2 = -1\). So substitute in for \(i^2\):\[ 1 + 2i + (-1) = 2i \]. Therefore, the simplified form of \((1+i)^2\) is \(2i\).

Step 3: Plot the Complex Number

On the complex plane, a complex number is represented as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. For \(2i\), the real part \(a\) is 0 and the imaginary part \(b\) is 2. This means the point \( (0, 2) \) should be plotted on the y-axis, at 2 units above the origin.

Key Concepts

complex numbers

Complex numbers are fascinating mathematical entities that extend the idea of numbers as we know them. They are numbers that comprise a real part and an imaginary part. The standard form of a complex number is expressed as \( a + bi \), where:

• \( a \) is the real part.
• \( b \) is the imaginary part.

The term "imaginary part" refers to the presence of the imaginary unit \( i \), where \( i \) is defined by the property \( i^2 = -1 \). This extension into the "imaginary" realm allows for the solution of equations that have no real roots, such as \( x^2 + 1 = 0 \).

In the case of the expression \((1+i)^2\), the initial complex number \(1 + i\) consists of a real component \(a = 1\) and an imaginary component \(b = 1\). Through manipulation like squaring, a complex number can transform into another, illustrating the dynamic nature of these unique numbers.

complex plane

The complex plane is a two-dimensional plane used to visualize complex numbers. It resembles the Cartesian coordinate system with a twist:

• The x-axis represents the real part of a complex number.
• The y-axis represents the imaginary part of a complex number.

Each complex number \( a + bi \) corresponds to a unique point \( (a, b) \) on this plane. For example, the number \( 2i \), derived from \((1+i)^2\), is expressed as \( (0, 2) \) on the complex plane.

To plot \( 2i \) on the complex plane, start at the origin \( (0, 0) \). Since the real part is 0, remain on the y-axis, moving up 2 units to position \( (0, 2) \). Hence, the geometric representation of \( 2i \) vividly captures the interplay between the real and imaginary components of complex numbers.

imaginary unit

The imaginary unit, denoted as \( i \), is the cornerstone of complex numbers. It is defined to satisfy the equation \( i^2 = -1 \). This peculiar property gives complex numbers their ability to exist in a dimension beyond the traditional number line.

In expressions like \((1+i)^2\), the imaginary unit plays a crucial role. As we expand the expression \( (1 + i)^2 = 1^2 + 2 \cdot 1 \cdot i + i^2 \), we see the influence of \( i \). By substituting \( i^2 = -1 \), the equation simplifies to \( 2i \)—showcasing how \( i \) allows us to simplify seemingly complex calculations into something manageable.

Therefore, \( i \) is more than just a placeholder; it is the key to unlocking complex arithmetic and enriching mathematical problem-solving.

binomial expansion

The binomial expansion is a formula used for expanding expressions raised to powers, such as \((a + b)^n\). This method is particularly useful when working with complex numbers. The formula for squaring a binomial is:

• \( (a + b)^2 = a^2 + 2ab + b^2 \)

To expand \((1+i)^2\), we identify \( a = 1 \) and \( b = i \). Substituting these into the formula, we have:

\( (1+i)^2 = 1^2 + 2 \cdot 1 \cdot i + i^2 \).
Simplifying further involves using the property of \( i \), where \( i^2 = -1 \):
\( 1 + 2i + (-1) = 2i \).

The binomial expansion not only simplifies calculations but also illustrates how polynomial structures interact with complex numbers, opening doors to further mathematical exploration and comprehension.