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Chapter 7: Problem 19

Represent the complex number geometrically. $$(1+i)^{2}$$

Short Answer

Expert verified
Geometrically, \((1+i)^2\) is represented by the point (0, 2) on the complex plane.

Step by step solution

01

Expand the Complex Number

First, we need to find the value of \((1+i)^2\). Use the formula for squaring a binomial, \(a + b\)^2 = a^2 + 2ab + b^2\. In this case, substitute \(a = 1\) and \(b = i\).
02

Perform the Multiplication

Calculate each part of the expanded equation.\[(1+i)^2 = 1^2 + 2 \cdot 1 \cdot i + i^2\]This expands to:\[= 1 + 2i + i^2\]
03

Simplify Using Complex Numbers

Now simplify the expression by remembering that \(i^2 = -1\):\[1 + 2i + i^2 = 1 + 2i - 1\]Thus, we are left with \[= 2i\]
04

Represent on the Complex Plane

To represent the number geometrically, plot it on the complex plane. The real part is \(0\) and the imaginary part is \(2\), so plot the point \((0, 2)\) on the complex plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Plane
When dealing with complex numbers, the complex plane is a helpful tool. Think of it like a graph with two axes:
  • The horizontal axis represents the real part of a complex number.
  • The vertical axis stands for the imaginary part, introduced by the imaginary unit \(i\).
Each complex number corresponds to a unique point on this plane, marked by two components: a real part \((x)\) and an imaginary part \((y)\). For the exercise, the complex number \(2i\) is represented as the point \((0, 2)\). Geometrically, this means you move zero units along the real axis and two units along the imaginary axis. Visualizing complex numbers this way can make operations like addition and multiplication clearer, as they can be interpreted as movements in this 2D space.
It turns a potentially abstract concept into a tangible tool for mathematical intuition.
Imaginary Unit
The imaginary unit, denoted as \(i\), plays a crucial role in forming complex numbers. Specifically, it is defined by a simple yet intriguing property: \(i^2 = -1\). While initially confusing, this definition allows us to extend the familiar number system into a broader realm where every number has both a real and an imaginary part.
This is particularly useful in various fields, such as engineering and physics, where complex numbers are used to solve problems that are difficult to tackle using only real numbers.
  • In the exercise provided, \(i\) helps transform the expression \((1+i)^2\) into \(2i\).
  • Remember, it's this formulation that lets us handle mathematical concepts involving square roots of negative numbers.
In essence, the imaginary unit opens up a new dimension for mathematical exploration beyond the basic real numbers.
Binomial Expansion
Binomial expansion is a handy algebraic technique when dealing with expressions raised to powers. In simpler terms, it lets you expand expressions like \((a + b)^n\) using a known formula.
  • For \((a + b)^2\), the formula is \(a^2 + 2ab + b^2\).
  • This is exactly what was applied to \((1+i)^2\) in the problem at hand.
By substituting \(a = 1\) and \(b = i\), the expression expands and simplifies, giving a clear pathway to solve the problem.Keep in mind that binomial expansion simplifies complex number operations by breaking down powers into easily manageable components. For students, this method often provides a structured approach to reach the correct answer without getting lost in more intricate algebraic manipulation.

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