91Ó°ÊÓ

When faced with expressions like \((1+i)^{2}\) or \((1-i)^{2}\), the binomial theorem helps us expand these expressions.
The binomial theorem is powerful for expanding binomials raised to a power. The general form is:\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
For our specific problem, we use the simplified forms for squaring binomials:
\[ (a + b)^2 = a^2 + 2ab + b^2 \] and \[ (a - b)^2 = a^2 - 2ab + b^2 \]
First, identify \(a\) and \(b\) from the binomials \( (1+i) \text{and}\ (1-i)\). These are expanded as follows:
For \((1+i)^2\):
\[ a = 1 \text{ and } b = i \]
\[ (1+i)^2 = 1^2 + 2 \times 1 \times i + i^2 \= 1 + 2i + i^2 \]
Remember, \( i^2 = -1 \). So, substitute and simplify:
\[ 1 + 2i - 1 = 2i \]
For \(1-i)^2\):
\[ a = 1 \text{ and } b = i \]
\[ (1-i)^2 = 1^2 - 2 \times 1 \times i + i^2 = 1 - 2i + i^2 \]
Again, using \( i^2 = -1 \):
\[ 1 - 2i - 1 = -2i \]
Understanding the binomial theorem helps simplify complex expressions significantly, giving a structured approach to expanding binomials.

Complex numbers extend our understanding of regular numbers by introducing the imaginary unit \( i \), where \( i^2 = -1 \).
A complex number is expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit.
Here are some key points about complex numbers:

• The imaginary part \( bi \) involves the square root of -1.
• Complex numbers can be added, subtracted, multiplied, and divided just like real numbers, but they follow specific rules.
• Multiplying complex numbers uses the distributive property, just like polynomials.
• The result often involves simplifying terms with \( i^2 = -1 \).

In our exercise, we handled complex multiplication in steps:

• First, we expanded \( (1+i)^2 \) and \( (1-i)^2 \).
• We then obtained \( 2i \) and \( -2i \).

Finally, multiplying these results:
\( 2i \times -2i = -4i^2 \)
Since \( i^2 = -1 \), this simplifies to:
\( -4(-1) = 4 \)

The 'standard form' of a complex number is \( a + bi \), helping us present complex numbers consistently and clearly.
In this form:

• \