Problem 47 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ \frac{-2+i}{(1+i)^{2}} $$

Step-by-Step Solution

Answer

The standard form of the expression is \(-\frac{1}{2} - i\).

Step 1: Expand the Denominator

First, let's expand the denominator \((1+i)^2\). We need to apply the formula for the square of a binomial: \((a+b)^2 = a^2 + 2ab + b^2\). In this case, \(a=1\) and \(b=i\). Thus, we have:\[(1+i)^2 = 1^2 + 2 \cdot 1 \cdot i + i^2\]Since \(i^2 = -1\), this becomes:\[1 + 2i + (-1) = 1 + 2i - 1 = 2i\]The expanded denominator is \(2i\).

Step 2: Multiply Numerator and Denominator by the Conjugate

To simplify the fraction \(\frac{-2+i}{2i}\), multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2i\) is \(-2i\). So:\[\frac{-2+i}{2i} \times \frac{-2i}{-2i}\]Multiplying the numerator and the denominator by \(-2i\) gives:\[\frac{(-2+i)(-2i)}{(2i)(-2i)}\]

Step 3: Simplify the Denominator

Calculate the new denominator using the property \(a \cdot b = -b \cdot a\). Here, \(a = 2i\), \(b = -2i\):\[(2i)(-2i) = 4i^2 = 4(-1) = -4\]So the denominator simplifies to \(-4\).

Step 4: Simplify the Numerator

Next, expand and simplify the numerator:\[(-2+i)(-2i) = -2(-2i) + i(-2i) = 4i - 2i^2\]Since \(i^2 = -1\), we substitute to get:\[4i - 2(-1) = 4i + 2 = 2 + 4i\]

Step 5: Express the Result in Standard Form

Finally, simplify \(\frac{2+4i}{-4}\) and express it in standard form, \(a+bi\):Divide both the real, \(2\), and the imaginary part, \(4i\), by \(-4\):\[\frac{2}{-4} + \frac{4i}{-4} = -\frac{1}{2} - i\]Thus, the expression in standard form is \(-\frac{1}{2} - i\).

Key Concepts

Standard Form

Standard form for complex numbers is the way we typically represent complex numbers, which is in the form of \(a + bi\), where \(a\) and \(b\) are real numbers. It's important because it allows us to clearly see the real and imaginary parts of the complex number.

• \(a\) is the real part and it is located on the x-axis of the complex plane.
• \(bi\) is the imaginary part and it is located on the y-axis of the complex plane.

When we express a result in standard form, it means converting a potentially complicated expression into the \(a + bi\) format. This helps in comparing and simplifying calculations with complex numbers. The expression \(-\frac{1}{2} - i\) is in standard form because it clearly separates the real part, \(-\frac{1}{2}\), and the imaginary part, \(-i\).
Understanding how to rewrite into standard form is crucial for solving and simplifying any complex number expressions.

Imaginary Unit

The imaginary unit is represented by \(i\), which is the foundation of complex numbers. It is defined by the equation \(i^2 = -1\), which separates imaginary numbers from real numbers. The imaginary unit allows us to handle numbers that are not real, especially when we encounter situations like taking the square root of a negative number.

• \(i\) can be considered as a rotation operator on the complex plane.
• Multiplying by \(i\) rotates a point by 90 degrees counter-clockwise.

In complex numbers, the imaginary unit \(i\) provides a bridge to understand concepts that extend beyond the usual real number line. Using \(i\) can at first be a bit tricky, but it becomes natural as you remember the key identity \(i^2 = -1\). This identity provides the base for simplifying expressions involving \(i\), such as in the step where \(i^2 = -1\) allowed the calculation of the denominator in the exercise.

Conjugate

Conjugates are a pair of complex numbers that have the same real part but opposite imaginary parts. For example, the conjugate of \(a + bi\) is \(a - bi\). This property is useful when simplifying fractions with complex numbers, especially to eliminate the imaginary part in a denominator.

• The conjugate of \(2i\) is \(-2i\).
• Multiplication by the conjugate results in a real number, as in the exercise.

When you multiply a complex number by its conjugate, the result is a real number because the imaginary parts cancel out. This concept was used in the exercise to simplify the expression by removing the imaginary unit from the denominator, turning it into a real number denominator, \(-4\), thereby allowing the fraction to be more easily expressed in standard form. Understanding and using the conjugate is often key in making complex number arithmetic more manageable.