Problem 61 - Algebra 2

Question

Simplify. \(\frac{1+i}{3-2 i}\)

Step-by-Step Solution

Answer

The simplified form is \( \frac{1+5i}{13} \).

Step 1: Identify the complex fraction

The exercise provided is to simplify the complex fraction \( \frac{1+i}{3-2i} \). A complex fraction like this involves complex numbers in both the numerator and the denominator.

Step 2: Find the conjugate of the denominator

The denominator of the complex fraction is \( 3 - 2i \). The conjugate of this is \( 3 + 2i \). Conjugates of complex numbers are used to eliminate imaginary parts from the denominator.

Step 3: Multiply the numerator and the denominator by the conjugate of the denominator

Multiply both the numerator \( 1+i \) and the denominator \( 3-2i \) by the conjugate of the denominator, which is \( 3+2i \). This gives:\[\frac{(1+i)(3+2i)}{(3-2i)(3+2i)}\]

Step 4: Expand the terms in the numerator

Use the distributive property to expand \((1+i)(3+2i)\):\[1 \cdot 3 + 1 \cdot 2i + i \cdot 3 + i \cdot 2i = 3 + 2i + 3i + 2i^2\]Since \(i^2 = -1\), \(2i^2 = 2(-1) = -2\). Thus, the expression becomes:\[3 + 2i + 3i - 2 = 1 + 5i\]

Step 5: Simplify the denominator

The denominator \((3-2i)(3+2i)\) is a difference of squares, which simplifies as follows:\[ a^2 - b^2 = (a-b)(a+b) \] for complex numbers,\[ 3^2 - (2i)^2 = 9 - 4i^2 = 9 - 4(-1) = 9 + 4 = 13 \]

Step 6: Form the simplified fraction

The expanded numerator and simplified denominator give:\[ \frac{1+5i}{13} \]This represents the simplified form of the original complex fraction.

Key Concepts

Complex Fraction

A complex fraction involves fractions that have complex numbers as part of the numerator, the denominator, or both. In our example, you are tasked with simplifying \( \frac{1+i}{3-2i} \), which is a complex fraction because it has complex numbers in both parts. Complex numbers, typically expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit, can make a fraction seem daunting to simplify at first. In essence, to simplify a complex fraction, we aim to eliminate the imaginary unit from the denominator. This process usually involves using the conjugate of the denominator.

Conjugate

The conjugate of a complex number is crucial when simplifying a complex fraction. If you have a complex number \( a + bi \), the conjugate is \( a - bi \). Conjugates are helpful because they allow us to eliminate the imaginary part from the denominator of a fraction. For instance, in the exercise, the denominator \( 3 - 2i \) has a conjugate of \( 3 + 2i \). When you multiply the numerator and denominator of the fraction by the conjugate, such as \( \frac{1+i}{3-2i} \cdot \frac{3+2i}{3+2i} \), the imaginary units in the denominator cancel out, enabling a simplified form. This technique relies on the property that the product of a complex number and its conjugate is a real number.

Distributive Property

The distributive property is a fundamental algebraic concept that states \( a(b + c) = ab + ac \). It comes into play when simplifying complex fractions by enabling us to expand products containing complex numbers. For example, to handle the product \( (1+i)(3+2i) \), you apply the distributive property:

• \( 1 \times 3 = 3 \)
• \( 1 \times 2i = 2i \)
• \( i \times 3 = 3i \)
• \( i \times 2i = 2i^2 \)

Remember, \( i^2 \) equals \( -1 \), so we adjust \( 2i^2 \) to \( -2 \). Combining these results gives the simplified expression \( 1 + 5i \) from the expansion \( 3 + 2i + 3i - 2 \). Breaking down each step using the distributive property makes complex calculations more manageable.

Imaginary Number

Imaginary numbers are a type of number that incorporates the imaginary unit \( i \), where \( i^2 = -1 \). In the context of complex numbers, they typically accompany real numbers to form expressions like \( 1 + i \). Imaginary numbers allow us to extend the number system so that we can solve equations not solvable by real numbers alone, like \( x^2 + 1 = 0 \). When dealing with complex fractions, the goal is often to remove the imaginary unit from the denominator. This is achieved using conjugates, as multiplying a complex number by its conjugate results in a real number. In simplifying \( \frac{1+i}{3-2i} \), operations involving \( i \) result in transformations crucial for obtaining a real number in the denominator.

Difference of Squares

The difference of squares is an algebraic term describing the difference between two squared numbers: \( a^2 - b^2 \). In complex arithmetic, it's very useful, especially when simplifying complex fractions by removing the imaginary unit from the denominator. For example, in the denominator of \( \frac{1+i}{3-2i} \cdot \frac{3+2i}{3+2i} \), the expression \[ (3 - 2i)(3 + 2i) \] uses the difference of squares. Computing \[ 3^2 - (2i)^2 \] results in \( 9 - 4(-1) \) since \( (2i)^2 = -4 \). Thus, it simplifies to \( 9 + 4 = 13 \), providing a real number in the denominator and enabling the fraction to be simplified to \( \frac{1+5i}{13} \). The technique of using the difference of squares assures us the denominator will become a straightforward real number.