Problem 70 - Algebra A Combined Function

Question

Divide. Write your answers in the form \(a+b i\) $$ \frac{6+5 i}{6-5 i} $$

Step-by-Step Solution

Answer

\(\frac{11}{61} + \frac{60}{61}i\)

Step 1: Identify the Conjugate

To divide complex numbers, first identify the conjugate of the denominator. The conjugate of a complex number like \(6 - 5i\) is \(6 + 5i\).

Step 2: Multiply Numerator and Denominator by the Conjugate

Multiply both the numerator and the denominator by the conjugate of the denominator. This will make the denominator a real number. Thus, multiply \(\frac{6+5i}{6-5i}\) by \(\frac{6+5i}{6+5i}\).

Step 3: Apply Distribution to the Numerator

Calculate the new numerator by applying the distributive property: \((6+5i)(6+5i) = 6^2 + 2\cdot6\cdot5i + (5i)^2 = 36 + 60i - 25 = 11 + 60i\).

Step 4: Simplify the Denominator

Calculate the new denominator by applying the difference of squares: \((6-5i)(6+5i) = 6^2 - (5i)^2 = 36 + 25 = 61\).

Step 5: Write the Result in Standard Form

Combine the results to obtain the simplified expression as a standard complex number: \(\frac{11 + 60i}{61} = \frac{11}{61} + \frac{60}{61}i\).

Key Concepts

Conjugate of Complex Numbers

In complex numbers, the conjugate is a key concept that helps simplify expressions like division. The conjugate of a complex number is formed by changing the sign of the imaginary part. For example, the conjugate of a complex number represented as \( a + bi \) is \( a - bi \).
Using the conjugate is particularly helpful when dividing complex numbers. By multiplying the numerator and the denominator by the conjugate of the denominator, you transform the denominator into a real number. This makes further calculations simpler and allows you to express the result in the standard form of a complex number.
In the given exercise, the conjugate of the denominator \( 6 - 5i \) is \( 6 + 5i \). Multiplying both the numerator and denominator by this conjugate helps eliminate the imaginary part in the denominator.

Distributive Property

The distributive property is a useful algebraic principle that states: \( a(b + c) = ab + ac \). When working with complex numbers, this property lets you expand products of binomials effectively. The same rule applies even if one or both of the terms include complex numbers. This property is especially useful in conjunction with the conjugate when dividing complex numbers.
In our example, we use the distributive property when multiplying the new numerator: \( (6 + 5i)(6 + 5i) \). By following the distributive rule, you calculate:

• \( 6 \times 6 = 36 \)
• \( 6 \times 5i = 30i \)
• \( 5i \times 6 = 30i \)
• \( 5i \times 5i = (5i)^2 = -25 \)

Combining these products gives \( 36 + 60i - 25 \), which simplifies to \( 11 + 60i \). By distributing and simplifying, the expression remains manageable.

Difference of Squares

The difference of squares is a mathematical pattern where \( a^2 - b^2 \) can be factored into \( (a + b)(a - b) \). This is extremely useful in operations involving conjugates, especially when simplifying the denominator in division of complex numbers.
When you multiply a complex number by its conjugate, the product is a real number because the imaginary parts cancel out through this pattern. In our case, multiplying the conjugates \( (6-5i) \) and \( (6+5i) \) uses the difference of squares:

• First term: \( 6^2 = 36 \)
• Second term: \( (5i)^2 = -25 \)

Subtract the second term from the first, following \( a^2 - b^2 \), results in a real number \( 36 + 25 = 61 \). This conversion is what allows for the division of complex numbers to result in a standard form.

Complex Number Standard Form

Complex numbers can be fully expressed in a format known as the standard form, \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.
When solving division of complex numbers, the goal is to express the result in this standard form. This ensures clarity and aligns with mathematical conventions.
After simplifying the numerator \( 11 + 60i \) and having a real number in the denominator (\( 61 \)), the division can be carried out:

• The real part becomes \( \frac{11}{61} \)
• The imaginary part becomes \( \frac{60}{61}i \)

The final result is thus expressed as \( \frac{11}{61} + \frac{60}{61}i \), a tidy and easy-to-read standard form.