Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \frac{1-7 i}{6-2 i} $$

Step-by-Step Solution

Answer

The expression in the form \(a+bi\) is \(\frac{1}{2} - i\).

Step 1: Identify the Conjugate

To simplify the expression \( \frac{1-7i}{6-2i} \) and write it in the form \( a + bi \), we first need to multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 6-2i \) is \( 6+2i \).

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate \( 6+2i \).\[\frac{(1-7i)(6+2i)}{(6-2i)(6+2i)}\]

Step 3: Simplify the Denominator

Apply the formula for the difference of squares to the denominator:\[(6-2i)(6+2i) = 6^2 - (2i)^2 = 36 - (-4) = 36 + 4 = 40\]

Step 4: Expand the Numerator

Expand the numerator using the distributive property:\[(1-7i)(6+2i) = 1\times 6 + 1\times 2i - 7i \times 6 - 7i \times 2i \]This simplifies to:\[6 + 2i - 42i - 14i^2\]Recall that \(i^2 = -1\), so:\[-14i^2 = 14\].

Step 5: Combine Like Terms in the Numerator

Combine the real numbers and the imaginary numbers in the expanded expression:\[6 + 14 - 40i = 20 - 40i\]

Step 6: Combine Numerator and Denominator

Write the entire fraction with the simplified numerator and denominator:\[\frac{20 - 40i}{40}\]

Step 7: Simplify the Expression

Divide both the real and the imaginary parts by the denominator:\[\frac{20}{40} - \frac{40i}{40} = \frac{1}{2} - i\]Thus, \(a = \frac{1}{2}\) and \(b = -1\).

Key Concepts

Conjugate

The conjugate of a complex number is a nifty tool when it comes to simplifying divisions. A complex number has the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit which satisfies \(i^2 = -1\). The conjugate of this complex number is \(a - bi\). Simply put, you change the sign of the imaginary part.

• For example, the conjugate of \(6-2i\) is \(6+2i\).

By multiplying a complex number by its conjugate, you effectively get rid of the imaginary part in the denominator. This is because the product of a number and its conjugate results in a real number, using the difference of squares formula. This trick is essential for keeping expressions neat and simpler in mathematics.

Imaginary Numbers

Imaginary numbers are the backbone of complex numbers, encompassing all numbers that cannot be seen on the traditional real number line. The fundamental unit of imaginary numbers is \(i\), defined as the square root of \(-1\), which leads to the identity \(i^2 = -1\).

• These numbers arise particularly in the context of complex arithmetic.
• For instance, in our example equation, imaginary numbers appear as \(-7i\) in the numerator and \(-2i\) in the denominator.

Imaginary numbers help in making expressions like square roots of negative numbers possible, solving polynomial equations that lack real solutions, and have broad applications in engineering, physics, and beyond. They allow mathematical calculations to extend beyond the limitations of real numbers.

Real Numbers

Real numbers are the everyday numbers we use. They include both rational numbers (like 2, 1/2, or −3) and irrational numbers (like the square root of 2 or π). Real numbers can be located on the number line that we learned about in grade school.

• In the expression \(1 - 7i\), the number 1 is the real part.
• Similarly, in \(6 - 2i\), the number 6 is the real part.

Real numbers are fundamental for many mathematical operations and provide the reference point from which imaginary numbers extend. Understanding real numbers is crucial because they serve as the real part in a complex number \(a + bi\), where \(a\) is the real component and \(bi\) is the imaginary component.

Difference of Squares

The difference of squares is a mathematical formula often used to simplify expressions and solve equations. It states that for any two numbers \(a\) and \(b\), \((a-b)(a+b) = a^2 - b^2\). This concept is handy when simplifying complex fractions involving imaginary numbers.

• In the original problem's denominator, \((6-2i)(6+2i)\) equates to \(6^2 - (2i)^2\).
• This simplifies with the formula to \(36 - (-4) = 40\).

Using the difference of squares helps transform a complex denominator into a manageable real number, which then allows for straightforward division of the numerator, simplifying the expression's form into \(a + bi\). This method is key for writing any complex number division into a simpler, more interpretable form.