Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \frac{5-\sqrt{-121}}{1+\sqrt{-25}} $$

Step-by-Step Solution

Answer

The expression is \(-\frac{25}{13} - \frac{18}{13}i\)."

Step 1: Simplify the square roots of negative numbers

Recognize that square roots of negative numbers involve imaginary numbers. Here, \( \sqrt{-121} = 11i \) because \( \sqrt{121} = 11 \) and \( i = \sqrt{-1} \). Similarly, \( \sqrt{-25} = 5i \) since \( \sqrt{25} = 5 \) and \( i = \sqrt{-1} \). Substitute these back into the expression: \( \frac{5 - 11i}{1 + 5i} \).

Step 2: Multiply numerator and denominator by the conjugate of the denominator

To simplify the expression, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(1 + 5i\) is \(1 - 5i\). So multiply the expression by \(\frac{1 - 5i}{1 - 5i}\) to get:\[\frac{(5 - 11i)(1 - 5i)}{(1 + 5i)(1 - 5i)}\].

Step 3: Apply the formula for conjugates in the denominator

Use the formula for the product of conjugates in the denominator: \((a + bi)(a - bi) = a^2 + b^2\). This gives:\(1^2 + (5i)^2 = 1 + 25 (-1) = 1 + 25\). Therefore, the denominator becomes \(1 + 25 = 26\).

Step 4: Expand the numerator

Use the distributive property to expand the numerator:\((5 - 11i)(1 - 5i) = 5 \cdot 1 + 5 \cdot (-5i) - 11i \cdot 1 - 11i \cdot (-5i)\).This simplifies as follows:- \(5 \cdot 1 = 5\)- \(5 \cdot (-5i) = -25i\)- \(-11i \cdot 1 = -11i\)- \(-11i \cdot (-5i) = 55i^2\).Note that \(i^2 = -1\), so \(55i^2 = -55\).

Step 5: Simplify the expanded form

Combine the real and imaginary parts:- Real part: \(5 - 55 = -50\)- Imaginary part: \(-25i + (-11i) = -36i\).The numerator is \(-50 - 36i\).

Step 6: Write the final expression

Divide both the real and imaginary parts by \(26\) to express the division:\[\frac{-50 - 36i}{26} = \frac{-50}{26} + \frac{-36i}{26}\]. Simplify:- Real part: \(\frac{-50}{26} = -\frac{25}{13}\)- Imaginary part: \(\frac{-36}{26} = -\frac{18}{13}\).Thus, the expression in the form \(a + bi\) is \(-\frac{25}{13} - \frac{18}{13}i\).

Key Concepts

Imaginary Numbers

When dealing with the square roots of negative numbers, we find ourselves in the realm of imaginary numbers. This is because no real number squared can be negative. The concept of imaginary numbers is introduced through the imaginary unit, denoted as \( i \), which is defined as \( i = \sqrt{-1} \). This allows us to express square roots of negative numbers in terms of \( i \).

• An example from the original exercise is \( \sqrt{-121} = 11i \), because \( \sqrt{121} = 11 \) and multiplying by \( i \) accounts for the negative under the root.
• Similarly, \( \sqrt{-25} = 5i \) because \( \sqrt{25} = 5 \).

Imaginary numbers are essential for expressing certain complex expressions in a standard form. They help us manipulate and simplify expressions involving negative square roots.

Conjugate

Conjugates are an essential concept when simplifying complex numbers, especially for division. The conjugate of a complex number \( a + bi \) is \( a - bi \). Multiplying a complex number by its conjugate results in a real number.

• This technique is particularly useful for eliminating the imaginary part from the denominator of a fraction involving complex numbers.
• In the original exercise, multiplying by the conjugate \( 1 - 5i \) simplified the expression.
• The denominator becomes a real number: \( (1 + 5i)(1 - 5i) = 1^2 + (-5i)^2 = 1 + 25 = 26 \).

Using conjugates allows us to work with complex numbers in a more manageable form.

Real Part

The real part of a complex number is the component without the imaginary unit \( i \). In the standard form \( a + bi \), \( a \) represents this real part.

• For instance, in our simplified expression \(-\frac{25}{13} - \frac{18}{13}i\), the real part is \(-\frac{25}{13}\).
• The real part is obtained after performing operations on the complex expression, focusing only on terms without \( i \).
• In the final step of the solution, the real portion came from combining the terms \( 5 \) and \(-55 \) from the expanded form, resulting in \(-50 \), which is then divided by \( 26 \) to simplify.

Mastering the identification of the real part is crucial for expressing and simplifying complex numbers.

Imaginary Part

The imaginary part of a complex number is the coefficient of the imaginary unit \( i \) in the expression. In \( a + bi \), the imaginary part is \( b \). Recognizing and working with the imaginary part is key in understanding and simplifying complex numbers.

• In our example, the imaginary part in the expression \(-\frac{25}{13} - \frac{18}{13}i\) is \(-\frac{18}{13}\).
• From the expanded form, both \(-25i\) and \(-11i\) contribute to the imaginary component, yielding a sum of \(-36i \).
• This combined figure is ultimately divided by \( 26 \) to express it in the simplified form.

The imaginary part tells us how a complex number lines up along the imaginary axis and plays a big role in calculations involving imaginary units.