Problem 34 - Intermediate Algebra

Question

Write each number in the form a \(+b i.\) a. \(88-\sqrt{-98}\) b. \(-2+\sqrt{-35}\)

Step-by-Step Solution

Answer

a. The number in the form a + bi is 88 - 7√2i; b. It is -2 + √35i.

Step 1: Identify Real and Imaginary Parts

For each number, determine the real part and the imaginary part. Recall that the imaginary unit is represented as \( i \) and is defined as the square root of -1.

Step 2: Simplify the Imaginary Part

For square roots of negative numbers, use the relation \( \sqrt{-1} = i \). For instance, \( \sqrt{-98} = \sqrt{98} \cdot \sqrt{-1} = \sqrt{98} \cdot i \). Calculate the square root of the positive part, \( \sqrt{98} = \sqrt{2 \cdot 49} = 7 \sqrt{2} \). Thus, \( \sqrt{-98} = 7 \sqrt{2} i \).

Step 3: Express the Expression in a + bi Form (Part a)

Substitute back the simplified imaginary expression into the original expression. For part a, substitute into \( 88 - \sqrt{-98} \) to get \( 88 - 7 \sqrt{2} i \).

Step 4: Simplify the Imaginary Part (Part b)

Similarly for \( \sqrt{-35} \), express it as \( \sqrt{35} \cdot i \). Since \( \sqrt{35} \) can't be broken down further, leave it as is. So \( \sqrt{-35} = \sqrt{35} i \).

Step 5: Express the Expression in a + bi Form (Part b)

For part b, replace \( \sqrt{-35} \) in \( -2 + \sqrt{-35} \) to get \(-2 + \sqrt{35} i \).

Key Concepts

Imaginary Unit

The imaginary unit, often denoted by the letter \( i \), is a fundamental concept in complex numbers. It represents the square root of \(-1\), a number that doesn't have a real value. Instead, we use \( i \) to work with square roots of negative numbers.

• \( i^2 = -1 \), which is the defining characteristic of the imaginary unit.
• The use of \( i \) allows us to extend real numbers into complex numbers, enabling calculations involving square roots of negative values.

Understanding the imaginary unit is key because it forms the basis of expressing complex numbers and allows for simplifying expressions containing negative square roots. In practice, whenever we see a negative under a square root, we can extract \( i \). For example, \( \sqrt{-9} = 3i \). This simplification is crucial when constructing complex numbers.

Simplifying Square Roots

When working with square roots of negative numbers, simplifying them into a form involving the imaginary unit \( i \) makes them more manageable. The process generally involves the following steps:

• Identify the negative under the square root.
• Express the square root as a product: the square root of the positive counterpart and the square root of \(-1\).

For example, to simplify \( \sqrt{-98} \):1. Recognize \( \sqrt{-98} \) as \( \sqrt{98} \times \sqrt{-1} \).2. Replace \( \sqrt{-1} \) with \( i \).3. Simplify \( \sqrt{98} \) by finding its factorization. Here, \( 98 = 2 \times 49 \), so it simplifies to \( 7 \sqrt{2} \).Putting it all together gives us \( \sqrt{-98} = 7 \sqrt{2} i \). This process turns a complex root problem into a simpler form where real and imaginary parts can be clearly identified.

Expressing Numbers in a + bi Form

Expressing complex numbers in the form \( a + bi \) is a standard way to present them, where \( a \) is the real part and \( b \) is the coefficient of the imaginary part \( i \). This form makes complex numbers easy to work with, particularly when adding, subtracting, or comparing them.

• The term \( a \) represents the real number component.
• The term \( bi \) signifies the imaginary part.

To express numbers like \( 88 - \sqrt{-98} \) in \( a + bi \) form:1. Simplify the imaginary part, as shown previously.2. Combine the simplified imaginary part with the real number.In this example, we simplify to: \( 88 - 7\sqrt{2}i \). Another example: transforming \( -2 + \sqrt{-35} \) into \( a + bi \) form results in: \(-2 + \sqrt{35}i \). Mastering this form helps in visual representation and calculation within the complex number system.