Problem 31 - Intermediate Algebra

Question

Write each number in the form a \(+b i.\) a. \(1+\sqrt{-25}\) b. \(-3+\sqrt{-8}\)

Step-by-Step Solution

Answer

a) \(1 + 5i\); b) \(-3 + 2\sqrt{2}i\)

Step 1: Simplify the square root of the negative number

For both parts, start by simplifying the square root of the negative number using the property \( \sqrt{-x} = i\sqrt{x} \).- For part (a), \( \sqrt{-25} = i\sqrt{25} = 5i \).- For part (b), \( \sqrt{-8} = i\sqrt{8} = i \cdot 2\sqrt{2} = 2\sqrt{2}i \).

Step 2: Write the numbers in a + bi form

Combine the simplified square root with the given real number to write each expression in the form \( a + bi \).- For part (a), the expression becomes \( 1 + 5i \).- For part (b), the expression becomes \( -3 + 2\sqrt{2}i \).

Key Concepts

Understanding Imaginary Numbers

Have you ever come across a math problem that involves the square root of a negative number and wondered how to solve it? This is where imaginary numbers come in. An imaginary number is a type of complex number. It's defined as the square root of a negative number and is denoted by the symbol \( i \), which stands for the imaginary unit.

• \( i \) itself is defined as \( \sqrt{-1} \).
• Imaginary numbers extend the concept of numbers beyond the real numbers found on the number line.
• For example, in the expression \( \sqrt{-9} \), we rewrite it using \( i \): \( \sqrt{-9} = i \cdot 3 \) because \( \sqrt{9} = 3 \).

Imaginary numbers can be added to real numbers, creating what we call complex numbers.

Square Roots of Negative Numbers

Why can't we have a real number as the square root of a negative number? This is an excellent question! The square root of a negative number isn't defined within the set of real numbers. However, using imaginary numbers, we can find what's called the principal square root.

• The principal square root of a negative number \( -x \) is given by \( i\sqrt{x} \).
• This means, for \( \sqrt{-36} \), it becomes \( i\sqrt{36} = 6i \), because \( \sqrt{36} = 6 \) and \( i \) accounts for the negative sign.

This approach allows us to deal with negative square roots easily and is useful in various fields such as engineering and physics.

Expressing Numbers in a + bi Form

When we talk about complex numbers, we're often referring to them in the form \( a + bi \), where \( a \) is the real part, and \( b \) is the imaginary part.

• Every complex number can be plotted on a plane, with the real part \( a \) along the horizontal axis and the imaginary part \( b \) along the vertical axis.
• For example, to convert \( 1+\sqrt{-25} \) to \( a + bi \) form, we first find \( \sqrt{-25} = i\sqrt{25} = 5i \). Thus, the expression becomes \( 1 + 5i \).
• Similarly, for \(-3+\sqrt{-8}\), it becomes \(-3 + 2\sqrt{2}i\) after simplifying.

This representation helps in performing arithmetic with complex numbers and gaining a better visual understanding of them.