Problem 19 - Algebra 2 and Trigonometry

Question

In \(19-34,\) write each sum or difference in terms of \(i\) $$ \sqrt{-100}+\sqrt{-81} $$

Step-by-Step Solution

Answer

The sum is \( 19i \).

Step 1: Identify imaginary roots

Recall that the square root of a negative number involves the imaginary unit 'i', where \[ i = \sqrt{-1} \]. Therefore, \( \sqrt{-100} \) and \( \sqrt{-81} \) can be written in terms of \( i \).

Step 2: Express as imaginary numbers

Rewrite \( \sqrt{-100} \) and \( \sqrt{-81} \) using the imaginary unit \( i \):\[ \sqrt{-100} = \sqrt{100} \cdot \sqrt{-1} = 10i \]\[ \sqrt{-81} = \sqrt{81} \cdot \sqrt{-1} = 9i \]

Step 3: Calculate the sum

Add the two imaginary numbers \( 10i \) and \( 9i \): \[ 10i + 9i = 19i \].

Step 4: Write the result in terms of i

Combine the terms obtained:The sum \( \sqrt{-100} + \sqrt{-81} \) is expressed as:\[ 19i \].

Key Concepts

Understanding the Imaginary Unit

The imaginary unit, commonly denoted as \( i \), is a fundamental concept in mathematics. It serves as the building block of complex numbers. The defining property of \( i \) is that it represents the square root of \(-1\). This means that \( i \) satisfies the equation \( i^2 = -1 \). This unusual property allows \( i \) to operate in ways that real numbers do not, providing a method for expressing and working with square roots of negative numbers.

• \( i \) is a symbol used to express numbers that can't be handled by real numbers alone.
• Understanding \( i \) opens up a wide range of possibilities in mathematics, including engineering and physics.

To simplify square roots of negative numbers, they can be broken into two parts: the square root of the positive component and \( i \), the square root of \(-1\). For example, \( \sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i \). By mastering this logic, students can navigate through problems involving imaginary numbers and perform operations such as addition, subtraction, and multiplication on them.

Exploring Square Roots in the Context of Negative Numbers

A square root asks the question: what number, when multiplied by itself, will give the original number? This is straightforward for positive numbers, but for negative numbers, we step into the realm of the imaginary unit.
Normally, the square root of 4 is 2, because \( 2 \times 2 = 4 \). But negative numbers work differently.

• Real numbers don't have real square roots for negative numbers.
• This is where the imaginary unit \( i \) becomes vital.

To find the square root of a negative number, split it into \( \sqrt{n} \cdot \sqrt{-1} \), with \( \sqrt{-1} \) equaling \( i \). For example, \( \sqrt{-25} \) becomes \( \sqrt{25} \cdot \sqrt{-1} = 5i \). By approaching square roots in this way, you can handle numerous mathematical problems involving both real and imaginary quantities smoothly.

The Fascinating World of Imaginary Numbers

Imaginary numbers, while sounding fictional, are an essential aspect of mathematics. They emerge from the necessity to address the square roots of negative numbers. Without imaginary numbers, many algebraic equations would lack solutions in the real number system.

• Imaginary numbers are numbers that include the imaginary unit \( i \), such as \( 3i \) or \(-7i \).
• They allow us to extend the one-dimensional number line into a complex plane, offering greater mathematical flexibility.

These numbers are crucial in complex number calculations, which incorporate both real and imaginary components. Complex numbers are expressed in the form \( a + bi \), where \( a \) is a real number, and \( b \) is an imaginary number.
Operations with imaginary numbers, such as addition, subtraction, and multiplication, follow similar rules to those for real numbers but incorporate the unique property of \( i \). For example, adding \( 5i + 12i \) equals \( 17i \), just like you would add \( 5 + 12 = 17 \) with real numbers. Understanding and utilizing imaginary numbers enables deeper exploration of equations and phenomena across various fields.