Simplifying Imaginary Numbers: A Step-by-Step Guide

Imaginary numbers, often a source of mystery in mathematics, are actually quite straightforward once you understand their basic principles. This comprehensive guide will walk you through simplifying various expressions involving imaginary numbers. Whether you're tackling square roots of negative numbers or complex expressions with the imaginary unit 'i', this article will provide you with clear explanations and step-by-step solutions. Let's dive in and unravel the world of imaginary numbers!

Understanding Imaginary Numbers

Before we jump into simplification, let's define what imaginary numbers are. In mathematics, the imaginary unit is denoted by i, which is defined as the square root of -1, i.e., i = √(-1). Imaginary numbers are multiples of i, such as 2i, -5i, or i√3. These numbers, when combined with real numbers, form complex numbers.

The concept of imaginary numbers arises from the need to solve equations that have no real solutions. For instance, the equation x² + 1 = 0 has no real solutions because any real number squared is non-negative. To address such scenarios, mathematicians introduced the concept of imaginary numbers, expanding the number system to include complex numbers. Understanding this foundation is crucial for simplifying expressions effectively. Complex numbers are typically written in the form a + bi, where a is the real part and bi is the imaginary part. This form allows us to perform arithmetic operations on complex numbers in a structured manner.

Simplifying Square Roots of Negative Numbers

To simplify the square root of a negative number, we first express the number as a product of -1 and a positive number. For example, to simplify √(-48), we rewrite it as √(-1 * 48). Then, we can use the property √(ab) = √(a) * √(b) to separate the square root into √(-1) * √(48). Since √(-1) is defined as i, we have i√(48). Finally, we simplify the square root of the positive number (if possible). In this case, √(48) can be simplified to √(16 * 3) = 4√3. Therefore, √(-48) simplifies to 4i√3. This process involves identifying perfect square factors within the radicand (the number under the square root) and extracting them. By breaking down the problem into smaller, manageable steps, we can systematically simplify complex expressions involving imaginary numbers.

Step-by-Step Simplification Examples

Let’s tackle the given problems one by one, providing a clear step-by-step solution for each.

1. Simplify $\sqrt{-48}$

As discussed above, we break down √(-48) as follows:

√(-48) = √(-1 * 48)

= √(-1) * √(48)

= i√(48)

Now, simplify √(48) by finding its perfect square factors:

√(48) = √(16 * 3) = √(16) * √(3) = 4√3

Thus, √(-48) = 4i√3.

2. Simplify $\sqrt{-63}$

Similarly, we simplify √(-63):

√(-63) = √(-1 * 63)

= √(-1) * √(63)

= i√(63)

Now, simplify √(63):

√(63) = √(9 * 7) = √(9) * √(7) = 3√7

Thus, √(-63) = 3i√7.

3. Simplify $\sqrt{-72}$

Simplifying √(-72):

√(-72) = √(-1 * 72)

= √(-1) * √(72)

= i√(72)

Simplify √(72):

√(72) = √(36 * 2) = √(36) * √(2) = 6√2

Thus, √(-72) = 6i√2.

4. Simplify $\sqrt{-24}$

Simplifying √(-24):

√(-24) = √(-1 * 24)

= √(-1) * √(24)

= i√(24)

Simplify √(24):

√(24) = √(4 * 6) = √(4) * √(6) = 2√6

Thus, √(-24) = 2i√6.

5. Simplify $\sqrt{-84}$

Simplifying √(-84):

√(-84) = √(-1 * 84)

= √(-1) * √(84)

= i√(84)

Simplify √(84):

√(84) = √(4 * 21) = √(4) * √(21) = 2√21

Thus, √(-84) = 2i√21.

6. Simplify $\sqrt{-99}$

Simplifying √(-99):

√(-99) = √(-1 * 99)

= √(-1) * √(99)

= i√(99)

Simplify √(99):

√(99) = √(9 * 11) = √(9) * √(11) = 3√11

Thus, √(-99) = 3i√11.

Simplifying Products of Imaginary Numbers

When multiplying imaginary numbers, it’s crucial to remember that i² = -1. This identity is fundamental for simplifying expressions involving products of imaginary numbers.

7. Simplify $\sqrt{-23} \cdot \sqrt{-46}$

√(-23) * √(-46) = (i√23) * (i√46)

= i² * √(23 * 46)

= -1 * √(23 * 23 * 2)

= -1 * 23√2

Thus, √(-23) * √(-46) = -23√2.

8. Simplify $\sqrt{-6} \cdot \sqrt{-3}$

√(-6) * √(-3) = (i√6) * (i√3)

= i² * √(6 * 3)

= -1 * √(18)

= -1 * √(9 * 2)

= -1 * 3√2

Thus, √(-6) * √(-3) = -3√2.

9. Simplify $\sqrt{-5} \cdot \sqrt{-10}$

√(-5) * √(-10) = (i√5) * (i√10)

= i² * √(5 * 10)

= -1 * √(50)

= -1 * √(25 * 2)

= -1 * 5√2

Thus, √(-5) * √(-10) = -5√2.

Simplifying Expressions with i

10. Simplify $(3 i)(-2 i)(5 i)$

(3i)(-2i)(5i) = 3 * -2 * 5 * i * i * i

= -30 * i³

Since i² = -1, then i³ = i² * i = -1 * i = -i

So, -30 * i³ = -30 * (-i) = 30i.

11. Simplify $i^{11}$

To simplify powers of i, we look for patterns. We know that:

i¹ = i

i² = -1

i³ = -i

i⁴ = 1

This pattern repeats every four powers. To simplify i¹¹, we divide the exponent (11) by 4:

11 ÷ 4 = 2 remainder 3

This means i¹¹ is equivalent to i³:

i¹¹ = i³ = -i.

12. Simplify $4 i(-6)$

4i(-6) = 4 * -6 * i

= -24i.

Conclusion

Simplifying imaginary numbers may seem challenging at first, but with a clear understanding of the basic principles and step-by-step approach, it becomes quite manageable. Remember the key concepts: i = √(-1) and i² = -1. These concepts, combined with the ability to break down square roots and identify patterns in powers of i, will enable you to simplify a wide range of expressions involving imaginary numbers. By practicing these techniques, you'll not only master the simplification of imaginary numbers but also deepen your understanding of the broader realm of complex numbers.

For more in-depth information and examples, you can explore resources like Khan Academy's Complex Numbers, which offers a comprehensive overview and practice exercises.