Imaginary numbers, a fascinating subset of complex numbers, introduce us to a realm where creativity meets mathematical precision. Rooted in the imaginary unit ii—where \(i^2 = -1\) —these numbers play a pivotal role in solving equations that defy real-number solutions. In this exploration, we will dive into the definition, basic operations, applications, and graphical representation of imaginary numbers.
Definition of Imaginary Numbers
An imaginary number is expressed as bi, where b is a real number and i is the imaginary unit. The general form of an imaginary number is xi, where x is any real number. Imaginary numbers are a subset of complex numbers, specifically those with a real part of 0.
Imaginary Unit (i):
The imaginary unit ii is the cornerstone of imaginary numbers, defined as i = \(\sqrt{-1}\). Its unique property of \(i^2 = -1\) provides a solution to equations that would otherwise have no real roots.
Basic Operations with Imaginary Numbers:
Addition and Subtraction:
Imaginary numbers are added or subtracted by combining their coefficients.
Example: \(2i + 3i = 5i/)
Multiplication:
Multiplying imaginary numbers involves leveraging the fact that \(i^2 = -1\)
Example: \(4i \times 2i = -8\)
Division:
Division requires multiplying the numerator and denominator by the conjugate of the denominator.
Example: \(\frac{3i}{2 – i}\)
Complex Numbers:
Imaginary numbers find their place within the broader set of complex numbers, expressed as \(a + bi\), where a and b are real numbers. Imaginary numbers occur as a special case when the real part (a) is zero.
Examples on Imaginary Numbers
1. Example: \(2i + 4i\)
Solution: \(2i + 4i = 6i\)
Explanation: Combine the imaginary parts.
2. Example: \((4 – 2i) – (2 + 4i)\)
Solution: \((5 – 2i) – (3 + 4i) = 2 – 6i\)
(4−2i)−(2+4i)=4−2i−2−4i
Combine the real parts: 4 – 2 = 2
Combine the imaginary parts: (-2i – 4i) = -6
So, (4 – 2i) – (2 + 4i) = 2−6i
3. Example: \((2i) \times (4 – i)\)
Solution: \((2i) \times (4 – i) = 8i – 2i^2 = 8i + 2\)
Explanation: Apply the distributive property and \(i^2 = -1\)