Imaginary numbers are a subset of complex numbers that involve the imaginary unit, denoted by i, where i = \(\sqrt{-1}\). These numbers are crucial in mathematics and engineering, especially when dealing with problems that have no real number solutions. Let’s explore imaginary numbers in more detail:
Table of Contents
Definition of Imaginary Numbers
An imaginary number is expressed in the form 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 Unit (i):
The imaginary unit is the fundamental building block of imaginary numbers. It is defined as i = \(\sqrt{-1}\) . The square of ii results in −1, providing a unique solution to equations with no real roots.
Basic Operations with Imaginary Numbers:
Addition and Subtraction:
Imaginary numbers are added or subtracted by combining their coefficients.
Example: \(3i + 4i = 7i/)
Multiplication:
Multiplying imaginary numbers involves leveraging the fact that \(i^2 = -1\)
Example: \(4i \times 3i = -12\)
Division:
Division requires multiplying the numerator and denominator by the conjugate of the denominator.
Example: \(\frac{3i}{2 – i}\)
Complex Numbers
Imaginary numbers are part of the broader set of complex numbers, which include both real and imaginary components. A complex number is expressed as \(a + bi\), where aa and b are real numbers. Imaginary numbers are a special case when a = 0.
Examples on Imaginary Numbers
1. Example: \(3i + 2i\)
Solution: \(3i + 2i = 5i\)
Explanation: Combine the imaginary parts.
2. Example: \((5 – 2i) – (3 + 4i)\)
Solution: \((5 – 2i) – (3 + 4i) = 2 – 6i\)
Explanation: Subtract the imaginary parts.
3. Example: \((2i) \times (4 – i)\)
Solution: \((2i) \times (4 – i) = 8i – 2i^2 = 8i + 2\)
Explanation: Apply the distributive property and \(i^2 = -1\)
4. Example: \( \frac{3i}{2 – i}\)
Solution: Multiply numerator and denominator by the conjugate of the denominator.
Result: \(\frac{3i(2 + i)}{5} = \frac{6i + 3i^2}{5}\) = \(-\frac{3}{5} + \frac{6}{5}i\)
