Imaginary number

Complex conjugate picture

Rotations on the complex plane

Imaginary numbers are a concept in mathematics that extend the real number system, ℝ, to the complex number system, ℂ, which provides a solution to equations that cannot be solved using only real numbers. The basic unit of imaginary numbers is the square root of -1, denoted as i, where i² = -1. Imaginary numbers, when combined with real numbers, form complex numbers, which can be expressed in the form a + bi, where a and b are real numbers.

Definition

An imaginary number is defined as any real number multiplied by i. For example, 2i is an imaginary number, as is -3i. The term "imaginary" may suggest that these numbers are fictitious or not of real value; however, they are a fundamental concept in mathematics and have practical applications in various fields such as engineering, physics, and signal processing.

Historical Background

The concept of imaginary numbers dates back to the 16th century, when mathematicians were trying to solve cubic equations. Italian mathematician Gerolamo Cardano is credited with one of the earliest uses of imaginary numbers, although he did not fully understand their properties or potential applications. It was not until the 18th century that mathematicians like Leonhard Euler and Carl Friedrich Gauss began to develop a more comprehensive theory of complex numbers, which includes both real and imaginary components.

Mathematical Operations

Imaginary numbers follow the same rules for addition, subtraction, and multiplication as real numbers, with the additional rule that i² = -1. Division of imaginary numbers involves multiplying the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator.

Addition and Subtraction

Adding or subtracting imaginary numbers is straightforward and involves combining like terms. For example, (3i + 2i) = 5i, and (5i - 2i) = 3i.

Multiplication

Multiplying imaginary numbers involves applying the distributive property and using the fact that i² = -1. For example, (2i)(3i) = 6i² = -6.

Division

To divide imaginary numbers, multiply the numerator and denominator by the conjugate of the denominator to make the denominator real. For example, to divide 2i by 3i, multiply both by the conjugate of 3i, which is -3i, resulting in (2i)(-3i)/(3i)(-3i) = -6/-9 = 2/3.

Applications

Imaginary numbers have applications in various scientific fields. In electrical engineering, they are used to describe the phase difference between voltage and current in an AC circuit. In quantum mechanics, complex numbers are essential for describing the state of quantum systems. They are also used in signal processing, control theory, and in solving differential equations that model physical processes.

See Also

• Complex number
• Real number
• Leonhard Euler
• Carl Friedrich Gauss
• Signal processing
• Quantum mechanics