Complex Numbers

In mathematics, a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is an imaginary unit that satisfies the equation i2 = -1. The number a is called the real part of the complex number, and b is called the imaginary part.

Understanding the imaginary unit i

The imaginary unit i has a unique property that when it is squared, it equals -1. This can be confusing at first because no real number has this property. To clarify:

i = √(-1) i² = -1

Form of complex numbers

A complex number can be written as:

z = a + bi

Where:

• z is a complex number.
• a is the real part, Real(z).
• b is the imaginary part, Imag(z).

Example of complex number: 3 + 4i

Graphical representation

Complex numbers can be represented graphically on a two-dimensional plane called the complex plane. The horizontal axis represents the real part, and the vertical axis represents the imaginary part.

In the above diagram, the vector from the origin to the point (3, 4) represents the complex number 3 + 4i. The horizontal distance is the real component, and the vertical distance is the imaginary component.

Complex conjugates

The complex conjugate of a complex number is obtained by changing the sign of the imaginary part. For a complex number z = a + bi, its complex conjugate is a - bi.

Example:

If z = 3 + 4i, then the conjugate of z, denoted as z̅, is 3 - 4i.

Arithmetic operations on complex numbers

Add

To add two complex numbers, you add their corresponding real parts and imaginary parts. For the complex numbers z₁ = a + bi and z₂ = c + di, the sum is:

z₁ + z₂ = (a + c) + (b + d)i

Example:

z₁ = 3 + 4i, z₂ = 1 + 2i z₁ + z₂ = (3 + 1) + (4 + 2)i = 4 + 6i

Subtraction

To subtract two complex numbers, you subtract their corresponding real parts and imaginary parts. For the complex numbers z₁ = a + bi and z₂ = c + di, the difference is:

z₁ - z₂ = (a - c) + (b - d)i

Example:

z₁ = 3 + 4i, z₂ = 1 + 2i z₁ - z₂ = (3 - 1) + (4 - 2)i = 2 + 2i

Multiplication

To multiply two complex numbers, you distribute each part and use the property i² = -1. For the complex numbers z₁ = a + bi and z₂ = c + di, the product is:

z₁ * z₂ = (a + bi)(c + di) = (ac - bd) + (ad + bc)i

Example:

z₁ = 3 + 4i, z₂ = 1 + 2i z₁ * z₂ = (3 * 1 - 4 * 2) + (3 * 2 + 4 * 1)i = (3 - 8) + (6 + 4)i = -5 + 10i

Division

To divide one complex number by another, multiply the numerator and denominator by the conjugate of the denominator. For the complex numbers z₁ = a + bi and z₂ = c + di, the quotient is:

z₁ / z₂ = [(a + bi)(c - di)] / [(c + di)(c - di)] = [(ac + bd) + (bc - ad)i] / (c² + d²)

Example:

z₁ = 3 + 4i, z₂ = 1 + 2i z₁ / z₂ = [(3 + 4i)(1 - 2i)] / (1² + 2²) = [(3*1 + 4*2) + (4*1 - 3*2)i] / (1 + 4) = (3 + 8 + i(4 - 6)) / 5 = (11 - 2i) / 5 = 11/5 - (2/5)i

Modulus and argument of a complex number

Module

The modulus of a complex number z = a + bi, denoted by |z|, is the distance from the origin to the point (a, b) in the complex plane. It is given as follows:

|z| = √(a² + b²)

Example:

z = 3 + 4i |z| = √(3² + 4²) = √(9 + 16) = √25 = 5

Logic

The argument of a complex number is the angle that the line joining the origin to the point (a, b) makes with the positive real axis. It is usually denoted by arg(z).

For a complex number z = a + bi, the argument θ can be found using:

θ = tan-1 (b / a)

Example:

z = 3 + 4i θ = tan-1 (4 / 3)

Polar form of complex numbers

A complex number can also be represented in polar form as z = r (cos θ + i sin θ) or more compactly, using the Euler formula, as z = reiθ, where r is the modulus and θ is the argument.

Example:

z = 3 + 4i |z| = 5, and θ = tan-1 (4/3) Polar form: z = 5 (cos θ + i sin θ)

Applications of complex numbers

Complex numbers have wide applications in various fields such as engineering, physics, and applied mathematics.

• Electrical engineering: Used to analyze AC circuits, where resistance, inductance, and capacitance are represented as complex impedances.
• Fluid dynamics: Models of fluid flow often involve complex numbers, especially in two-dimensional flow.
• Quantum mechanics: Wave functions are expressed using complex numbers.

Conclusion

Complex numbers extend the concept of one-dimensional numbers to the two-dimensional complex plane by using an imaginary component in addition to the real part. Understanding complex numbers helps provide solutions to certain equations that do not have a solution in the real number system and serves as a gateway to more advanced topics in mathematics and its applications.

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