Complex Number

1. Algebra of the Complex Number System

Algebra of the Complex Number System

The complex number system follows specific algebraic operations similar to real numbers, with additional properties due to the imaginary unit $i = \sqrt{-1}$​. Let's break them down:

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1. Addition and Subtraction of Complex Numbers

Let $z_1 = x_1 + i y_1$z1​=x1​+iy1​ and $z_2 = x_2 + i y_2$z2​=x2​+iy2​ be two complex numbers.

• Addition is defined as:

  $$z_1 + z_2 = (x_1 + i y_1) + (x_2 + i y_2) = (x_1 + x_2) + i(y_1 + y_2)$$z1​+z2​=(x1​+iy1​)+(x2​+iy2​)=(x1​+x2​)+i(y1​+y2​)

• Subtraction follows similarly:

  $$z_1 - z_2 = (x_1 + i y_1) - (x_2 + i y_2) = (x_1 - x_2) + i(y_1 - y_2)$$z1​−z2​=(x1​+iy1​)−(x2​+iy2​)=(x1​−x2​)+i(y1​−y2​)

This means that complex addition and subtraction are done component-wise for real and imaginary parts.

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2. Multiplication of Complex Numbers

Multiplication follows the distributive property:

$$z_1 \cdot z_2 = (x_1 + i y_1) \cdot (x_2 + i y_2)$$z1​⋅z2​=(x1​+iy1​)⋅(x2​+iy2​)

Expanding using the distributive law:

$$x_1 x_2 + i x_1 y_2 + i x_2 y_1 + i^2 y_1 y_2$$x1​x2​+ix1​y2​+ix2​y1​+i2y1​y2​

Since $i^2 = -1$i2=−1, we simplify to:

$$(x_1 x_2 - y_1 y_2) + i (x_1 y_2 + x_2 y_1)$$(x1​x2​−y1​y2​)+i(x1​y2​+x2​y1​)

Thus, multiplication of two complex numbers results in another complex number.

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3. Conjugate of a Complex Number

The conjugate of a complex number $z = x + i y$z=x+iy is denoted as:

$$\overline{z} = x - i y$$z=x−iy

Multiplying a complex number by its conjugate gives:

$$z \cdot \overline{z} = (x + i y)(x - i y)$$z⋅z=(x+iy)(x−iy)

Expanding using the difference of squares:

$$x^2 - i x y + i x y - i^2 y^2$$x2−ixy+ixy−i2y2

Since $i^2 = -1$i2=−1, this simplifies to:

$$x^2 + y^2$$x2+y2

which is the modulus squared of $z$z:

$$z \cdot \overline{z} = |z|^2$$z⋅z=∣z∣2

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4. Rationalization (Finding the Reciprocal of a Complex Number)

The reciprocal of a complex number $z$z is given by:

$$z^{-1} = \frac{1}{z}$$z−1=z1​

To rationalize the denominator, multiply by the conjugate:

$$z^{-1} = \frac{1}{z} \times \frac{\overline{z}}{\overline{z}} = \frac{\overline{z}}{|z|^2}$$z−1=z1​×zz​=∣z∣2z​

This means:

$$\frac{1}{z} = \frac{x - i y}{x^2 + y^2}$$z1​=x2+y2x−iy​

This is useful in division.

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5. Quotient (Division) of Complex Numbers

To divide two complex numbers:

$$\frac{z_1}{z_2} = \frac{x_1 + i y_1}{x_2 + i y_2}$$z2​z1​​=x2​+iy2​x1​+iy1​​

Multiply by the conjugate of the denominator:

$$\frac{z_1}{z_2} = \frac{(x_1 + i y_1)(x_2 - i y_2)}{(x_2 + i y_2)(x_2 - i y_2)}$$z2​z1​​=(x2​+iy2​)(x2​−iy2​)(x1​+iy1​)(x2​−iy2​)​

Since the denominator simplifies to $|z_2|^2 = x_2^2 + y_2^2$∣z2​∣2=x22​+y22​, we get:

$$\frac{z_1}{z_2} = \frac{x_1 x_2 + y_1 y_2}{|z_2|^2} + i \frac{y_1 x_2 - x_1 y_2}{|z_2|^2}$$z2​z1​​=∣z2​∣2x1​x2​+y1​y2​​+i∣z2​∣2y1​x2​−x1​y2​​

This expresses the quotient in standard form.

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6. Polar Form of a Complex Number

From Figure (3.1) (which you referenced), a complex number can also be expressed in polar form using:

$$x = r \cos\theta, \quad y = r \sin\theta$$x=rcosθ,y=rsinθ

where:

• $r = |z| = \sqrt{x^2 + y^2}$         r=∣z∣=x2+y2 is the modulus (magnitude) of $z$z.

• $\theta = \tan^{-1} \left(\frac{y}{x}\right)$θ=tan−1(xy​) is the argument (angle) of $z$z.

Thus, we can write a complex number as:

$$z = r (\cos\theta + i\sin\theta)$$z=r(cosθ+isinθ)

This is called the trigonometric (polar) form of a complex number.