Complex Number

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.