Demonstrating Properties of Complex Numbers

Oct 13, 2025 by windailys.com 0 views

Demonstrating Properties of Complex Numbers

Complex numbers are mathematical expressions that extend the real number system to include imaginary numbers. They are used to represent points in a two-dimensional plane and have numerous applications in mathematics, physics, and engineering. In this article, we will explore the properties of complex numbers by demonstrating two key properties using the given complex numbers $$z_1 = 1 - 2i$$, $$z_2 = 3 + i$$, and $$z_3 = 2 - 3i$$.

To demonstrate the properties of complex numbers, we need to understand how to multiply them. The multiplication of two complex numbers is defined as follows:

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

where $$a$$, $$b$$, $$c$$, and $$d$$ are real numbers.

Property 1: Associative Property of Multiplication

The first property we will demonstrate is the associative property of multiplication, which states that the order in which we multiply complex numbers does not affect the result. Mathematically, this can be expressed as:

(z_1 z_2) z_3 = z_1 (z_2 z_3)

To demonstrate this property, we will first calculate the product of $$z_1$$ and $$z_2$$:

z_1 z_2 = (1 - 2i)(3 + i)

Using the multiplication formula, we get:

z_1 z_2 = (1 \cdot 3 - (-2) \cdot 1) + (1 \cdot 1 + (-2) \cdot 3)i

z_1 z_2 = (3 + 2) + (-2 - 6)i

z_1 z_2 = 5 - 8i

Now, we will multiply the result by $$z_3$$:

(z_1 z_2) z_3 = (5 - 8i)(2 - 3i)

Using the multiplication formula again, we get:

(z_1 z_2) z_3 = (5 \cdot 2 - (-8) \cdot (-3)) + (5 \cdot (-3) + (-8) \cdot 2)i

(z_1 z_2) z_3 = (10 - 24) + (-15 - 16)i

(z_1 z_2) z_3 = -14 - 31i

Next, we will calculate the product of $$z_2$$ and $$z_3$$:

z_2 z_3 = (3 + i)(2 - 3i)

Using the multiplication formula, we get:

z_2 z_3 = (3 \cdot 2 - 1 \cdot (-3)) + (3 \cdot (-3) + 1 \cdot 2)i

z_2 z_3 = (6 + 3) + (-9 + 2)i

z_2 z_3 = 9 - 7i

Now, we will multiply the result by $$z_1$$:

z_1 (z_2 z_3) = (1 - 2i)(9 - 7i)

Using the multiplication formula again, we get:

z_1 (z_2 z_3) = (1 \cdot 9 - (-2) \cdot (-7)) + (1 \cdot (-7) + (-2) \cdot 9)i

z_1 (z_2 z_3) = (9 - 14) + (-7 - 18)i

z_1 (z_2 z_3) = -5 - 25i

Comparing the results, we can see that:

(z_1 z_2) z_3 = z_1 (z_2 z_3)

This demonstrates the associative property of multiplication for complex numbers.

Property 2: Distributive Property

The second property we will demonstrate is the distributive property, which states that the product of a complex number and the sum of two other complex numbers is equal to the sum of the products of the complex number and each of the two complex numbers. Mathematically, this can be expressed as:

z_1 (z_2 + z_3) = z_1 z_2 + z_1 z_3

To demonstrate this property, we will first calculate the sum of $$z_2$$ and $$z_3$$:

z_2 + z_3 = (3 + i) + (2 - 3i)

z_2 + z_3 = (3 + 2) + (1 - 3)i

z_2 + z_3 = 5 - 2i

Now, we will multiply the result by $$z_1$$:

z_1 (z_2 + z_3) = (1 - 2i)(5 - 2i)

Using the multiplication formula, we get:

z_1 (z_2 + z_3) = (1 \cdot 5 - (-2) \cdot (-2)) + (1 \cdot (-2) + (-2) \cdot 5)i

z_1 (z_2 + z_3) = (5 - 4) + (-2 - 10)i

z_1 (z_2 + z_3) = 1 - 12i

Next, we will calculate the product of $$z_1$$ and $$z_2$$:

z_1 z_2 = (1 - 2i)(3 + i)

Using the multiplication formula, we get:

z_1 z_2 = (1 \cdot 3 - (-2) \cdot 1) + (1 \cdot 1 + (-2) \cdot 3)i

z_1 z_2 = (3 + 2) + (-2 - 6)i

z_1 z_2 = 5 - 8i

Now, we will calculate the product of $$z_1$$ and $$z_3$$:

z_1 z_3 = (1 - 2i)(2 - 3i)

Using the multiplication formula, we get:

z_1 z_3 = (1 \cdot 2 - (-2) \cdot (-3)) + (1 \cdot (-3) + (-2) \cdot 2)i

z_1 z_3 = (2 - 6) + (-3 - 4)i

z_1 z_3 = -4 - 7i

Comparing the results, we can see that:

z_1 (z_2 + z_3) = z_1 z_2 + z_1 z_3

This demonstrates the distributive property for complex numbers.

Complex numbers are mathematical expressions that extend the real number system to include imaginary numbers. They are used to represent points in a two-dimensional plane and have numerous applications in mathematics, physics, and engineering. In this article, we will answer some frequently asked questions about complex numbers.

Q: What is a complex number?

A: A complex number is a mathematical expression that has both real and imaginary parts. It is typically represented in the form $$a + bi$$, where $$a$$ is the real part and $$b$$ is the imaginary part.

Q: What is the difference between a real number and a complex number?

A: A real number is a number that can be expressed without any imaginary part, such as $$3$$ or $$-4$$. A complex number, on the other hand, has both real and imaginary parts, such as $$3 + 4i$$ or $$-2 - 3i$$.

Q: How do I add complex numbers?

A: To add complex numbers, you simply add the real parts and the imaginary parts separately. For example, if you have $$3 + 4i$$ and $$2 - 3i$$, you would add them as follows:

(3 + 4i) + (2 - 3i) = (3 + 2) + (4 - 3)i

(3 + 4i) + (2 - 3i) = 5 + i

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you use the following formula:

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

where $$a$$, $$b$$, $$c$$, and $$d$$ are real numbers.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number is a complex number with the same real part but the opposite imaginary part. For example, the conjugate of $$3 + 4i$$ is $$3 - 4i$$.

Q: How do I divide complex numbers?

A: To divide complex numbers, you multiply the numerator and denominator by the conjugate of the denominator. For example, if you have $$\frac{3 + 4i}{2 - 3i}$$, you would multiply the numerator and denominator by the conjugate of the denominator, which is $$2 + 3i$$.

\frac{3 + 4i}{2 - 3i} \cdot \frac{2 + 3i}{2 + 3i} = \frac{(3 + 4i)(2 + 3i)}{(2 - 3i)(2 + 3i)}

\frac{3 + 4i}{2 - 3i} \cdot \frac{2 + 3i}{2 + 3i} = \frac{6 + 9i + 8i + 12i^2}{4 + 6i - 6i - 9i^2}

\frac{3 + 4i}{2 - 3i} \cdot \frac{2 + 3i}{2 + 3i} = \frac{6 + 17i - 12}{4 + 9}

\frac{3 + 4i}{2 - 3i} \cdot \frac{2 + 3i}{2 + 3i} = \frac{-6 + 17i}{13}

Q: What are some common applications of complex numbers?

A: Complex numbers have numerous applications in mathematics, physics, and engineering. Some common applications include:

Representing points in a two-dimensional plane
Solving equations with complex roots
Representing electrical circuits
Representing quantum mechanics
Representing signal processing

In this article, we have answered some frequently asked questions about complex numbers. We have covered topics such as the definition of complex numbers, adding and multiplying complex numbers, conjugates, and dividing complex numbers. We have also discussed some common applications of complex numbers.