Complex Numbers Representation

Oct 13, 2025 by leolavanderia.com.br 149 views

Introduction

Complex numbers are an extension of the real number system, which includes a new element, the imaginary unit, denoted by i. The imaginary unit is defined as the square root of -1, i.e., i = √(-1). Complex numbers have numerous applications in mathematics, physics, engineering, and other fields. In this article, we will focus on the multiplication of complex numbers, specifically the result of multiplying two given complex numbers, z1 and z2.

Complex Numbers Representation

Complex numbers can be represented in the form a + bi, where a and b are real numbers, and i is the imaginary unit. The complex number z1 is given by:

z1 = 4(cos(2π/3) + i sin(2π/3))

Similarly, the complex number z2 is given by:

z2 = 2√2(cos(π/12) + i sin(π/12))

Multiplication of Complex Numbers

The multiplication of complex numbers is a fundamental operation in complex analysis. Given two complex numbers z1 = a + bi and z2 = c + di, the product z1z2 is defined as:

z1z2 = (a + bi)(c + di)

Using the distributive property, we can expand the product as:

z1z2 = ac + adi + bci + bdi^2

Since i^2 = -1, we can simplify the expression as:

z1z2 = (ac - bd) + (ad + bc)i

Multiplication of z1 and z2

Now, let's apply the multiplication formula to the given complex numbers z1 and z2. We have:

z1 = 4(cos(2π/3) + i sin(2π/3)) z2 = 2√2(cos(π/12) + i sin(π/12))

Using the multiplication formula, we get:

z1z2 = (4)(2√2)(cos(2π/3)cos(π/12) - sin(2π/3)sin(π/12)) + (4)(2√2)(cos(2π/3)sin(π/12) + sin(2π/3)cos(π/12))i

Simplifying the Expression

To simplify the expression, we can use the trigonometric identities:

cos(A + B) = cos(A)cos(B) - sin(A)sin(B) sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

Using these identities, we can rewrite the expression as:

z1z2 = 8√2(cos(2π/3 + π/12) + i sin(2π/3 + π/12))

Evaluating the Trigonometric Functions

Now, let's evaluate the trigonometric functions:

cos(2π/3 + π/12) = cos(5π/4) sin(2π/3 + π/12) = sin(5π/4)

Using the unit circle or a trigonometric table, we can find the values of these functions:

cos(5π/4) = -1/√2 sin(5π/4) = 1/√2

Final Result

Substituting these values into the expression, we get:

z1z2 = 8√2(-1/√2 + i 1/√2)

Simplifying the expression, we get:

z1z2 = -8 + 8i

Conclusion

In this article, we have shown how to multiply two complex numbers, z1 and z2, using the multiplication formula. We have also simplified the expression using trigonometric identities and evaluated the trigonometric functions. The final result is a complex number in the form a + bi, where a and b are real numbers. This result can be used in various applications, such as solving equations, finding roots of polynomials, and representing periodic phenomena.

References

[1] "Complex Numbers" by MathWorld
[2] "Complex Analysis" by MIT OpenCourseWare
[3] "Trigonometry" by Khan Academy

Glossary

Complex number: A number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit.
Imaginary unit: The square root of -1, denoted by i.
Trigonometric function: A function that relates the angles of a triangle to the ratios of the lengths of its sides.
Unit circle: A circle with a radius of 1, centered at the origin of a coordinate system.
Trigonometric identity: A mathematical statement that relates two or more trigonometric functions.
Complex Numbers Multiplication: A Comprehensive Guide ===========================================================

Q&A: Complex Numbers Multiplication

Q: What is the formula for multiplying complex numbers?

A: The formula for multiplying complex numbers is:

z1z2 = (a + bi)(c + di) = (ac - bd) + (ad + bc)i

Q: How do I simplify the expression after multiplying complex numbers?

A: To simplify the expression, you can use the trigonometric identities:

cos(A + B) = cos(A)cos(B) - sin(A)sin(B) sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

Q: What are some common trigonometric identities used in complex analysis?

A: Some common trigonometric identities used in complex analysis include:

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

Q: How do I evaluate the trigonometric functions in complex analysis?

A: To evaluate the trigonometric functions, you can use the unit circle or a trigonometric table. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate system.

Q: What is the unit circle, and how is it used in complex analysis?

A: The unit circle is a circle with a radius of 1, centered at the origin of a coordinate system. It is used to evaluate the trigonometric functions in complex analysis.

Q: How do I find the values of the trigonometric functions using the unit circle?

A: To find the values of the trigonometric functions using the unit circle, you can use the following steps:

Draw a circle with a radius of 1, centered at the origin of a coordinate system.
Label the points on the circle with the corresponding angles.
Use the coordinates of the points to find the values of the trigonometric functions.

Q: What are some common applications of complex numbers in mathematics and physics?

A: Some common applications of complex numbers in mathematics and physics include:

Solving equations and finding roots of polynomials
Representing periodic phenomena, such as sound waves and light waves
Modeling population growth and decay
Analyzing electrical circuits and signal processing

Q: How do I use complex numbers to solve equations and find roots of polynomials?

A: To use complex numbers to solve equations and find roots of polynomials, you can follow these steps:

Write the equation or polynomial in the form a + bi.
Use the quadratic formula or other methods to find the roots of the equation or polynomial.
Simplify the roots using the trigonometric identities.

Q: What are some common mistakes to avoid when working with complex numbers?

A: Some common mistakes to avoid when working with complex numbers include:

Forgetting to simplify the expression after multiplying complex numbers
Using the wrong trigonometric identities or formulas
Not evaluating the trigonometric functions correctly
Not using the unit circle or trigonometric table to find the values of the trigonometric functions

Q: How do I check my work when working with complex numbers?

A: To check your work when working with complex numbers, you can follow these steps:

Simplify the expression after multiplying complex numbers.
Evaluate the trigonometric functions using the unit circle or trigonometric table.
Use the trigonometric identities to simplify the expression.
Check your work by plugging in the values of the trigonometric functions into the expression.

Conclusion

In this article, we have provided a comprehensive guide to complex numbers multiplication, including the formula, simplification, and evaluation of trigonometric functions. We have also answered some common questions and provided tips and tricks for working with complex numbers. By following these guidelines and practicing with examples, you can become proficient in complex numbers multiplication and apply it to various applications in mathematics and physics.