Introduction to Complex Numbers: Product and Division

Relevance of the Topic

The study of complex numbers is essential to Mathematics, as it extends the set of real numbers in a way that allows the solution of expressions that would not be possible without it. The introduction of this topic represents a paradigm shift and the opening of a vast field of mathematical investigation.

Contextualization

By the time we reach the 3rd year of High School, we already have a solid foundation on real numbers and their operations. Moreover, we have probably already encountered the notion of imaginary numbers, which are the square root of negative real numbers, such as -1.

Complex numbers are a natural expansion of real numbers, as they allow the addition of a second dimension, the imaginary part. Thus, complex numbers are formed by a real part and an imaginary part added together.

At this point, when studying the product and division of complex numbers, we are exploring the richness of the set of complex numbers, deepening our understanding of their operations and properties, and strengthening our ability to solve more complex problems.

Theoretical Development

• Complex Numbers: Complex numbers z are in the form z = a + bi, where a and b are real numbers and i is the imaginary unit, defined by i² = -1. The real part of z, represented by Re(z), is the number a, and the imaginary part, represented by Im(z), is the number bi.

• Product of Complex Numbers: The product of two complex numbers z1 = a1 + b1i and z2 = a2 + b2i is calculated using the distributive property of multiplication:

  z1 * z2 = (a1 + b1i) * (a2 + b2i) = a1a2 + a1b2i + a2b1i + b1b2i² = a1a2 + (a1b2 + a2b1)i + b1b2i²

  Remember that i² = -1, so we have i² * b1b2 = -b1b2. Therefore, the final product is in the form:

  z1 * z2 = a1a2 - b1b2 + (a1b2 + a2b1)i

• Division of Complex Numbers: The division of two complex numbers z1 = a1 + b1i and z2 = a2 + b2i is done by multiplying the numerator and denominator by the conjugate of the denominator (a2 - b2i):

  (z1/z2) = (a1 + b1i) / (a2 + b2i) = (a1 + b1i) * (a2 - b2i) / (a2 + b2i) * (a2 - b2i) = (a1a2 - b1b2 + (a1b2 - a2b1)i) / (a2a2 - b2b2) = (a1a2 - b1b2 + (a1b2 - a2b1)i) / (a2² + b2²)

  Therefore, the final quotient is in the form:

  (z1/z2) = (a1a2 + b1b2) / (a2² + b2²) + (a2b1 - a1b2)i / (a2² + b2²)

  Note that the denominator is not zero. If it is, the division is undefined.

Detailed Summary

Key Points

• Composition of Complex Numbers: Complex numbers are formed by a real part and an imaginary part, where the imaginary unit i is the square root of -1. Therefore, a complex number can be represented as z = a + bi, where a is the real part and b is the imaginary part.

• Operation of Product of Complex Numbers: The product of two complex numbers is performed considering that i² = -1. Thus, the multiplication operation between the real and imaginary terms of the two complex numbers is carried out, and in the end, the i² part is replaced by -1:

  z1 * z2 = a1a2 - b1b2 + (a1b2 + a2b1)i

• Operation of Division of Complex Numbers: The division of two complex numbers z1 and z2 is performed by multiplying the numerator and denominator by the conjugate of the denominator. After the multiplication, the expression is simplified to represent the complex number in the form a + bi:

  (z1/z2) = (a1a2 + b1b2) / (a2² + b2²) + (a2b1 - a1b2)i / (a2² + b2²)

Conclusions

• The domain of complex numbers and their operations allows dealing with a wider range of mathematical expressions. The operations of product and division between complex numbers are essential in this expansion.

• The product operation between complex numbers includes the multiplication of the real and imaginary terms, as well as the manipulation of the i² expression to represent the result in the form a + bi.

• The division operation between complex numbers is performed by multiplying the numerator and denominator by the conjugate of the denominator. This ensures that the denominator is a real number, which facilitates the simplification of the expression.

Suggested Exercises

1. Perform the product of the complex numbers z1 = 3 + 2i and z2 = -1 + 5i.

2. Divide the complex numbers z1 = 6 - 4i by z2 = 2 + i. Check if the division is undefined.

3. Write the complex number z3 = 4 - 7i in polar form (modulus-argument).