91Ó°ÊÓ

Complex numbers are a fascinating class of numbers that extend the real numbers with the inclusion of an imaginary unit, often represented as \(i\), where \(i^2 = -1\). A complex number can be expressed in the standard form: \(a + bi\), where \(a\) and \(b\) are real numbers. The component \(a\) is referred to as the real part, whereas \(b\) is known as the imaginary part.

In the context of the exercise, the subset \(\mathbb{Z} + \mathbb{Z}i\) refers specifically to complex numbers with integer components. This means both the real part \(a\) and the imaginary part \(b\) are integers. Complex numbers with such integer components form a unique and interesting subset of the complex plane.

It's important to note that complex numbers follow specific arithmetic rules, such as commutativity and associativity. This exercise explores such properties in terms of multiplication, which is determined by the formula:

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

The concept of integer components revolves around the elements \(a\) and \(b\) in the complex number \(a + bi\). Integer components mean that these elements belong specifically to the set of whole numbers, both positive and negative, including zero.

In the domain of complex numbers, having integer components allows for certain algebraic properties to manifest meaningfully. For example, when multiplying numbers in the set \(\mathbb{Z} + \mathbb{Z}i\), any resulting number also naturally has integer components. This adherence to integer values is significant because:

• It ensures the closure property under multiplication, essential for the structure to be considered a semigroup.
• It ensures that operations respect the discrete nature of this subset.
• It preserves integer property across multiplication given the additive form \((a+b)\) and solutions such as \((ac-bd)\) remain within integer form.

In mathematics, especially within the context of algebraic structures, the multiplicative identity is a critical component. This is an element which, when multiplied by any other element in the set, leaves the other element unchanged. For complex numbers, the multiplicative identity is \(1\), represented as \(1 + 0i\).

In the subset \(\mathbb{Z} + \mathbb{Z}i\), the multiplicative identity plays the same fundamental role as within the larger set of complex numbers. For any complex number \(z = a + bi\), multiplying it by \(1 + 0i\) results in:

• \(z \times (1 + 0i) = a + bi\)
• This verifies that \(1 + 0i\) functions as the identity.

This property is crucial for verifying the subgroup as a semigroup, as the presence of an identity element solidifies the group structure definition.

Invertible elements within a set are those that have a multiplicative inverse, which means they can be multiplied by another element to result in the multiplicative identity, \(1\). This aspect is central to understanding semigroups and groups in algebra.

In the subset \(\mathbb{Z} + \mathbb{Z}i\), finding invertible elements is akin to finding numbers where both the real and imaginary parts of its inverse are integers. Solving \((a + bi) \times (c + di) = 1\) in this subset leads to the conclusion that:

• The numbers are \(\pm 1\) and \(\pm i\).
• These are the only elements whose inverses also belong to \(\mathbb{Z} + \mathbb{Z}i\).

For example, the inverse of \(1 + 0i\) is itself, while the inverse of \(i\) is \(-i\). Recognizing these invertible elements helps identify the units within the semigroup, reinforcing the group's algebraic properties.