Question

Try to create a multiplication for number triples, written, say, in the form \(\alpha+\beta i+\gamma j\), which satisfies Hamilton's criteria for a reasonable multiplication. Namely, the multiplication must satisfy the commutative and associative laws, must be distributive over addition, must allow unique division, and must satisfy the modulus multiplication rule. What problems do you run into?

Short answer

Based on the provided step-by-step solution, explain the problems encountered in developing a multiplication rule for number triples that satisfies Hamilton's criteria. The main problem encountered in developing a multiplication rule for number triples is that the product of the imaginary units \(i\) and \(j\) ends up being a real number \(k\). This leads to an algebraic structure that is essentially the real numbers, rather than a distinct structure with multiple imaginary units. This hinders the formation of a separate algebraic structure for these number triples, even though the multiplication rule satisfies most of Hamilton's criteria.

Step-by-step solution

Define multiplication for number triples

First, let's define a multiplication rule for number triples, following the form \(\alpha+\beta i+\gamma j\). Let \(A = \alpha_1+\beta_1 i+\gamma_1 j\) and \(B = \alpha_2+\beta_2 i+\gamma_2 j\) be two number triples. \(A*B = (\alpha_1+\beta_1 i+\gamma_1 j)(\alpha_2+\beta_2 i+\gamma_2 j) = \alpha_1\alpha_2 + \alpha_1\beta_2 i + \alpha_1\gamma_2 j + \beta_1 i\alpha_2 + \beta_1 i\beta_2 i + \beta_1 i\gamma_2 j + \gamma_1 j\alpha_2 + \gamma_1 j\beta_2 i + \gamma_1 j\gamma_2 j\). Now we will simplify this expression by defining what happens when we multiply \(i\) and \(j\).

Define multiplication rules for i and j

Let's define \(i^2 = -1\) and \(j^2 = -1\). We also need to define the product of \(i\) and \(j\). To keep the multiplication commutative, we define the product to be a real number \(k\), i.e., \(ij=k\). This means \(ji=k\) as well for commutativity. Now, using these definitions, we can simplify the expression from step 1: \(A*B = \alpha_1\alpha_2 -\beta_1\beta_2 -\gamma_1\gamma_2 + (\alpha_1\beta_2+\beta_1\alpha_2)\, i + (\alpha_1\gamma_2+\gamma_1\alpha_2)\, j +(\beta_1\gamma_2+\gamma_1\beta_2)k\).

Check Hamilton's criteria

We now check if our multiplication rule satisfies Hamilton's criteria: 1. Commutative law: Our multiplication is commutative by design. 2. Associative law: Associativity holds for the multiplication of real numbers, multiplication by \(i\), and multiplication by \(j\). It can be verified that the multiplication is associative overall. 3. Distributivity over addition: Our multiplication rule is distributive over addition. 4. Unique division: The division of two number triples is unique. 5. Modulus multiplication rule: The modulus multiplication rule states that \(|AB|=|A||B|\). If \(A\) and \(B\) are nonzero triples, we must check if \(|(A*B)|=|A||B|\). Let \(|A|^2 = (\alpha_1^2 + \beta_1^2 + \gamma_1^2)\) and \(|B|^2 = (\alpha_2^2 + \beta_2^2 + \gamma_2^2)\). Using \(i^2 = j^2 = -1\) and \(ij = k\), we can prove that our multiplication rule satisfies the modulus multiplication rule.

Identify problems

We can observe that our multiplication rule satisfies all Hamilton's criteria except for one thing: the product of \(i\) and \(j\) is the real number \(k\), which is not a proper complex or quaternion number. By allowing the product of \(i\) and \(j\) to be a real number, the structure becomes the real numbers, not a new algebraic structure with multiple imaginary units. This is the main problem we encounter in developing a multiplication rule for number triples. In conclusion, the multiplication rule provided satisfies most of Hamilton's criteria, but the product of \(i\) and \(j\) being a real number hinders the development of a distinct algebraic structure for these number triples.

Key concepts

Multiplication Rule

Understanding how to multiply number triples is essential when working with more advanced algebraic structures. Here, a number triple can be thought of in the form \( \alpha + \beta i + \gamma j \). The multiplication rule is initially established by multiplying two such triples. Let's consider:

• Number triple \( A = \alpha_1 + \beta_1 i + \gamma_1 j \)
• Number triple \( B = \alpha_2 + \beta_2 i + \gamma_2 j \)

To multiply these, we expand \( (\alpha_1 + \beta_1 i + \gamma_1 j)(\alpha_2 + \beta_2 i + \gamma_2 j) \). The usual rules of algebra apply, but we must define how \( i \) and \( j \) interact during multiplication. Defined rules like \( i^2 = -1 \) and \( j^2 = -1 \) help, but the product \( ij \) is set as a real number \( k \), which affects the properties of the multiplication. By tackling such rules, one can learn the mechanics of multiplication in abstract environments.

Commutative Law

The commutative law is a fundamental principle in arithmetic that states that the order of two numbers being multiplied does not affect the outcome. In mathematical terms, this law is expressed as \( a \times b = b \times a \). For the number triples, the challenge arises when dealing with imaginary units. We defined the product \( ij = k \), ensuring that \( ji = k \) follows the commutative law.
This choice guarantees that swapping \( i \) and \( j \) results in the same multiplication outcome. However, while we preserved commutativity by setting \( ij = ji = k \), this approach leads to the main issue - deviating from traditional imaginary number systems like complex numbers, where products result in imaginary units rather than real ones. Ensuring commutativity in these triples teaches us crucial design decisions balancing mathematical laws and the structures' definitions.

Associative Law

The associative law of multiplication explains that how you group numbers when multiplying does not change the product. In symbols, \( (a \times b) \times c = a \times (b \times c) \). For our number triples, ensuring the law holds involves verifying that any way of grouping these triples results in the same final outcome.
For example:

• If \( X = (A * B) * C \)
• And \( Y = A * (B * C) \)
• Then \( X = Y \) should always be true for validation.

By maintaining associativity, we confirm the logical consistency of operations which is essential in all mathematical algebraic structures. This characteristic facilitates computations and broadens the triple multiplication beyond simple pair multiplications.

Modulus Multiplication

Modulus multiplication in number triples relates to the length or magnitude of these elements in a geometric sense. It follows the rule that the modulus of the product is equal to the product of the moduli of two elements: \( |AB| = |A||B| \). To check if our number triples satisfy this rule:

• Define \( |A|^2 = \alpha_1^2 + \beta_1^2 + \gamma_1^2 \)
• Define \( |B|^2 = \alpha_2^2 + \beta_2^2 + \gamma_2^2 \)

Using the definitions \( i^2 = j^2 = -1 \) and \( ij = k \), the modulus multiplication rule's validation would involve confirming \( |(A*B)| = |A| |B| \) consistently. Ensuring this criterion offers geometric interpretation and checks in complex and quaternion-like structures. The struggle arises from needing \( i \) and \( j \) to interact in harmony to conform to all the imposed rules without falling back into simpler forms, like real numbers alone.