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Show that there is no consistent way to define multiplication on objects of the form a+bi+cj, where a,b, and c are real numbers, if all of the following properties are to hold:

- closure under multiplication;
- the usual associative and distributive laws;
- i ² =−1;
- a+bi+cj=0 if and only if a,b, and c are all 0 .

User Freedoo
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Final answer:

Multiplication cannot be consistently defined on objects of the form a+bi+cj, given the properties of closure, associativity, distribution, and the square of i. The assumption of closure under multiplication leads to a contradiction when considering the multiplication of two objects that have a real part and an imaginary part of zero. Therefore, there is no consistent way to define multiplication on these objects.

Step-by-step explanation:

To show that there is no consistent way to define multiplication on objects of the form a+bi+cj, where a, b, and c are real numbers, we can use the properties given.

Let's assume that we can define multiplication consistently and examine the consequences of that assumption.

If we multiply two such objects, (a+bi+cj)(x+yi+zj), using the usual associative and distributive laws, we get:

(a+bi+cj)(x+yi+zj) = (ax + bxi + cxj) + (ayi - by + cyj) + (azi - bzi + cz),

where we used the fact that i^2 = -1.

Now, if we assume that these objects form a field (a mathematical structure that satisfies all the properties of numbers), then we also require closure under multiplication, meaning that the result of any multiplication also belongs to the set of these objects.

However, consider the multiplication of two such objects that have a real part and an imaginary part is zero: (0+bi+cj)(0+yi+zj).

According to the given property, this should be true if and only if b = c = 0.

But if b = c = 0, then the imaginary part of the product is zero, and this violates the closure property, which states that the result of any multiplication should still be an object of the form a+bi+cj.

Therefore, based on the given properties and the assumption of closure under multiplication, we can conclude that there is no consistent way to define multiplication on objects of the form a+bi+cj.

User Amos Batto
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