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Re: Just what is equality in mathematics, anyway?
Posted:
Apr 16, 2012 10:17 PM
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I think Joe pointed you in the right direction with isomorphism. Rudin even makes a point of saying that the arithmetic is preserved.
Bob Hansen
On Apr 16, 2012, at 7:24 PM, Paul Tanner <upprho@gmail.com> wrote:
> Yes he is replacing - you are looking at the wrong equation. Look again:
>
> "1.29. Theorem. If a and b are real, then
> (a,b) = a + bi.
>
> Proof.
> a + bi = (a,0) + (b,0)(0,1) = (a,0) + (0,b) = (a,b)."
>
> Look at the first equation, the leftmost equation. He is replacing a
> with (a,0) and is replacing b with (b,0) and is replacing i with
> (0,1). Look again at what Rudin says in the textbook. Here is the
> identification quote again, following his Theorem 1.26:
>
> "Re: Just what is equality in mathematics, anyway?"
> http://mathforum.org/kb/message.jspa?messageID=7769795
>
> In Chapter I titled *The Real and Complex Number Systems* on page 13
> he gives Theorem 1.26, which are the two equalities
> (a,0) + (b,0) = (a + b,0)
> and
> (a,0)(b,0) = (ab,0).
> After saying that the proofs are trivial, he then writes the following:
>
> "Theorem 1.26 shows that the complex numbers of the form (a,0) have
> the same arithmetic properties as the corresponding real numbers a. We
> can therefore identify (a,0) with a. This identification gives us the
> real field as a subfield of the complex field."
>
> On Mon, Apr 16, 2012 at 6:36 PM, Robert Hansen <bob@rsccore.com> wrote:
>>
>> He isn't substituting (replacing) there Paul, he just skipped the (trivial) addition step.
>>
>> (a, 0) + (0, b) = (a+0, 0+b) = (a, b)
>>
>> I don't think "identify" means what you think it does.
>>
>> The whole point of this section is to define the set of complex numbers as a field at the outset, without starting with "i" and show that it is the unique definitions for addition and multiplication in this field that equate (0, 1) to i. He is familiarizing the student with the set of complex numbers as a field.
>>
>> Bob Hansen
>>
>>
>>
>> On Apr 16, 2012, at 2:59 PM, Paul Tanner <upprho@gmail.com> wrote:
>>
>>> Proof.
>>> a + bi = (a,0) + (b,0)(0,1) = (a,0) + (0,b) = (a,b)."
>>>
>>> Note that he uses his prior statement that we can identify (a,0) with
>>> a to replace real a and b with complexes (a,0) and (b,0) in the first
>>> equality in the proof, using the replacement property of logic with
>>> respect to either equivalent or equal well-formed formulas.
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