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Regarding mathematical operators... 
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#1
Dec2312, 07:59 PM

P: 273

One basic operator is addition. In order to add the any number x number of times, multiplication was invented. In order to multiply any number x number of times, exponentiation was invented. What if we want to raise a number to a power x number of times? How come we didn't invent that?
Also, why is it that addition and multiplication use symbols, but exponents are just simply superscripted? Why do we have the opposite operations of addition and multiplication but there is no opposite operation of exponentiation? Edit: I realize now that the opposite operation of exponentiation is the radical. 


#2
Dec2312, 08:50 PM

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This in not correct:



#3
Dec2312, 09:07 PM

P: 273




#4
Dec2312, 09:33 PM

P: 273

Regarding mathematical operators...
Also, why is it that AA^{1} = I i.e. the inverse of a matrix behaves like a reciprocal (multiplicative inverse) in regular algebra? Shouldn't it be called the reciprocal or at the very least, the multiplicative inverse?



#5
Dec2312, 09:58 PM

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#6
Dec2312, 10:14 PM

P: 273

Thank you for answering my questions Mandelbroth. I think you may have missed post #4 though.
Also, a little bummed out that hyperoperations already exist since I thought I was onto something new. I got really excited and even wrote my own notation. 


#7
Dec2312, 11:01 PM

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#8
Dec2412, 12:08 PM

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#9
Dec2412, 05:48 PM

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#10
Dec2412, 07:02 PM

Mentor
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I think the OP his questions would very likely be cleared up by an abstract algebra course. 


#11
Dec2412, 07:31 PM

P: 273

So as micromass said, there is the additive inverse and the multiplicative inverse of a matrix. So why is it that when we say inverse of a matrix, we are referring to the multiplicative inverse? Shouldn't it be called the reciprocal? I think it's misleading.



#12
Dec2412, 07:45 PM

Sci Advisor
P: 820

Not all matrices have multiplicative inverses. So when we use "inverse" we mean multiplicative because it's significant. ##\left(\frac{a}{b}\right)^{1} = \frac{b}{a}##. In fields every nonzero element has a multiplicative inverse, so we don't use use the term "inverse" as it isn't significant. Later in your studies you'll come across rings, which is where this nomenclature originates from. It's make more sense then. 


#13
Dec2412, 09:36 PM

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#14
Dec2412, 10:45 PM

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Makes sense.



#15
Dec2512, 12:03 AM

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Take any real number other than zero. Call it a. It has an inverse (meaning here multiplicative inverse) a^{1}. This allows the equation, aa^{1}=1. Also, a^{1}a=1. Reciprocal is a multiplicative inverse of a Real Number. That is for Real Numbers. Matrices are not always like that. Take any square matrix. MAYBE it has a multiplicative inverse and maybe it does not. What if you have a square matrix, A. Then it might or might not have an inverse, A^{1}. If it does, then AA^{1}=I, and A^{1}A=I. Sometimes, there is a matrix B for which AB=I, but BA=\=I; or that BA=I but AB=\=I. For matrix multiplication, AB and BA are not always the same. 


#16
Dec2512, 03:34 AM

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#17
Dec2512, 04:40 AM

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